This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2025.2.7)  15 OCT 2025 17:21
entering extended mode
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**filon-quadrature/filon-quadrature.html
(./filon-quadrature/filon-quadrature.html
LaTeX2e <2023-11-01> patch level 1
L3 programming layer <2024-01-22>

! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.1 <
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You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.7     <
         title>Filon quadrature</title>
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.9     <
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You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

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Overfull \hbox (20.0pt too wide) in paragraph at lines 9--12
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! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.13     <
          !-- Theme used for syntax highlighted code -->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
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Missing character: There is no i in font nullfont!
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Missing character: There is no t in font nullfont!
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Missing character: There is no n in font nullfont!
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Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
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Missing character: There is no / in font nullfont!
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Missing character: There is no s in font nullfont!
Missing character: There is no " in font nullfont!
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! You can't use `macro parameter character #' in horizontal mode.
l.17         --uswds-red-cool-20: #
                                   ecbec6;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no e in font nullfont!
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Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
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Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.18         --uswds-gold-20: #
                               dec69a;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no 6 in font nullfont!
Missing character: There is no 9 in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
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Missing character: There is no - in font nullfont!
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Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.19         --uswds-green-cool-20: #
                                     b4d0b9;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no b in font nullfont!
Missing character: There is no 4 in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no 9 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.20         --uswds-blue-cool-20: #
                                    adcfdc;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.21         --uswds-indigo-20: #
                                 c5c5f3;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no c in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no 3 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.22         --uswds-violet-warm-20: #
                                      e2bee4;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no e in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no 4 in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no 4 in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ( in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no ) in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no 6 in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no : in font nullfont!
! You can't use `macro parameter character #' in horizontal mode.
l.36         background-color: #
                                fff;
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

Missing character: There is no f in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no k in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 13--46
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.47             -
                  ->
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no U in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no N in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
! Misplaced alignment tab character &.
l.48 ... style="text-align: center;">UoN Applied &
                                                   Industrial Maths | 2025 O...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

Missing character: There is no I in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no M in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no | in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 0 in font nullfont!
Missing character: There is no 2 in font nullfont!
Missing character: There is no 5 in font nullfont!
Missing character: There is no O in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no 1 in font nullfont!
Missing character: There is no 7 in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no > in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 47--49
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.50             #
                   Filon Quadrature
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.50             # F
                    ilon Quadrature
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no F in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no Q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 50--51
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.52             <
                  p style="text-align: center;">Dave Smith</p>
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no < in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no = in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no ; in font nullfont!
Missing character: There is no " in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no / in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 52--55
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.56             #
                  # Quadrature
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.56             ##
                    Quadrature
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.56             ## Q
                     uadrature
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no Q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no * in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 56--58
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.59             I
                  mportant because analytical evaluation of integrals is usu...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no I in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 59--60
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.61             e
                  g. if
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <7> on input line 62.
LaTeX Font Info:    External font `cmex10' loaded for size
(Font)              <5> on input line 62.
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 61--68
[]
 []


Overfull \hbox (30.02084pt too wide) in paragraph at lines 61--68
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (74.77917pt too wide) in paragraph at lines 61--68
[] [] \OT1/cmr/m/n/10 +
 []


Overfull \hbox (20.89308pt too wide) in paragraph at lines 61--68
[] \OT1/cmr/m/n/10 +
 []


Overfull \hbox (5.00002pt too wide) in paragraph at lines 61--68
\OT1/cmr/m/n/10 2$
 []


Overfull \hbox (46.06877pt too wide) in paragraph at lines 61--68
[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\OML
/cmm/m/it/10 dx$
 []

! You can't use `macro parameter character #' in vertical mode.
l.69             #
                  # Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.69             ##
                    Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.69             ## A
                     pproximation
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 69--71
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.72             #
                  ## Constant approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.72             ##
                   # Constant approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.72             ###
                     Constant approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.72             ### C
                      onstant approximation
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no C in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!

! LaTeX Error: Bad math environment delimiter.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.75             for some $c\in\[
                                 a,b\]$.
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.


! LaTeX Error: Bad math environment delimiter.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.75             for some $c\in\[a,b\]
                                      $.
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 72--76
[]
 []


Overfull \hbox (42.80562pt too wide) in paragraph at lines 72--76
\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) =
 []


Overfull \hbox (32.09033pt too wide) in paragraph at lines 72--76
\OML/cmm/m/it/10 f[] \OT1/cmr/m/n/10 :=
 []


Overfull \hbox (18.07762pt too wide) in paragraph at lines 72--76
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c\OT1/cmr/m/n/10 )$
 []


Overfull \hbox (13.77196pt too wide) in paragraph at lines 72--76
\OML/cmm/m/it/10 c \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (14.02196pt too wide) in paragraph at lines 72--76
\OML/cmm/m/it/10 a; b$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.77             T
                  hen
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 77--79
[]
 []


Overfull \hbox (42.80562pt too wide) in paragraph at lines 77--79
\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/c
msy/m/n/10 ^^Y
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 77--79
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.80             T
                  herefore,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 80--82
[]
 []


Overfull \hbox (56.62428pt too wide) in paragraph at lines 80--82
[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\OML
/cmm/m/it/10 dx \OMS/cmsy/m/n/10 ^^Y
 []


Overfull \hbox (15.95836pt too wide) in paragraph at lines 80--82
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (27.93182pt too wide) in paragraph at lines 80--82
\OML/cmm/m/it/10 a\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 f[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.83             B
                  ut how do we choose $c$?
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no B in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ? in font nullfont!
Missing character: There is no N in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no I in font nullfont!
Missing character: There is no V in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 83--85
[]
 []


Overfull \hbox (4.32756pt too wide) in paragraph at lines 83--85
\OML/cmm/m/it/10 c$
 []


Overfull \hbox (5.55557pt too wide) in paragraph at lines 83--85
\OMS/cmsy/m/n/10 9$
 []


Overfull \hbox (13.77196pt too wide) in paragraph at lines 83--85
\OML/cmm/m/it/10 c \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (19.57753pt too wide) in paragraph at lines 83--85
\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 a; b\OT1/cmr/m/n/10 ]$
 []

! You can't use `macro parameter character #' in vertical mode.
l.86             #
                  ## Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.86             ##
                   # Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.86             ###
                     Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.86             ### E
                      rror
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
! Undefined control sequence.
l.87 ... (b-a) \left(\max_{\xi\in[a,b]}\left\lvert
                                                   f(\xi)\right\rvert - \min...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.87 ...b-a) \left(\max_{\xi\in[a,b]}\left\lvert f
                                                  (\xi)\right\rvert - \min_{...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.87 ...\xi\in[a,b]}\left\lvert f(\xi)\right\rvert
                                                   - \min_{\xi\in[a,b]}\left...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   -
l.87 ...i\in[a,b]}\left\lvert f(\xi)\right\rvert -
                                                   \min_{\xi\in[a,b]}\left\l...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.87 ...ight\rvert - \min_{\xi\in[a,b]}\left\lvert
                                                   f(\xi)\right\rvert \right)$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.87 ...ht\rvert - \min_{\xi\in[a,b]}\left\lvert f
                                                  (\xi)\right\rvert \right)$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.87 ...\xi\in[a,b]}\left\lvert f(\xi)\right\rvert
                                                   \right)$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \right 
l.87 ...a,b]}\left\lvert f(\xi)\right\rvert \right
                                                  )$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 86--88
[]
 []


Overfull \hbox (29.69449pt too wide) in paragraph at lines 86--88
\OMS/cmsy/m/n/10 E[] ^^T
 []


Overfull \hbox (15.95836pt too wide) in paragraph at lines 86--88
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (131.13661pt too wide) in paragraph at lines 86--88
\OML/cmm/m/it/10 a\OT1/cmr/m/n/10 ) []$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.89             B
                  y MVT,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no B in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no M in font nullfont!
Missing character: There is no V in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 89--90
[]
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.91             $
                  \displaystyle \mathcal E_{\mathrm{rect}} \leq (b-a) \max_{...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

! Undefined control sequence.
l.91 ...} \leq (b-a) \max_{\xi\in[a,b]}\left\lvert
                                                   f'(\xi)\right\rvert$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.91 ...\leq (b-a) \max_{\xi\in[a,b]}\left\lvert f
                                                  '(\xi)\right\rvert$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.91 ...xi\in[a,b]}\left\lvert f'(\xi)\right\rvert
                                                  $.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   $
l.91 ...i\in[a,b]}\left\lvert f'(\xi)\right\rvert$
                                                  .
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!

Overfull \hbox (49.69449pt too wide) in paragraph at lines 91--94
[]$\OMS/cmsy/m/n/10 E[] ^^T
 []


Overfull \hbox (15.95836pt too wide) in paragraph at lines 91--94
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (60.32236pt too wide) in paragraph at lines 91--94
\OML/cmm/m/it/10 a\OT1/cmr/m/n/10 ) [] []$
 []

! You can't use `macro parameter character #' in vertical mode.
l.95             #
                  # Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.95             ##
                    Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.95             ## A
                     pproximation
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 95--97
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.98             #
                  ## Piecewise constant approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.98             ##
                   # Piecewise constant approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.98             ###
                     Piecewise constant approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.98             ### P
                      iecewise constant approximation
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no P in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no L in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 98--100
[]
 []


Overfull \hbox (19.33565pt too wide) in paragraph at lines 98--100
\OML/cmm/m/it/10 m \OT1/cmr/m/n/10 =
 []


Overfull \hbox (16.95259pt too wide) in paragraph at lines 98--100
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a \OT1/cmr/m/n/10 +
 []


Overfull \hbox (18.18059pt too wide) in paragraph at lines 98--100
\OML/cmm/m/it/10 b\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 2$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.101             D
                   efine
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no D in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
! Misplaced alignment tab character &.
l.102 ...thrm{rect}2}(x) := \begin{cases} f(c_1) &
                                                   \text{if }a \leq x \leq m...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! Undefined control sequence.
l.102 ...ect}2}(x) := \begin{cases} f(c_1) & \text
                                                  {if }a \leq x \leq m \\\\ ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Misplaced alignment tab character &.
l.102 ...& \text{if }a \leq x \leq m \\\\ f(c_2) &
                                                   \text{if }m \leq x \leq b...
I can't figure out why you would want to use a tab mark
here. If you just want an ampersand, the remedy is
simple: Just type `I\&' now. But if some right brace
up above has ended a previous alignment prematurely,
you're probably due for more error messages, and you
might try typing `S' now just to see what is salvageable.

! Undefined control sequence.
l.102 ...t{if }a \leq x \leq m \\\\ f(c_2) & \text
                                                  {if }m \leq x \leq b.\end{...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! Undefined control sequence.
l.103 ...isplaystyle a := \begin{cases} b \& \text
                                                  {if }c \\\\ d \& \text{if ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.103 ...n{cases} b \& \text{if }c \\\\ d \& \text
                                                  {if }e.\end{cases}$-->
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!

! LaTeX Error: Bad math environment delimiter.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.104             $\exists$ $c_1\in\[
                                     a,m\]$, $c_2\in\[m,b\]$.
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.


! LaTeX Error: Bad math environment delimiter.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.104             $\exists$ $c_1\in\[a,m\]
                                          $, $c_2\in\[m,b\]$.
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

Missing character: There is no , in font nullfont!

! LaTeX Error: Bad math environment delimiter.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.104 ...     $\exists$ $c_1\in\[a,m\]$, $c_2\in\[
                                                  m,b\]$.
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.


! LaTeX Error: Bad math environment delimiter.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.104 ...$\exists$ $c_1\in\[a,m\]$, $c_2\in\[m,b\]
                                                  $.
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

Missing character: There is no . in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 101--107
[]
 []


Overfull \hbox (49.56953pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) :=
 []


Overfull \hbox (57.3558pt too wide) in paragraph at lines 101--107
[] \OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c[]\OT1/cmr/m/n/10 )[]\OML/cmm/m/it/10 a \
OMS/cmsy/m/n/10 ^^T
 []


Overfull \hbox (16.27078pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^T
 []


Overfull \hbox (8.78014pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 m[] 
 []


Overfull \hbox (51.31679pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c[]\OT1/cmr/m/n/10 )[]\OML/
cmm/m/it/10 m \OMS/cmsy/m/n/10 ^^T
 []


Overfull \hbox (16.27078pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 x \OMS/cmsy/m/n/10 ^^T
 []


Overfull \hbox (7.06944pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 b:$
 []


Overfull \hbox (18.61919pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 a \OT1/cmr/m/n/10 :=
 []


Overfull \hbox (35.34767pt too wide) in paragraph at lines 101--107
[] \OT1/cmr/m/n/10 &[]\OML/cmm/m/it/10 c[] 
 []


Overfull \hbox (29.8341pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 d\OT1/cmr/m/n/10 &[]\OML/cmm/m/it/10 e:$
 []


Overfull \hbox (5.55557pt too wide) in paragraph at lines 101--107
\OMS/cmsy/m/n/10 9$
 []


Overfull \hbox (18.25809pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 c[] \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (18.51044pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 a; m$
 []


Overfull \hbox (18.25809pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 c[] \OMS/cmsy/m/n/10 2
 []


Overfull \hbox (17.5162pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 m; b$
 []


Overfull \hbox (46.79175pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/c
msy/m/n/10 ^^Y
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 101--107
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.108             T
                   herefore,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 108--109
[] 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.110             $
                   \displaystyle \int_a^b f(x) dx \approx \frac{b-a}2(f(c_1)...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no . in font nullfont!

Overfull \hbox (76.62428pt too wide) in paragraph at lines 110--111
[]$[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\
OML/cmm/m/it/10 dx \OMS/cmsy/m/n/10 ^^Y
 []


Overfull \hbox (58.43013pt too wide) in paragraph at lines 110--111
[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c[]\OT1/
cmr/m/n/10 ) +
 []


Overfull \hbox (26.45265pt too wide) in paragraph at lines 110--111
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 c[]\OT1/cmr/m/n/10 ))$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.112             B
                   ut how do we choose $c_j$?
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no B in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no ? in font nullfont!
Missing character: There is no S in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 112--114
[]
 []


Overfull \hbox (8.54112pt too wide) in paragraph at lines 112--114
\OML/cmm/m/it/10 c[]$
 []

! You can't use `macro parameter character #' in vertical mode.
l.115             #
                   ## Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.115             ##
                    # Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.115             ###
                      Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.115             ### E
                       rror
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
! Undefined control sequence.
l.116 ...(m-a) \left(\max_{\xi\in[a,m]}\left\lvert
                                                   f(\xi)\right\rvert - \min...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.116 ...-a) \left(\max_{\xi\in[a,m]}\left\lvert f
                                                  (\xi)\right\rvert - \min_{...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.116 ...xi\in[a,m]}\left\lvert f(\xi)\right\rvert
                                                   - \min_{\xi\in[a,m]}\left...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   -
l.116 ...\in[a,m]}\left\lvert f(\xi)\right\rvert -
                                                   \min_{\xi\in[a,m]}\left\l...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.116 ...ght\rvert - \min_{\xi\in[a,m]}\left\lvert
                                                   f(\xi)\right\rvert \right...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.116 ...t\rvert - \min_{\xi\in[a,m]}\left\lvert f
                                                  (\xi)\right\rvert \right) ...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.116 ...xi\in[a,m]}\left\lvert f(\xi)\right\rvert
                                                   \right) + (b-m) \left(\ma...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \right 
l.116 ...,m]}\left\lvert f(\xi)\right\rvert \right
                                                  ) + (b-m) \left(\max_{\xi\...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.116 ...(b-m) \left(\max_{\xi\in[m,b]}\left\lvert
                                                   f(\xi)\right\rvert - \min...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.116 ...-m) \left(\max_{\xi\in[m,b]}\left\lvert f
                                                  (\xi)\right\rvert - \min_{...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.116 ...xi\in[m,b]}\left\lvert f(\xi)\right\rvert
                                                   - \min_{\xi\in[m,b]}\left...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   -
l.116 ...\in[m,b]}\left\lvert f(\xi)\right\rvert -
                                                   \min_{\xi\in[m,b]}\left\l...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.116 ...ght\rvert - \min_{\xi\in[m,b]}\left\lvert
                                                   f(\xi)\right\rvert \right...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.116 ...t\rvert - \min_{\xi\in[m,b]}\left\lvert f
                                                  (\xi)\right\rvert \right) ...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.116 ...xi\in[m,b]}\left\lvert f(\xi)\right\rvert
                                                   \right) \leq \mathcal E_{...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   \right 
l.116 ...,b]}\left\lvert f(\xi)\right\rvert \right
                                                  ) \leq \mathcal E_{\mathrm...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 115--119
[]
 []


Overfull \hbox (33.68062pt too wide) in paragraph at lines 115--119
\OMS/cmsy/m/n/10 E[] ^^T
 []


Overfull \hbox (20.44684pt too wide) in paragraph at lines 115--119
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 m \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (146.07332pt too wide) in paragraph at lines 115--119
\OML/cmm/m/it/10 a\OT1/cmr/m/n/10 ) [] +
 []


Overfull \hbox (15.95836pt too wide) in paragraph at lines 115--119
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (150.70331pt too wide) in paragraph at lines 115--119
\OML/cmm/m/it/10 m\OT1/cmr/m/n/10 ) [] \OMS/cmsy/m/n/10 ^^T
 []


Overfull \hbox (19.13898pt too wide) in paragraph at lines 115--119
\OMS/cmsy/m/n/10 E[]$
 []

! You can't use `macro parameter character #' in vertical mode.
l.120             #
                   # Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.120             ##
                     Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.120             ## A
                      pproximation
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 120--122
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.123             #
                   ## Trapezium rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.123             ##
                    # Trapezium rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.123             ###
                      Trapezium rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.123             ### T
                       rapezium rule
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 123--125
[]
 []


Overfull \hbox (36.54405pt too wide) in paragraph at lines 123--125
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a; f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a\OT1/c
mr/m/n/10 ))$
 []


Overfull \hbox (34.55559pt too wide) in paragraph at lines 123--125
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b; f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b\OT1/c
mr/m/n/10 ))$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.126             T
                   hen
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 126--128
[]
 []


Overfull \hbox (40.54172pt too wide) in paragraph at lines 126--128
\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 ) \OMS/c
msy/m/n/10 ^^Y
 []


Overfull \hbox (19.46533pt too wide) in paragraph at lines 126--128
\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.129             T
                   herefore,
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no , in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 129--130
[] 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.131             $
                   \displaystyle \int_a^b f(x) dx \approx (b-a)\frac{f(b)+f(...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no . in font nullfont!

Overfull \hbox (76.62428pt too wide) in paragraph at lines 131--132
[]$[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\
OML/cmm/m/it/10 dx \OMS/cmsy/m/n/10 ^^Y
 []


Overfull \hbox (15.95836pt too wide) in paragraph at lines 131--132
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b \OMS/cmsy/m/n/10 ^^@
 []


Overfull \hbox (60.87459pt too wide) in paragraph at lines 131--132
\OML/cmm/m/it/10 a\OT1/cmr/m/n/10 )[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.133             N
                   ote: This is the same integral as the $f_{\mathrm{rect}2}...

You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no N in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 133--136
[]
 []


Overfull \hbox (22.74316pt too wide) in paragraph at lines 133--136
\OML/cmm/m/it/10 f[]$
 []


Overfull \hbox (19.3692pt too wide) in paragraph at lines 133--136
\OML/cmm/m/it/10 c[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (5.28589pt too wide) in paragraph at lines 133--136
\OML/cmm/m/it/10 a$
 []


Overfull \hbox (19.3692pt too wide) in paragraph at lines 133--136
\OML/cmm/m/it/10 c[] \OT1/cmr/m/n/10 =
 []


Overfull \hbox (4.29166pt too wide) in paragraph at lines 133--136
\OML/cmm/m/it/10 b$
 []

! You can't use `macro parameter character #' in vertical mode.
l.137             #
                   # Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.137             ##
                     Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.137             ## A
                      pproximation
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 137--139
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.140             #
                   ## Trapezium rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.140             ##
                    # Trapezium rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.140             ###
                      Trapezium rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.140             ### T
                       rapezium rule
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no T in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 140--142
[]
 []


Overfull \hbox (36.54405pt too wide) in paragraph at lines 140--142
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a; f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a\OT1/c
mr/m/n/10 ))$
 []


Overfull \hbox (34.55559pt too wide) in paragraph at lines 140--142
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b; f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b\OT1/c
mr/m/n/10 ))$ 
 []

! You can't use `macro parameter character #' in vertical mode.
l.143             #
                   ## Multiple trapezia
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.143             ##
                    # Multiple trapezia
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.143             ###
                      Multiple trapezia
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.143             ### M
                       ultiple trapezia
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no M in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no , in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no T in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no n in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 143--146
[]
 []


Overfull \hbox (36.54405pt too wide) in paragraph at lines 143--146
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a; f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 a\OT1/c
mr/m/n/10 ))$
 []


Overfull \hbox (43.53255pt too wide) in paragraph at lines 143--146
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 m; f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 m\OT1/c
mr/m/n/10 ))$
 []


Overfull \hbox (34.55559pt too wide) in paragraph at lines 143--146
\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b; f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 b\OT1/c
mr/m/n/10 ))$
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.147             $
                   $\begin{align*}
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.


Overfull \hbox (20.0pt too wide) in paragraph at lines 147--147
[] 
 []


! LaTeX Error: Environment align* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.147             $$\begin{align*}
                                  
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.


! LaTeX Error: \begin{document} ended by \end{align*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.150             \end{align*}
                              $$
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.


Overfull \hbox (322.82301pt too wide) detected at line 150
[][] \OML/cmm/m/it/10 f\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 )\OML
/cmm/m/it/10 dx\OT1/cmr/m/n/10 & \OMS/cmsy/m/n/10 ^^Y [] ^^B [][] [] \OT1/cmr/m
/n/10 & = [] [] \OML/cmm/m/it/10 ;
 []

Missing character: There is no w in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (16.3171pt too wide) in paragraph at lines 150--153
 $\OML/cmm/m/it/10 h \OT1/cmr/m/n/10 =
 []


Overfull \hbox (16.50432pt too wide) in paragraph at lines 150--153
[]$ 
 []


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.154             W
                   ith $N$ equal subdivisions
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no W in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 154--155
[]
 []


Overfull \hbox (9.12497pt too wide) in paragraph at lines 154--155
\OML/cmm/m/it/10 N$
 []


LaTeX Warning: Command \^ invalid in math mode on input line 156.


LaTeX Warning: Command \^ invalid in math mode on input line 156.

LaTeX Font Info:    Trying to load font information for +cmr on input line 156.

LaTeX Font Info:    No file cmr.fd. on input line 156.

LaTeX Font Warning: Font shape `/cmr/m/n' undefined
(Font)              using `/cmr/m/n' instead on input line 156.

! Corrupted NFSS tables.
wrong@fontshape ...message {Corrupted NFSS tables}
                                                  error@fontshape else let f...
l.156 ...x \frac h2 \left(f(a)+2\sum\_{j=1}\^{N-1}
                                                  f(m_j)+f(b)\right).
This error message was generated by an \errmessage
command, so I can't give any explicit help.
Pretend that you're Hercule Poirot: Examine all clues,
and deduce the truth by order and method.


LaTeX Font Warning: Font shape `/cmr/m/n' undefined
(Font)              using `OT1/cmr/m/n' instead on input line 156.

! Please use \mathaccent for accents in math mode.
\add@accent ... \let \ignorespaces \relax \accent 
                                                  #1 #2\egroup \ifmmode \els...
l.156 ...x \frac h2 \left(f(a)+2\sum\_{j=1}\^{N-1}
                                                  f(m_j)+f(b)\right).
I'm changing \accent to \mathaccent here; wish me luck.
(Accents are not the same in formulas as they are in text.)

! Missing { inserted.
<to be read again> 
                   \edef 
l.156 ...x \frac h2 \left(f(a)+2\sum\_{j=1}\^{N-1}
                                                  f(m_j)+f(b)\right).
A left brace was mandatory here, so I've put one in.
You might want to delete and/or insert some corrections
so that I will find a matching right brace soon.
(If you're confused by all this, try typing `I}' now.)

! Missing } inserted.
<inserted text> 
                }
l.156 ...f(a)+2\sum\_{j=1}\^{N-1}f(m_j)+f(b)\right
                                                  ).
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.


Overfull \hbox (246.03336pt too wide) detected at line 157
[][] \OML/cmm/m/it/10 f\/cmr/m/n/10 (\OML/cmm/m/it/10 x\/cmr/m/n/10 )\OML/cmm/m
/it/10 dx \OMS/cmsy/m/n/10 ^^Y [] [] \OML/cmm/m/it/10 :
 []

Missing character: There is no < in font nullfont!
Missing character: There is no ! in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no . in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no : in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no > in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
Missing character: There is no - in font nullfont!
! You can't use `macro parameter character #' in vertical mode.
l.161             #
                   # Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.161             ##
                     Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.161             ## A
                      pproximation
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no A in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no x in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no f in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no w in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no m in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no g in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no c in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no y in font nullfont!
Missing character: There is no . in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 161--163
[]
 []

! You can't use `macro parameter character #' in vertical mode.
l.164             #
                   ## Multiple trapezia
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.164             ##
                    # Multiple trapezia
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.164             ###
                      Multiple trapezia
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.164             ### M
                       ultiple trapezia
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no M in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no p in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no z in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no W in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no t in font nullfont!
Missing character: There is no h in font nullfont!
Missing character: There is no e in font nullfont!
Missing character: There is no q in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no a in font nullfont!
Missing character: There is no l in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no u in font nullfont!
Missing character: There is no b in font nullfont!
Missing character: There is no d in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no v in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no s in font nullfont!
Missing character: There is no i in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no n in font nullfont!
Missing character: There is no s in font nullfont!

Overfull \hbox (20.0pt too wide) in paragraph at lines 164--166
[]
 []


Overfull \hbox (9.12497pt too wide) in paragraph at lines 164--166
\OML/cmm/m/it/10 N$
 []


LaTeX Warning: Command \^ invalid in math mode on input line 167.


LaTeX Warning: Command \^ invalid in math mode on input line 167.

! Please use \mathaccent for accents in math mode.
\add@accent ... \let \ignorespaces \relax \accent 
                                                  #1 #2\egroup \ifmmode \els...
l.167 ...x \frac h2 \left(f(a)+2\sum\_{j=1}\^{N-1}
                                                  f(m_j)+f(b)\right).
I'm changing \accent to \mathaccent here; wish me luck.
(Accents are not the same in formulas as they are in text.)

! Missing { inserted.
<to be read again> 
                   \edef 
l.167 ...x \frac h2 \left(f(a)+2\sum\_{j=1}\^{N-1}
                                                  f(m_j)+f(b)\right).
A left brace was mandatory here, so I've put one in.
You might want to delete and/or insert some corrections
so that I will find a matching right brace soon.
(If you're confused by all this, try typing `I}' now.)

! Missing } inserted.
<inserted text> 
                }
l.167 ...f(a)+2\sum\_{j=1}\^{N-1}f(m_j)+f(b)\right
                                                  ).
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.


Overfull \hbox (246.03336pt too wide) detected at line 168
[][] \OML/cmm/m/it/10 f\/cmr/m/n/10 (\OML/cmm/m/it/10 x\/cmr/m/n/10 )\OML/cmm/m
/it/10 dx \OMS/cmsy/m/n/10 ^^Y [] [] \OML/cmm/m/it/10 :
 []

! You can't use `macro parameter character #' in vertical mode.
l.170             #
                   ## Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.170             ##
                    # Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in vertical mode.
l.170             ###
                      Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.


! LaTeX Error: Missing \begin{document}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.170             ### E
                       rror
You're in trouble here.  Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Missing character: There is no E in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no r in font nullfont!
Missing character: There is no o in font nullfont!
Missing character: There is no r in font nullfont!
! Undefined control sequence.
l.171 ...a)h^2}{12} \max\_{\xi\in[a,b]}\left\lvert
                                                   f\^{\prime\prime}(\xi) \r...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.171 ...h^2}{12} \max\_{\xi\in[a,b]}\left\lvert f
                                                  \^{\prime\prime}(\xi) \rig...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.


LaTeX Warning: Command \^ invalid in math mode on input line 171.


LaTeX Warning: Command \^ invalid in math mode on input line 171.

! Missing $ inserted.
<inserted text> 
                $
l.171 ...\xi\in[a,b]}\left\lvert f\^{\prime\prime}
                                                  (\xi) \right\rvert$.
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Improper \spacefactor.
<recently read> \spacefactor 
                             
l.171 ...\xi\in[a,b]}\left\lvert f\^{\prime\prime}
                                                  (\xi) \right\rvert$.
You can refer to \spacefactor only in horizontal mode;
you can refer to \prevdepth only in vertical mode; and
neither of these is meaningful inside \write. So
I'm forgetting what you said and using zero instead.

! Extra }, or forgotten $.
\add@accent ... \accent@spacefactor \spacefactor }
                                                  \let \maybe@load@fontshape...
l.171 ...\xi\in[a,b]}\left\lvert f\^{\prime\prime}
                                                  (\xi) \right\rvert$.
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Please use \mathaccent for accents in math mode.
\add@accent ... \let \ignorespaces \relax \accent 
                                                  #1 #2\egroup \ifmmode \els...
l.171 ...\xi\in[a,b]}\left\lvert f\^{\prime\prime}
                                                  (\xi) \right\rvert$.
I'm changing \accent to \mathaccent here; wish me luck.
(Accents are not the same in formulas as they are in text.)

! Missing { inserted.
<to be read again> 
                   \edef 
l.171 ...\xi\in[a,b]}\left\lvert f\^{\prime\prime}
                                                  (\xi) \right\rvert$.
A left brace was mandatory here, so I've put one in.
You might want to delete and/or insert some corrections
so that I will find a matching right brace soon.
(If you're confused by all this, try typing `I}' now.)

! Extra }, or forgotten $.
\UseTextAccent  ...e@text@encoding \@curr@enc #3}}
                                                  
l.171 ...\xi\in[a,b]}\left\lvert f\^{\prime\prime}
                                                  (\xi) \right\rvert$.
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Undefined control sequence.
l.171 ...lvert f\^{\prime\prime}(\xi) \right\rvert
                                                  $.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   $
l.171 ...vert f\^{\prime\prime}(\xi) \right\rvert$
                                                  .
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Extra \right.
<to be read again> 
                   $
l.171 ...vert f\^{\prime\prime}(\xi) \right\rvert$
                                                  .
I'm ignoring a \right that had no matching \left.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.175             #
                   # Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.175             ##
                     Approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.178             #
                   ## Multiple rectangles
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.178             ##
                    # Multiple rectangles
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.178             ###
                      Multiple rectangles
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! Undefined control sequence.
l.181 ...rror $\mathcal E\_{\mathrm{rect}}\lesssim
                                                   (b-a)h\left\lVert f' \rig...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.181 ...{\mathrm{rect}}\lesssim (b-a)h\left\lVert
                                                   f' \right\rVert_\infty$ f...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.181 ...mathrm{rect}}\lesssim (b-a)h\left\lVert f
                                                  ' \right\rVert_\infty$ for...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.181 ...lesssim (b-a)h\left\lVert f' \right\rVert
                                                  _\infty$ for $h$ the subin...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   _
l.181 ...esssim (b-a)h\left\lVert f' \right\rVert_
                                                  \infty$ for $h$ the subint...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.183             #
                   ## Multiple trapezia
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.183             ##
                    # Multiple trapezia
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.183             ###
                      Multiple trapezia
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! Undefined control sequence.
l.186 ...Error $\mathcal E\_{\mathrm{tpz}}\lesssim
                                                   (b-a)h^2\left\lVert f'' \...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.186 ...\mathrm{tpz}}\lesssim (b-a)h^2\left\lVert
                                                   f'' \right\rVert_\infty$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.186 ...athrm{tpz}}\lesssim (b-a)h^2\left\lVert f
                                                  '' \right\rVert_\infty$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.186 ...ssim (b-a)h^2\left\lVert f'' \right\rVert
                                                  _\infty$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   _
l.186 ...sim (b-a)h^2\left\lVert f'' \right\rVert_
                                                  \infty$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.188             #
                   ## Next step?
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.188             ##
                    # Next step?
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.188             ###
                      Next step?
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! Undefined control sequence.
l.193 ...error $\mathcal E\_{\mathrm{Smp}}\lesssim
                                                   (b-a)h^3\left\lVert f''' ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.193 ...\mathrm{Smp}}\lesssim (b-a)h^3\left\lVert
                                                   f''' \right\rVert_\infty$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.193 ...athrm{Smp}}\lesssim (b-a)h^3\left\lVert f
                                                  ''' \right\rVert_\infty$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.193 ...sim (b-a)h^3\left\lVert f''' \right\rVert
                                                  _\infty$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   _
l.193 ...im (b-a)h^3\left\lVert f''' \right\rVert_
                                                  \infty$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.197             #
                   # Simpson's rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.197             ##
                     Simpson's rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.200             #
                   ## Quadratic approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.200             ##
                    # Quadratic approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.200             ###
                      Quadratic approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! Missing $ inserted.
<inserted text> 
                $
l.205             \begin{pmatrix} a
                                   ^2 \& a \& 1 \\\\ m^2 \& m \& 1 \\\\ b^2 ...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Missing $ inserted.
<inserted text> 
                $
l.205 ...& m \& 1 \\\\ b^2 \& b \& 1 \end{pmatrix}
                                                   \begin{pmatrix} \rho \\\\...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Missing $ inserted.
<inserted text> 
                $
l.205 ...b \& 1 \end{pmatrix} \begin{pmatrix} \rho
                                                   \\\\ \sigma \\\\ \tau \en...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Missing $ inserted.
<inserted text> 
                $
l.205 ... \rho \\\\ \sigma \\\\ \tau \end{pmatrix}
                                                   = \begin{pmatrix} f(a) \\...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Missing $ inserted.
<inserted text> 
                $
l.205 ...\\ \tau \end{pmatrix} = \begin{pmatrix} f
                                                  (a) \\\\ f(m) \\\\ f(b) \e...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Missing $ inserted.
<inserted text> 
                $
l.205 ...x} f(a) \\\\ f(m) \\\\ f(b) \end{pmatrix}
                                                  
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.


! LaTeX Error: Environment align* undefined.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.209             $$\begin{align*}
                                  
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.210             \int
                      _a^b f(x) dx \&\approx \frac \rho3(a^3-b^3) + \frac \s...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.


! LaTeX Error: \begin{document} ended by \end{align*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.212             \end{align*}
                              $$
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Missing $ inserted.
<inserted text> 
                $
l.212             \end{align*}
                              $$
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.214             #
                   ## Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.214             ##
                    # Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.214             ###
                      Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! Undefined control sequence.
l.215 ...      $\mathcal E_{\mathrm{Smp}} \lesssim
                                                   (b-a)h^4\left\lVert f\^{(...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.215 ...mathrm{Smp}} \lesssim (b-a)h^4\left\lVert
                                                   f\^{(4)} \right\rVert_\in...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.215 ...thrm{Smp}} \lesssim (b-a)h^4\left\lVert f
                                                  \^{(4)} \right\rVert_\inft...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.


LaTeX Warning: Command \^ invalid in math mode on input line 215.


LaTeX Warning: Command \^ invalid in math mode on input line 215.

! Please use \mathaccent for accents in math mode.
\add@accent ... \let \ignorespaces \relax \accent 
                                                  #1 #2\egroup \ifmmode \els...
l.215 ...p}} \lesssim (b-a)h^4\left\lVert f\^{(4)}
                                                   \right\rVert_\infty$ wher...
I'm changing \accent to \mathaccent here; wish me luck.
(Accents are not the same in formulas as they are in text.)

! Missing { inserted.
<to be read again> 
                   \edef 
l.215 ...p}} \lesssim (b-a)h^4\left\lVert f\^{(4)}
                                                   \right\rVert_\infty$ wher...
A left brace was mandatory here, so I've put one in.
You might want to delete and/or insert some corrections
so that I will find a matching right brace soon.
(If you're confused by all this, try typing `I}' now.)

! Missing } inserted.
<inserted text> 
                }
l.215 ...sssim (b-a)h^4\left\lVert f\^{(4)} \right
                                                  \rVert_\infty$ where $h$ i...
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Undefined control sequence.
l.215 ...(b-a)h^4\left\lVert f\^{(4)} \right\rVert
                                                  _\infty$ where $h$ is the ...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   _
l.215 ...b-a)h^4\left\lVert f\^{(4)} \right\rVert_
                                                  \infty$ where $h$ is the i...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.219             #
                   # Simpson's rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.219             ##
                     Simpson's rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.222             #
                   ## Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.222             ##
                    # Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.222             ###
                      Error
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! Undefined control sequence.
l.223 ...      $\mathcal E_{\mathrm{Smp}} \lesssim
                                                   (b-a)h^4\left\lVert f\^{(...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.223 ...mathrm{Smp}} \lesssim (b-a)h^4\left\lVert
                                                   f\^{(4)} \right\rVert_\in...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.223 ...thrm{Smp}} \lesssim (b-a)h^4\left\lVert f
                                                  \^{(4)} \right\rVert_\infty$
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.


LaTeX Warning: Command \^ invalid in math mode on input line 223.


LaTeX Warning: Command \^ invalid in math mode on input line 223.

! Please use \mathaccent for accents in math mode.
\add@accent ... \let \ignorespaces \relax \accent 
                                                  #1 #2\egroup \ifmmode \els...
l.223 ...p}} \lesssim (b-a)h^4\left\lVert f\^{(4)}
                                                   \right\rVert_\infty$
I'm changing \accent to \mathaccent here; wish me luck.
(Accents are not the same in formulas as they are in text.)

! Missing { inserted.
<to be read again> 
                   \edef 
l.223 ...p}} \lesssim (b-a)h^4\left\lVert f\^{(4)}
                                                   \right\rVert_\infty$
A left brace was mandatory here, so I've put one in.
You might want to delete and/or insert some corrections
so that I will find a matching right brace soon.
(If you're confused by all this, try typing `I}' now.)

! Missing } inserted.
<inserted text> 
                }
l.223 ...sssim (b-a)h^4\left\lVert f\^{(4)} \right
                                                  \rVert_\infty$
I've inserted something that you may have forgotten.
(See the <inserted text> above.)
With luck, this will get me unwedged. But if you
really didn't forget anything, try typing `2' now; then
my insertion and my current dilemma will both disappear.

! Undefined control sequence.
l.223 ...(b-a)h^4\left\lVert f\^{(4)} \right\rVert
                                                  _\infty$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   _
l.223 ...b-a)h^4\left\lVert f\^{(4)} \right\rVert_
                                                  \infty$
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Undefined control sequence.
l.225 ...      $\mathcal E_{\mathrm{Smp}} \lesssim
                                                   (b-a)h^3\left\lVert f\^{\...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.225 ...mathrm{Smp}} \lesssim (b-a)h^3\left\lVert
                                                   f\^{\prime\prime\prime} \...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   f
l.225 ...thrm{Smp}} \lesssim (b-a)h^3\left\lVert f
                                                  \^{\prime\prime\prime} \ri...
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.


LaTeX Warning: Command \^ invalid in math mode on input line 225.


LaTeX Warning: Command \^ invalid in math mode on input line 225.

! Missing $ inserted.
<inserted text> 
                $
l.225 ...-a)h^3\left\lVert f\^{\prime\prime\prime}
                                                   \right\rVert_\infty$.
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Improper \spacefactor.
<recently read> \spacefactor 
                             
l.225 ...-a)h^3\left\lVert f\^{\prime\prime\prime}
                                                   \right\rVert_\infty$.
You can refer to \spacefactor only in horizontal mode;
you can refer to \prevdepth only in vertical mode; and
neither of these is meaningful inside \write. So
I'm forgetting what you said and using zero instead.

! Extra }, or forgotten $.
\add@accent ... \accent@spacefactor \spacefactor }
                                                  \let \maybe@load@fontshape...
l.225 ...-a)h^3\left\lVert f\^{\prime\prime\prime}
                                                   \right\rVert_\infty$.
I've deleted a group-closing symbol because it seems to be
spurious, as in `$x}$'. But perhaps the } is legitimate and
you forgot something else, as in `\hbox{$x}'. In such cases
the way to recover is to insert both the forgotten and the
deleted material, e.g., by typing `I$}'.

! Please use \mathaccent for accents in math mode.
\add@accent ... \let \ignorespaces \relax \accent 
                                                  #1 #2\egroup \ifmmode \els...
l.225 ...-a)h^3\left\lVert f\^{\prime\prime\prime}
                                                   \right\rVert_\infty$.
I'm changing \accent to \mathaccent here; wish me luck.
(Accents are not the same in formulas as they are in text.)

! Missing { inserted.
<to be read again> 
                   \edef 
l.225 ...-a)h^3\left\lVert f\^{\prime\prime\prime}
                                                   \right\rVert_\infty$.
A left brace was mandatory here, so I've put one in.
You might want to delete and/or insert some corrections
so that I will find a matching right brace soon.
(If you're confused by all this, try typing `I}' now.)

! Undefined control sequence.
l.225 ...Vert f\^{\prime\prime\prime} \right\rVert
                                                  _\infty$.
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing delimiter (. inserted).
<to be read again> 
                   _
l.225 ...ert f\^{\prime\prime\prime} \right\rVert_
                                                  \infty$.
I was expecting to see something like `(' or `\{' or
`\}' here. If you typed, e.g., `{' instead of `\{', you
should probably delete the `{' by typing `1' now, so that
braces don't get unbalanced. Otherwise just proceed.
Acceptable delimiters are characters whose \delcode is
nonnegative, or you can use `\delimiter <delimiter code>'.

! Extra \right.
<to be read again> 
                   _
l.225 ...ert f\^{\prime\prime\prime} \right\rVert_
                                                  \infty$.
I'm ignoring a \right that had no matching \left.

! Missing $ inserted.
<inserted text> 
                $
l.233             f_
                    {\mathrm{cubic}}(x) = f_{\mathrm{Smp}}(x) + C\begin{pmat...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Undefined control sequence.
<argument> \text 
                 
l.233 ...{\mathrm{Smp}}(x) + C\begin{pmatrix}\text
                                                  {cubic which evaluates}\\\...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<recently read> \text 
                      
l.233 ...rix}\text{cubic which evaluates}\\\\\text
                                                  {to } 0 \text{ at } x=a,m,...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.233 ...ic which evaluates}\\\\\text{to } 0 \text
                                                  { at } x=a,m,b.\end{pmatrix}
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing $ inserted.
<inserted text> 
                $
l.240 
      
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.250             #
                   # Simpson's rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.250             ##
                     Simpson's rule
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.256             #
                   ## Composite quadratic approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.256             ##
                    # Composite quadratic approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! You can't use `macro parameter character #' in restricted horizontal mode.
l.256             ###
                      Composite quadratic approximation
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn't ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! Missing $ inserted.
<inserted text> 
                $
l.259             \frac{h}3
                           \left(f(a) + 4\sum\_{j=1}\^{N}f(m_\{2j-1}) + 2\su...
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.

! Extra }, or forgotten $.
\frac  #1#2->{\begingroup #1\endgroup \over #2}
                                               
l.259             \frac{h}3
                           \left(f(a) + 4\sum\_{j=1}\^{N}f(m_\{2j-1}) + 2\su...
(That makes 100 errors; please try again.) 
Here is how much of TeX's memory you used:
 42 strings out of 474222
 855 string characters out of 5748734
 1936975 words of memory out of 5000000
 22393 multiletter control sequences out of 15000+600000
 558069 words of font info for 36 fonts, out of 8000000 for 9000
 1141 hyphenation exceptions out of 8191
 23i,20n,32p,382b,160s stack positions out of 10000i,1000n,20000p,200000b,200000s

!  ==> Fatal error occurred, no output PDF file produced!