Mathematica 9 is now available
Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2000
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2000

[Date Index] [Thread Index] [Author Index]

Search the Archive

Problem with definite integrals having symbolic limits

  • To: mathgroup at smc.vnet.net
  • Subject: [mg21850] Problem with definite integrals having symbolic limits
  • From: "Wolter Kaper" <kaper at chem.uva.nl>
  • Date: Sun, 30 Jan 2000 18:53:31 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Dear MathGroup,

I discovered what I think is an inconsistency and would like to check my
opinion and offer a suggestion to Wolfram.
When you do a "definite" integration with symbolic limits (instead of
concrete numbers), the result you get may not be unconditionally true. The
necessary conditions are not stated by Mathematica, although
"GenerateConditions" is set to True.

The notebook added demonstrates this problem. I suggest that a warning
message should be given, that describes the conditions for the stated
result.

With best regards,
Wolter Kaper
dept. of Chemistry
Univ. of Amsterdam

Notebook follows


(***********************************************************************

                    Mathematica-Compatible Notebook

This notebook can be used on any computer system with Mathematica 4.0,
MathReader 4.0, or any compatible application. The data for the notebook
starts with the line containing stars above.

To get the notebook into a Mathematica-compatible application, do one of
the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the application;

* Copy the data starting with the line of stars above to the
  clipboard, then use the Paste menu command inside the application.

Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode.  Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing the
word CacheID, otherwise Mathematica-compatible applications may try to
use invalid cache data.

For more information on notebooks and Mathematica-compatible
applications, contact Wolfram Research:
  web: http://www.wolfram.com
  email: info at wolfram.com
  phone: +1-217-398-0700 (U.S.)

Notebook reader applications are available free of charge from
Wolfram Research.
***********************************************************************)

(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[     56107,       1779]*)
(*NotebookOutlinePosition[     56895,       1809]*)
(*  CellTagsIndexPosition[     56821,       1803]*)
(*WindowFrame->Normal*)



Notebook[{
Cell["A problem with definite integration", "Title"],

Cell[TextData[{
  StyleBox["Summary:",
    FontWeight->"Bold"],
  " When you do a \"definite\" integration with symbolic limits (instead of
\
concrete numbers), the result you get may not be unconditionally true. The \
necessary conditions are not stated by ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", although \"GenerateConditions\" is set to True."
}], "Text"],

Cell[CellGroupData[{

Cell[TextData[{
  StyleBox["Demonstration/Explanation",
    FontWeight->"Bold"],
  ". The ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " book explains that even when you can find the indefinite form of an \
integral, you will often not get the right answer for a definit integral if
\
you just subtract values of the limits at each point. ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  ", however, knows about this possibility and gives the right answer if you
\
supply the limits in numerical form. (Open to see an example)"
}], "Text"],

Cell["Example:", "Text",
  CellTags->"Example"],

Cell[CellGroupData[{

Cell[BoxData[
    \(F[x_] = \[Integral]\(1\/x\^2\) \[DifferentialD]x\)], "Input"],

Cell[BoxData[
    \(\(-\(1\/x\)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(F[2] - F[\(-2\)]\)], "Input"],

Cell[BoxData[
    \(\(-1\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\[Integral]\_\(-2\)\%2\( 1\/x\^2\) \[DifferentialD]x\)], "Input"],

Cell[BoxData[
    InterpretationBox["\[Infinity]",
      DirectedInfinity[ 1]]], "Output"]
}, Open  ]],

Cell["\<\
The last answer is correct. It is not equal to -1 because the two limits are
\
situated on either side of a discontinuity in F.\
\>", "Text"]
}, Closed]],

Cell["\<\
Now I want to determine the value of the definite integral as a function of
\
its end point.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Int[
        xend_] =
ntegral]\_\(-2\)\%xend\( 
              1\/x\^2\) \[DifferentialD]x\)], "Input"],

Cell[BoxData[
    \(\(-\(1\/2\)\) - 1\/xend\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Int[xend_] = 
      Integrate[1\/x\^2, {x, \(-2\), xend}, \ 
        GenerateConditions \[Rule] True]\)], "Input"],

Cell[BoxData[
    \(\(-\(\(2 + xend\)\/\(2\ xend\)\)\)\)], "Output"]
}, Open  ]],

Cell["\<\
This result is of limited validity, it holds only for xend < 0, although no \
condition is stated. 
If I insert a positive value...\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Int[2]\)], "Input"],

Cell[BoxData[
    \(\(-1\)\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "...I get the old wrong result again (compare with ",
  ButtonBox["Example",
    ButtonData:>"Example",
    ButtonStyle->"Hyperlink"],
  ").\n\nBecause \"GenerateConditions\" is True (the ",
  StyleBox["default valu",
    FontSlant->"Italic"],
  "e!) and \"xend\" may be termed a parameter of the definite integral, I \
think the condition \"",
  StyleBox["xend < 0",
    FontWeight->"Bold"],
  "\" should be stated together with the formula ",
  Cell[BoxData[
      \(\(-\(1\/2\)\) - 1\/xend\)]],
  ".\n\nThis would:\n- be more consistent than say nothing\n- increase the \
usefulness of the program for unattended student-use in education. "
}], "Text"],

Cell[CellGroupData[{

Cell["Where did I meet the problem", "Subsection"],

Cell["\<\
In a thermodynamics course for chemistry students, I asked them to calculate \
the work done and the heat taken up by a reversible heat engine, that \
contains 1 mole of a Van der Waals gas, and follows a cyclic process. The \
changes of Volume and Temperature are described by the following parametric \
curve\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(T = 475*Sin[\(-t\)] + 750\)], "Input",
  GeneratedCell->False],

Cell[BoxData[
    \(750 - 475\ Sin[t]\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(V = 1.1*Cos[\(-t\)] + 1.3\)], "Input",
  GeneratedCell->False],

Cell[BoxData[
    \(\(\(1.3`\)\(\[InvisibleSpace]\)\) + 1.1`\ Cos[t]\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(ParametricPlot[{T, V}, \ {t, 0, 2*Pi}]\)], "Input",
  GeneratedCell->False],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Courier
%%IncludeFont: Courier
/Courier findfont 10  scalefont  setfont
% Scaling calculations
-0.25188 0.00100251 0.0147151 0.245252 [
[.34962 .00222 -9 -9 ]
[.34962 .00222 9 0 ]
[.55013 .00222 -9 -9 ]
[.55013 .00222 9 0 ]
[.75063 .00222 -12 -9 ]
[.75063 .00222 12 0 ]
[.95113 .00222 -12 -9 ]
[.95113 .00222 12 0 ]
[.13662 .13734 -18 -4.5 ]
[.13662 .13734 0 4.5 ]
[.13662 .25997 -6 -4.5 ]
[.13662 .25997 0 4.5 ]
[.13662 .38259 -18 -4.5 ]
[.13662 .38259 0 4.5 ]
[.13662 .50522 -6 -4.5 ]
[.13662 .50522 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.34962 .01472 m
.34962 .02097 L
s
[(600)] .34962 .00222 0 1 Mshowa
.55013 .01472 m
.55013 .02097 L
s
[(800)] .55013 .00222 0 1 Mshowa
.75063 .01472 m
.75063 .02097 L
s
[(1000)] .75063 .00222 0 1 Mshowa
.95113 .01472 m
.95113 .02097 L
s
[(1200)] .95113 .00222 0 1 Mshowa
.125 Mabswid
.19925 .01472 m
.19925 .01847 L
s
.24937 .01472 m
.24937 .01847 L
s
.2995 .01472 m
.2995 .01847 L
s
.39975 .01472 m
.39975 .01847 L
s
.44987 .01472 m
.44987 .01847 L
s
.5 .01472 m

.5 .01847 L
s
.60025 .01472 m
.60025 .01847 L
s
.65038 .01472 m
.65038 .01847 L
s
.7005 .01472 m
.7005 .01847 L
s
.80075 .01472 m
.80075 .01847 L
s
.85088 .01472 m
.85088 .01847 L
s
.901 .01472 m
.901 .01847 L
s
.099 .01472 m
.099 .01847 L
s
.04887 .01472 m
.04887 .01847 L
s
.25 Mabswid
0 .01472 m
1 .01472 L
s
.14912 .13734 m
.15537 .13734 L
s
[(0.5)] .13662 .13734 1 0 Mshowa
.14912 .25997 m
.15537 .25997 L
s
[(1)] .13662 .25997 1 0 Mshowa
.14912 .38259 m
.15537 .38259 L
s
[(1.5)] .13662 .38259 1 0 Mshowa
.14912 .50522 m
.15537 .50522 L
s
[(2)] .13662 .50522 1 0 Mshowa
.125 Mabswid
.14912 .03924 m
.15287 .03924 L
s
.14912 .06377 m
.15287 .06377 L
s
.14912 .08829 m
.15287 .08829 L
s
.14912 .11282 m
.15287 .11282 L
s
.14912 .16187 m
.15287 .16187 L
s
.14912 .18639 m
.15287 .18639 L
s
.14912 .21092 m
.15287 .21092 L
s
.14912 .23544 m
.15287 .23544 L
s
.14912 .28449 m
.15287 .28449 L
s
.14912 .30902 m
.15287 .30902 L
s
.14912 .33354 m
.15287 .33354 L
s
.14912 .35807 m
.15287 .35807 L
s
.14912 .40712 m
.15287 .40712 L
s
.14912 .43164 m
.15287 .43164 L
s
.14912 .45617 m
.15287 .45617 L
s
.14912 .48069 m
.15287 .48069 L
s
.14912 .52974 m
.15287 .52974 L
s
.14912 .55427 m
.15287 .55427 L
s
.14912 .57879 m
.15287 .57879 L
s
.14912 .60332 m
.15287 .60332 L
s
.25 Mabswid
.14912 0 m
.14912 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
.5 .60332 m
.49631 .60331 L
.49295 .60329 L
.48907 .60325 L
.4854 .60319 L
.47889 .60305 L
.47182 .60285 L
.46404 .60255 L
.4558 .60215 L
.44107 .60124 L
.42591 .60003 L
.41178 .59865 L
.37993 .5946 L
.34803 .58921 L
.31948 .58318 L
.25809 .56592 L
.20875 .54697 L
.16012 .52249 L
.11981 .49598 L
.0839 .46472 L
.06916 .44844 L
.05544 .43023 L
.04552 .41407 L
.03673 .39596 L
.03336 .38731 L
.03031 .37797 L
.02814 .36986 L
.02628 .36099 L
.02543 .35578 L
.0248 .35096 L
.02435 .34634 L
.02417 .34401 L
.02401 .34143 L
.02391 .33911 L
.02385 .33699 L
.02381 .33466 L
.02381 .33349 L
.02382 .33221 L
.02385 .33004 L
.02392 .32772 L
.02402 .32554 L
.02414 .32355 L
.02446 .31945 L
.02494 .31493 L
.02633 .30583 L
.0281 .29743 L
.03017 .2896 L
.03632 .2721 L
.04519 .25361 L
.05563 .23658 L
Mistroke
.06927 .21851 L
.0999 .18725 L
.13677 .15909 L
.18274 .13236 L
.236 .10902 L
.29095 .09115 L
.319 .08401 L
.34941 .07761 L
.37853 .07269 L
.40548 .06913 L
.43491 .0663 L
.45122 .06518 L
.45872 .06478 L
.4667 .06443 L
.47351 .06418 L
.47713 .06408 L
.48103 .06398 L
.4846 .06391 L
.48788 .06385 L
.491 .06381 L
.49429 .06378 L
.49789 .06377 L
.50173 .06377 L
.50535 .06378 L
.50864 .06381 L
.51251 .06386 L
.5167 .06393 L
.5243 .06412 L
.53172 .06436 L
.53967 .0647 L
.554 .06551 L
.57098 .06678 L
.58639 .06824 L
.61953 .0724 L
.6504 .07757 L
.70821 .09092 L
.75857 .107 L
.80567 .12668 L
.85231 .15204 L
.89018 .17889 L
.92392 .21066 L
.94879 .24335 L
.95837 .26043 L
.96287 .27019 L
.96653 .27949 L
.96938 .28809 L
.97192 .29749 L
.97371 .30606 L
.97494 .314 L
.97544 .31844 L
Mistroke
.9758 .3226 L
.97595 .32498 L
.97606 .32717 L
.9761 .32837 L
.97614 .32968 L
.97617 .33093 L
.97618 .33208 L
.97619 .33328 L
.97619 .33459 L
.97617 .33582 L
.97615 .33695 L
.97609 .33909 L
.97599 .34136 L
.97585 .3438 L
.97568 .34607 L
.97516 .35124 L
.97456 .35586 L
.97387 .36013 L
.97188 .36976 L
.96962 .37822 L
.9667 .38714 L
.9604 .40244 L
.95164 .41904 L
.92874 .45094 L
.89626 .48315 L
.85795 .51146 L
.81091 .53788 L
.76139 .55904 L
.71135 .57529 L
.68121 .58302 L
.65196 .58921 L
.62481 .59389 L
.5956 .59783 L
.57212 .60021 L
.55869 .60126 L
.54609 .60205 L
.5345 .60261 L
.52389 .60298 L
.51803 .60313 L
.51493 .60319 L
.51164 .60324 L
.50847 .60328 L
.5056 .6033 L
.50271 .60331 L
.5 .60332 L
Mfstroke
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  ImageSize->{288, 177.938},
  ImageMargins->{{43, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40004P0000/A000`40O003h00Oogoo8Goo001N
Ool3
0004Ool20004Ool2000YOol20004Ool20004Ool2000TOol50002Ool20004Ool20004Ool2000R
Ool5
00000goo000000020003Ool20004Ool2000<Ool005ioo`04001oogoo0002Ool01000Oomoo`00
0Woo
00 at 007ooOol002Moo`04001oogoo0002Ool01000Oomoo`000Woo00@007ooOol002Eoo`05001o
ogoo
Ool00002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`03001oogoo025oo`05001oogooOol0
0005
Ool01000Oomoo`000Woo00 at 007ooOol000]oo`00GWoo00@007ooOol0009oo`04001oogoo0002
Ool0
1000Oomoo`009goo00 at 007ooOol0009oo`04001oogoo0002Ool01000Oomoo`009Goo00D007oo
Oomo
o`00009oo`04001oogoo0002Ool01000Oomoo`000Woo00<007ooOol08Goo00<007ooOol00Woo
00<0
07ooOol00Woo00 at 007ooOol0009oo`04001oogoo000;Ool005ioo`<000=oo`04001oogoo0002
Ool0
1000Oomoo`00:7oo0P000goo00 at 007ooOol0009oo`04001oogoo000UOol01@00Oomoogoo0000
0Woo
00 at 007ooOol0009oo`04001oogoo0002Ool00`00Oomoo`0QOol00`00Oomoo`03Ool01@00Oomo
ogoo
00000Woo00 at 007ooOol0009oo`03001oogoo00Uoo`00Ggoo00<007ooOol00Woo00 at 007ooOol0
009o
o`04001oogoo000WOol01000Oomoo`000Woo00 at 007ooOol0009oo`04001oogoo000UOol01@00
Oomo
ogoo00000Woo00 at 007ooOol0009oo`04001oogoo0002Ool00`00Oomoo`0QOol01@00Oomoogoo
0000
0Woo00 at 007ooOol0009oo`04001oogoo0002Ool00`00Oomoo`09Ool005moo`<000=oo`8000Ao
o`80
02Uoo`8000Aoo`8000Aoo`8002Eoo`8000Aoo`8000Aoo`8000Aoo`8002=oo`8000Aoo`8000Ao
o`80
00Aoo`8000aoo`00ogoo8Goo000_Ool00`00Oomoo`3^Ool002moo`03001oogoo0>ioo`00;goo
00<0
07ooOol0kWoo000_Ool00`00Oomoo`3^Ool000Moool0018000Qoo`0057oo00<007ooOol02goo
00<0
07ooOol02Woo00<007ooOol02goo00<007ooOol02goo00<007ooOol02Woo00<007ooOol02goo
00<0
07ooOol02goo00<007ooOol02Woo00<007ooOol02goo00<007ooOol02goo00<007ooOol02Woo
00<0
07ooOol02goo00<007ooOol02goo00<007ooOol02Woo00<007ooOol02goo00<007ooOol02goo
00<0
07ooOol02Woo00<007ooOol02goo00<007ooOol04goo000_Ool00`00Oomoo`0dOol00`00Oomo
o`0d
Ool00`00Oomoo`0cOol00`00Oomoo`0dOol00`00Oomoo`0COol002moo`03001oogoo0>ioo`00
;goo
00<007ooOol0kWoo000_Ool00`00Oomoo`3^Ool002moo`03001oogoo0>ioo`00;goo0P00kgoo
000_
Ool00`00Oomoo`3^Ool002moo`03001oogoo0>ioo`00;goo00<007ooOol0kWoo000_Ool00`00
Oomo
o`3^Ool002moo`03001oogoo0>ioo`00;goo0P00EWoo4000RGoo000_Ool00`00Oomoo`17Ool>
000@
Ool>001kOol002moo`03001oogoo041oo`L002aoo`L007Aoo`00;goo00<007ooOol0>7oo2000
>Woo
2000K7oo000_Ool00`00Oomoo`0bOol6001:Ool6001VOol002moo`03001oogoo02ioo` at 005Io
o` at 0
069oo`00;goo00<007ooOol0:Woo1000GWoo1000GWoo000_Ool2000WOol4001VOol4001JOol0
02mo
o`03001oogoo02=oo`<006ioo` at 005Ioo`00;goo00<007ooOol087oo0`00MGoo1000DWoo000_
Ool0
0`00Oomoo`0MOol3001lOol3001?Ool002moo`03001oogoo01Yoo`<0089oo` at 004]oo`00;goo
00<0
07ooOol05goo0`00RGoo0P00BGoo000_Ool00`00Oomoo`0DOol3002>Ool20017Ool002moo`80
01=o
o`8009=oo`8004Eoo`00;goo00<007ooOol03goo0`00Ugoo0P00 at goo000_Ool00`00Oomoo`0=
Ool2
002LOol20011Ool002moo`03001oogoo00Yoo`<00:1oo`8003moo`0077oo0P0017oo0P0017oo
0P00
1Goo00<007ooOol027oo0P00YGoo0`00?7oo000KOol01000Oomoo`0027oo00 at 007ooOol000Ao
o`03
001oogoo00Ioo`800:Yoo`8003Yoo`006goo00 at 007ooOol000]oo`03001oogoo009oo`<000Ao
o`80
0:ioo`8003Qoo`006goo00 at 007ooOol000Uoo`8000Eoo`03001oogoo009oo`800;9oo`8003Io
o`00
6goo00 at 007ooOol000Uoo`03001oogoo00Aoo`03001oogoo00800;Ioo`8003Aoo`0077oo0P00
2Woo
0`0017oo00<007oo0000^Woo0P00<Woo000_Ool2002mOol2000`Ool002eoo`<00<1oo`03001o
ogoo
02eoo`00;7oo00 at 007ooOol00<5oo`8002eoo`00:goo00 at 007ooOomoo`800<9oo`03001oogoo
02Yo
o`00:Goo0P0017oo00<007ooOol0`Woo00<007ooOol0:Goo000XOol00`00Oomoo`04Ool00`00
Oomo
o`33Ool00`00Oomoo`0XOol002Moo`03001oogoo00Eoo`03001oogoo0<Aoo`8002Qoo`009Woo
00<0
07ooOol01Woo00<007ooOol0aWoo00<007ooOol09Goo000TOol20009Ool00`00Oomoo`37Ool0
0`00
Oomoo`0TOol002=oo`03001oogoo00Uoo`03001oogoo0<Qoo`03001oogoo02=oo`008Woo00<0
07oo
Ool02Woo0P00bWoo0P008goo000QOol00`00Oomoo`0;Ool00`00Oomoo`3;Ool00`00Oomoo`0P
Ool0
021oo`03001oogoo00aoo`03001oogoo0<aoo`03001oogoo01moo`007goo00<007ooOol03Goo
00<0
07ooOol0cGoo00<007ooOol07Woo000NOol00`00Oomoo`0>Ool00`00Oomoo`3>Ool2000NOol0
01io
o`03001oogoo00ioo`03001oogoo0=1oo`03001oogoo01]oo`007Goo00<007ooOol03goo0P00
dWoo
00<007ooOol06Woo000LOol00`00Oomoo`0 at Ool00`00Oomoo`3BOol00`00Oomoo`0IOol001]o
o`03
001oogoo015oo`03001oogoo0=9oo`03001oogoo01Uoo`006Woo00<007ooOol04Woo00<007oo
Ool0
dgoo00<007ooOol067oo000IOol00`00Oomoo`0COol00`00Oomoo`3DOol00`00Oomoo`0GOol0
01Qo
o`03001oogoo01Aoo`03001oogoo0=Aoo`03001oogoo01Moo`005goo00<007ooOol05Goo00<0
07oo
Ool0eGoo00<007ooOol05Woo000FOol00`00Oomoo`0FOol2003GOol00`00Oomoo`0EOol001Eo
o`03
001oogoo01Moo`03001oogoo0=Ioo`03001oogoo01Eoo`005Goo00<007ooOol05goo00<007oo
Ool0
egoo00<007ooOol057oo000DOol00`00Oomoo`0HOol00`00Oomoo`3HOol00`00Oomoo`0COol0
01Ao
o`03001oogoo01Qoo`03001oogoo0=Qoo`03001oogoo01=oo`004goo00<007ooOol04Goo1 at 00
0goo
00<007ooOol0fGoo00<007ooOol04Woo000COol00`00Oomoo`0COol00`00Oomoo`03Ool00`00
Oomo
o`3IOol00`00Oomoo`0BOol0019oo`03001oogoo01Aoo`03001oogoo00=oo`<00=Yoo`03001o
ogoo
015oo`004Woo00<007ooOol057oo00<007ooOol00goo00<007ooOol0fWoo00<007ooOol04Goo
000A
Ool00`00Oomoo`0EOol00`00Oomoo`03Ool00`00Oomoo`3KOol00`00Oomoo`0 at Ool0015oo`03
001o
ogoo01Aoo`8000Eoo`03001oogoo0=]oo`03001oogoo011oo`004Goo00<007ooOol06goo00<0
07oo
Ool0g7oo00<007ooOol03goo000 at Ool00`00Oomoo`0LOol00`00Oomoo`3LOol00`00Oomoo`0?
Ool0
011oo`03001oogoo01aoo`03001oogoo0=aoo`03001oogoo00moo`003goo00<007ooOol07Goo
0P00
gWoo00<007ooOol03Woo000?Ool00`00Oomoo`0MOol00`00Oomoo`3MOol00`00Oomoo`0>Ool0
00mo
o`03001oogoo01eoo`03001oogoo0=ioo`03001oogoo00eoo`003Woo00<007ooOol07Woo00<0
07oo
Ool0gWoo00<007ooOol03Goo000>Ool00`00Oomoo`0NOol00`00Oomoo`3NOol00`00Oomoo`0=
Ool0
00ioo`03001oogoo01ioo`03001oogoo0=ioo`03001oogoo00eoo`003Woo00<007ooOol07Woo
0P00
ggoo00<007ooOol03Goo000=Ool00`00Oomoo`0OOol00`00Oomoo`3OOol00`00Oomoo`0<Ool0
00eo
o`03001oogoo01moo`03001oogoo0=moo`03001oogoo00aoo`003Goo00<007ooOol07goo00<0
07oo
Ool0ggoo00<007ooOol037oo000=Ool00`00Oomoo`0OOol00`00Oomoo`3OOol00`00Oomoo`0<
Ool0
00eoo`03001oogoo01moo`03001oogoo0=moo`03001oogoo00aoo`003Goo00<007ooOol07goo
00<0
07ooOol0ggoo00<007ooOol037oo000=Ool00`00Oomoo`0OOol2003POol00`00Oomoo`0<Ool0
00eo
o`03001oogoo01moo`03001oogoo0=moo`03001oogoo00aoo`003Goo00<007ooOol07goo00<0
07oo
Ool0ggoo00<007ooOol037oo000=Ool00`00Oomoo`0OOol00`00Oomoo`3OOol00`00Oomoo`0<
Ool0
00eoo`03001oogoo01moo`03001oogoo0=moo`03001oogoo00aoo`003Goo00<007ooOol07goo
00<0
07ooOol0ggoo00<007ooOol037oo000>Ool00`00Oomoo`0NOol00`00Oomoo`3NOol00`00Oomo
o`0=
Ool000ioo`03001oogoo01ioo`800=moo`03001oogoo00eoo`003Woo00<007ooOol07Woo00<0
07oo
Ool0gWoo00<007ooOol03Goo000>Ool00`00Oomoo`0NOol00`00Oomoo`3NOol00`00Oomoo`0=
Ool0
00ioo`03001oogoo01ioo`03001oogoo0=ioo`03001oogoo00eoo`003Woo00<007ooOol02Woo
1 at 00
0Woo0P0017oo0P001Goo00<007ooOol0gGoo00<007ooOol03Woo000?Ool00`00Oomoo`0;Ool0
0`00
Oomoo`07Ool01000Oomoo`0017oo00<007ooOol0gGoo00<007ooOol03Woo000?Ool00`00Oomo
o`0;
Ool00`00Oomoo`0:Ool00`00Oomoo`02Ool3003MOol00`00Oomoo`0>Ool0011oo`03001oogoo
00Yo
o`03001oogoo00Qoo`8000Eoo`03001oogoo0=aoo`03001oogoo00moo`0047oo00<007ooOol0
2Woo
00<007ooOol027oo00<007ooOol017oo00<007ooOol0g7oo00<007ooOol03goo000 at Ool00`00
Oomo
o`09Ool2000:Ool30004Ool00`00Oomoo`3LOol00`00Oomoo`0?Ool0015oo`03001oogoo01]o
o`03
001oogoo0=]oo`03001oogoo011oo`004Goo00<007ooOol06goo00<007ooOol0fgoo00<007oo
Ool0
47oo000BOol00`00Oomoo`0JOol00`00Oomoo`3KOol00`00Oomoo`0 at Ool0019oo`03001oogoo
01Yo
o`800=]oo`03001oogoo015oo`004goo00<007ooOol06Goo00<007ooOol0fGoo00<007ooOol0
4Woo
000COol00`00Oomoo`0IOol00`00Oomoo`3IOol00`00Oomoo`0BOol001Aoo`03001oogoo01Qo
o`03
001oogoo0=Qoo`03001oogoo01=oo`0057oo00<007ooOol067oo00<007ooOol0egoo00<007oo
Ool0
57oo000EOol00`00Oomoo`0GOol00`00Oomoo`3GOol00`00Oomoo`0DOol001Ioo`03001oogoo
01Io
o`03001oogoo0=Ioo`03001oogoo01Eoo`005goo00<007ooOol05Goo0P00eWoo00<007ooOol0
5Woo
000HOol00`00Oomoo`0DOol00`00Oomoo`3EOol00`00Oomoo`0FOol001Qoo`03001oogoo01Ao
o`03
001oogoo0=Aoo`03001oogoo01Moo`006Goo00<007ooOol04goo00<007ooOol0dgoo00<007oo
Ool0
67oo000JOol00`00Oomoo`0BOol00`00Oomoo`3COol00`00Oomoo`0HOol001]oo`03001oogoo
015o
o`03001oogoo0=9oo`03001oogoo01Uoo`0077oo00<007ooOol047oo0P00dWoo00<007ooOol0
6Woo
000LOol00`00Oomoo`0 at Ool00`00Oomoo`3@Ool00`00Oomoo`0KOol001eoo`03001oogoo00mo
o`03
001oogoo0<moo`03001oogoo01aoo`007Woo00<007ooOol03Woo00<007ooOol0cWoo00<007oo
Ool0
7Goo000OOol00`00Oomoo`0=Ool00`00Oomoo`3=Ool00`00Oomoo`0NOol0021oo`03001oogoo
00ao
o`03001oogoo0<aoo`03001oogoo01moo`008Goo00<007ooOol02goo00<007ooOol0bgoo00<0
07oo
Ool087oo000ROol2000;Ool2003;Ool00`00Oomoo`0QOol002Aoo`03001oogoo00Qoo`03001o
ogoo
0<Uoo`03001oogoo029oo`009Goo00<007ooOol01goo00<007ooOol0agoo0P009Goo000VOol0
0`00
Oomoo`06Ool00`00Oomoo`36Ool00`00Oomoo`0UOol002Moo`03001oogoo00Eoo`03001oogoo
0<Eo
o`03001oogoo02Ioo`009goo100017oo00<007ooOol0`goo0P00:Goo000WOol01000Oomoo`00
17oo
00<007ooOol0`Woo00<007ooOol0:Goo000XOol00`00Oomoo`020002Ool30030Ool2000/Ool0
02Uo
o`04001oogooOol30031Ool00`00Oomoo`0/Ool002Moo`04001oogoo0004Ool00`00Oomoo`2m
Ool2
000_Ool002Qoo`8000Eoo`<00;]oo`80035oo`00;goo00<007ooOol00P00^7oo00<007ooOol0
<Goo
000_Ool00`00Oomoo`02Ool2002dOol2000dOol002moo`8000Eoo`800;5oo`03001oogoo03Ao
o`00
;goo00<007ooOol01Woo0P00[Goo0P00=goo000_Ool00`00Oomoo`08Ool2002YOol2000iOol0
02mo
o`03001oogoo00Yoo`800:Eoo`8003]oo`00;goo00<007ooOol037oo0P00X7oo0`00?Goo000_
Ool0
0`00Oomoo`0>Ool2002LOol20010Ool002moo`03001oogoo011oo`8009Moo`<0049oo`00;goo
0P00
4goo0`00TGoo0`00AGoo000_Ool00`00Oomoo`0EOol3002;Ool30018Ool002moo`03001oogoo
01Qo
o`8008Moo`8004]oo`00;goo00<007ooOol06Woo0`00PGoo0`00CGoo000_Ool00`00Oomoo`0M
Ool4
001jOol3001 at Ool002moo`03001oogoo025oo`<007Eoo`8005=oo`00;goo00<007ooOol097oo
0`00
KWoo1000EGoo000_Ool2000XOol4001VOol4001IOol002moo`03001oogoo02]oo` at 005ioo`@0
05eo
o`00;goo00<007ooOol0;goo1000EWoo1000HGoo000_Ool00`00Oomoo`0cOol6001:Ool6001U
Ool0
02moo`03001oogoo03Uoo`P003]oo`L006]oo`00;goo00<007ooOol0 at Goo1`00;Goo1`00LWoo
000_
Ool20019Ool]001iOol002moo`03001oogoo0>ioo`00;goo00<007ooOol0kWoo000_Ool00`00
Oomo
o`3^Ool002moo`03001oogoo0>ioo`00\
\>"],
  ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {225.11, -0.185526, 3.65776,
\
0.0149517}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell["\<\
With the help of some theory, students could deduce for themselves that the
\
heat dq per unit of the process parameter dt (not necessarily the time) is:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dqdt =
      Cv \[PartialD]\_t\ T + \(\(R*T\)\/\(V - b\)\) \[PartialD]\_t
              V\  /. \ {Cv \[Rule] 28.8, \ R \[Rule] 8.3145, \
          b ->  .00004267}\)], "Input"],

Cell[BoxData[
    \(\(-13680.`\)\ Cos[
          t] - \(9.145950000000001`\ \((750 - 475\ Sin[t])\)\ \
Sin[t]\)\/\(\(\(1.29995733`\)\(\[InvisibleSpace]\)\) + 1.1`\ Cos[t]\)\)], \
"Output"]
}, Open  ]],

Cell["\<\
Cv is the heat capacity at constant volume, R is the universal gas constant,
\
b is one of the Van der Waals constants. Values given are for carbon dioxide
\
gas.
The heat absorbed by the engine during one cycle should then be:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\[Integral]\_0\%\(2*\[Pi]\)dqdt \[DifferentialD]t\)], "Input"],

Cell[BoxData[
    \(\(\(13698.118108694445`\)\(\[InvisibleSpace]\)\) +
      0.`\ \[ImaginaryI]\)], "Output"]
}, Open  ]],

Cell["\<\
It has absorbed 13.7 kiloJoules.
Now I want to see how the heat taken up depends on the process parameter t.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(qt = \[Integral]\_0\%tend dqdt \[DifferentialD]t\)], "Input"],

Cell[BoxData[
    \(\(-4864.860375412931`\) + 4667.304754213977`\ tend +
      2487.181546158651`\ \[ImaginaryI]\ Log[\(1.2886469044942739`\ \
\((\(\(0.5520155234337794`\)\(\[InvisibleSpace]\)\) + 1.`\ \
\[ExponentialE]\^\(\[ImaginaryI]\
tend\))\)\)\/\(\(\(1.`\)\(\[InvisibleSpace]\
\)\) + 1.`\ \[ExponentialE]\^\(\[ImaginaryI]\ tend\)\)] -
      2487.181546158651`\ \[ImaginaryI]\ Log[\(0.7113530955057261`\ \
\((\(\(1.8115432583844022`\)\(\[InvisibleSpace]\)\) + 1.`\ \
\[ExponentialE]\^\(\[ImaginaryI]\
tend\))\)\)\/\(\(\(1.`\)\(\[InvisibleSpace]\
\)\) + 1.`\ \[ExponentialE]\^\(\[ImaginaryI]\ tend\)\)] +
      6235.875000000001`\ \
Log[\(\(1.1817793909090908`\)\(\[InvisibleSpace]\)\) + 1.`\ Cos[tend]] -
      17629.3875`\ Sin[tend]\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Plot[qt, \ {tend, \ 0, \ 2*\[Pi]}]\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .61803
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Courier
%%IncludeFont: Courier
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.151576 0.183497 1.07653e-005 [
[.17539 .171 -3 -9 ]
[.17539 .171 3 0 ]
[.32696 .171 -3 -9 ]
[.32696 .171 3 0 ]
[.47854 .171 -3 -9 ]
[.47854 .171 3 0 ]
[.63011 .171 -3 -9 ]
[.63011 .171 3 0 ]
[.78169 .171 -3 -9 ]
[.78169 .171 3 0 ]
[.93327 .171 -3 -9 ]
[.93327 .171 3 0 ]
[.01131 .07584 -36 -4.5 ]
[.01131 .07584 0 4.5 ]
[.01131 .29115 -30 -4.5 ]
[.01131 .29115 0 4.5 ]
[.01131 .3988 -30 -4.5 ]
[.01131 .3988 0 4.5 ]
[.01131 .50646 -30 -4.5 ]
[.01131 .50646 0 4.5 ]
[.01131 .61411 -30 -4.5 ]
[.01131 .61411 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.17539 .1835 m
.17539 .18975 L
s
[(1)] .17539 .171 0 1 Mshowa
.32696 .1835 m
.32696 .18975 L
s
[(2)] .32696 .171 0 1 Mshowa
.47854 .1835 m
.47854 .18975 L
s
[(3)] .47854 .171 0 1 Mshowa
.63011 .1835 m
.63011 .18975 L
s
[(4)] .63011 .171 0 1 Mshowa
.78169 .1835 m
.78169 .18975 L
s
[(5)] .78169 .171 0 1 Mshowa
.93327 .1835 m
.93327 .18975 L
s
[(6)] .93327 .171 0 1 Mshowa
.125 Mabswid
.05412 .1835 m
.05412 .18725 L
s
.08444 .1835 m
.08444 .18725 L
s
.11476 .1835 m
.11476 .18725 L
s
.14507 .1835 m
.14507 .18725 L
s
.2057 .1835 m
.2057 .18725 L
s
.23602 .1835 m
.23602 .18725 L
s
.26633 .1835 m
.26633 .18725 L
s
.29665 .1835 m
.29665 .18725 L
s
.35728 .1835 m
.35728 .18725 L
s
.38759 .1835 m
.38759 .18725 L
s
.41791 .1835 m
.41791 .18725 L
s
.44822 .1835 m
.44822 .18725 L
s
.50885 .1835 m
.50885 .18725 L
s
.53917 .1835 m
.53917 .18725 L
s
.56948 .1835 m
.56948 .18725 L
s
.5998 .1835 m
.5998 .18725 L
s
.66043 .1835 m
.66043 .18725 L
s
.69074 .1835 m
.69074 .18725 L
s
.72106 .1835 m
.72106 .18725 L
s
.75138 .1835 m
.75138 .18725 L
s
.81201 .1835 m
.81201 .18725 L
s
.84232 .1835 m
.84232 .18725 L
s
.87264 .1835 m
.87264 .18725 L
s
.90295 .1835 m
.90295 .18725 L
s
.96358 .1835 m
.96358 .18725 L
s
.9939 .1835 m
.9939 .18725 L
s
.25 Mabswid
0 .1835 m
1 .1835 L
s
.02381 .07584 m
.03006 .07584 L
s
[(-10000)] .01131 .07584 1 0 Mshowa
.02381 .29115 m
.03006 .29115 L
s
[(10000)] .01131 .29115 1 0 Mshowa
.02381 .3988 m
.03006 .3988 L
s
[(20000)] .01131 .3988 1 0 Mshowa
.02381 .50646 m
.03006 .50646 L
s
[(30000)] .01131 .50646 1 0 Mshowa
.02381 .61411 m
.03006 .61411 L
s
[(40000)] .01131 .61411 1 0 Mshowa
.125 Mabswid
.02381 .09737 m
.02756 .09737 L
s
.02381 .11891 m
.02756 .11891 L
s
.02381 .14044 m
.02756 .14044 L
s
.02381 .16197 m
.02756 .16197 L
s
.02381 .20503 m
.02756 .20503 L
s
.02381 .22656 m
.02756 .22656 L
s
.02381 .24809 m
.02756 .24809 L
s
.02381 .26962 m
.02756 .26962 L
s
.02381 .31268 m
.02756 .31268 L
s
.02381 .33421 m
.02756 .33421 L
s
.02381 .35574 m
.02756 .35574 L
s
.02381 .37727 m
.02756 .37727 L
s
.02381 .42033 m
.02756 .42033 L
s
.02381 .44187 m
.02756 .44187 L
s
.02381 .4634 m
.02756 .4634 L
s
.02381 .48493 m
.02756 .48493 L
s
.02381 .52799 m
.02756 .52799 L
s
.02381 .54952 m
.02756 .54952 L
s
.02381 .57105 m
.02756 .57105 L
s
.02381 .59258 m
.02756 .59258 L
s
.02381 .05431 m
.02756 .05431 L
s
.02381 .03278 m
.02756 .03278 L
s
.02381 .01125 m
.02756 .01125 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
.02381 .1835 m
.06244 .14547 L
.10458 .10524 L
.14415 .07171 L
.18221 .04563 L
.20178 .03511 L
.22272 .02623 L
.23293 .02282 L
.24402 .0198 L
.25449 .0176 L
.26416 .01613 L
.26882 .01561 L
.27139 .01538 L
.27383 .01519 L
.27609 .01504 L
.27856 .01491 L
.2808 .01482 L
.28189 .01479 L
.28289 .01476 L
.28418 .01474 L
.28486 .01473 L
.28558 .01472 L
.28679 .01472 L
.28748 .01472 L
.28812 .01472 L
.28932 .01473 L
.29041 .01474 L
.29168 .01477 L
.29287 .0148 L
.29552 .01489 L
.29797 .01501 L
.30351 .01538 L
.30815 .01581 L
.31314 .01637 L
.32218 .01767 L
.34278 .02181 L
.36282 .02712 L
.38392 .03366 L
.42203 .04684 L
.44328 .05497 L
.46259 .06372 L
.47223 .06903 L
.48267 .07586 L
.48746 .07944 L
.49254 .08359 L
.49687 .08744 L
.49806 .08854 L
.49933 .08976 L
.50053 .25916 L
.50163 .26025 L
Mistroke
.54139 .31221 L
.57964 .37732 L
.62034 .44732 L
.65952 .50613 L
.67953 .53142 L
.70115 .55465 L
.72029 .57149 L
.74127 .58584 L
.75092 .59099 L
.76125 .59549 L
.76696 .59754 L
.77216 .59913 L
.77709 .6004 L
.78233 .60149 L
.78706 .60225 L
.78957 .60258 L
.79227 .60286 L
.79474 .60306 L
.79582 .60313 L
.79701 .60319 L
.79806 .60324 L
.79917 .60327 L
.80035 .6033 L
.80145 .60332 L
.80213 .60332 L
.80285 .60332 L
.80414 .6033 L
.80486 .60329 L
.80553 .60327 L
.80706 .60322 L
.80839 .60316 L
.80961 .60308 L
.81234 .60288 L
.81477 .60264 L
.81742 .60232 L
.82223 .60158 L
.83161 .59959 L
.84198 .59656 L
.86298 .5879 L
.8823 .57722 L
.90315 .56314 L
.94425 .52925 L
.97619 .4992 L
Mfstroke
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  ImageSize->{288, 177.938},
  ImageMargins->{{43, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40004P0000/A000`40O003h00Oogoo8Goo003o
OolQ
Ool00?moob5oo`00ogoo8Goo003oOolQOol00?moob5oo`00ogoo8Goo003oOolQOol002Moo`03
001o
ogoo0?Ioo`009goo00<007ooOol0mWoo000WOol00`00Oomoo`3fOol002Moo`800?Moo`009goo
00<0
07ooOol0=Woo5 at 00Zgoo000WOol00`00Oomoo`0dOol2000EOol5002VOol002Moo`03001oogoo
02mo
o`D001aoo`800:Aoo`009goo00<007ooOol0;Goo0P008goo1000X7oo000WOol2000/Ool2000Y
Ool4
002LOol002Moo`03001oogoo02Uoo`8002moo`8009Yoo`009goo00<007ooOol09goo0P00<goo
0`00
Ugoo000WOol00`00Oomoo`0UOol2000hOol2002EOol002Moo`03001oogoo02=oo`8003aoo`<0
099o
o`009goo00<007ooOol08Woo00<007ooOol0?goo0P00T7oo000WOol2000QOol20014Ool3002=
Ool0
02Moo`03001oogoo01moo`03001oogoo04Moo`8008]oo`009goo00<007ooOol07Woo00<007oo
Ool0
BWoo0`00R7oo0007Ool50002Ool20004Ool20004Ool20004Ool20005Ool00`00Oomoo`0LOol2
001@
Ool20026Ool000Uoo`05001oogooOol00002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`04
001o
ogoo0002Ool00`00Oomoo`02Ool00`00Oomoo`0KOol00`00Oomoo`1BOol00`00Oomoo`23Ool0
00Uo
o`05001oogooOol00002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`04001oogoo0002Ool0
0`00
Oomoo`02Ool2000KOol00`00Oomoo`1DOol00`00Oomoo`22Ool00003Ool00000008000Aoo`05
001o
ogooOol00002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`04001oogoo0002Ool00`00Oomo
o`02
Ool00`00Oomoo`0IOol00`00Oomoo`1FOol00`00Oomoo`21Ool000Uoo`05001oogooOol00002
Ool0
1000Oomoo`000Woo00 at 007ooOol0009oo`04001oogoo0002Ool00`00Oomoo`02Ool00`00Oomo
o`0H
Ool00`00Oomoo`1HOol20021Ool000Qoo`8000Aoo`8000Aoo`8000Aoo`8000Aoo`8000Eoo`03
001o
ogoo01Ioo`8005eoo`03001oogoo07ioo`009goo00<007ooOol05Goo00<007ooOol0GGoo00<0
07oo
Ool0OWoo000WOol00`00Oomoo`0DOol00`00Oomoo`1NOol00`00Oomoo`1nOol002Moo`8001Ao
o`03
001oogoo05moo`03001oogoo07ioo`009goo00<007ooOol04Woo00<007ooOol0H7oo00<007oo
Ool0
OWoo000WOol00`00Oomoo`0AOol00`00Oomoo`1QOol00`00Oomoo`1nOol002Moo`03001oogoo
011o
o`03001oogoo069oo`03001oogoo07ioo`009goo00<007ooOol03goo00<007ooOol0Hgoo00<0
07oo
Ool0OWoo000WOol2000?Ool00`00Oomoo`1TOol00`00Oomoo`1nOol002Moo`03001oogoo00eo
o`03
001oogoo06Eoo`03001oogoo07ioo`009goo00<007ooOol02goo0P00J7oo00<007ooOol0OWoo
000W
Ool00`00Oomoo`0:Ool00`00Oomoo`1XOol00`00Oomoo`1nOol002Moo`03001oogoo00Uoo`03
001o
ogoo06Uoo`03001oogoo07ioo`009goo0P002Goo00<007ooOol05Woo1 at 008Woo10008goo0P00
17oo
00<007ooOol07goo0`008Woo0P0097oo0`004Goo000WOol00`00Oomoo`07Ool00`00Oomoo`0I
Ool0
0`00Oomoo`0ROol00`00Oomoo`0SOol01000Oomoo`000goo00<007ooOol087oo00<007ooOol0
87oo
00 at 007ooOol002=oo`04001oogoo000@Ool002Moo`03001oogoo00Ioo`03001oogoo01Yoo`03
001o
ogoo02=oo`03001oogoo02Aoo`03001oogoo009oo`03001oogoo01eoo`D002Aoo`03001oogoo
025o
o`04001oogoo000 at Ool002Moo`03001oogoo00Eoo`03001oogoo01]oo`03001oogoo02Aoo`03
001o
ogoo02Aoo`05001oogooOol0000OOol01000Oomoo`008goo0P0097oo0`004Goo000WOol00`00
Oomo
o`04Ool00`00Oomoo`0LOol00`00Oomoo`0ROol01000Oomoo`008Woo00 at 007ooOol000=oo`03
001o
ogoo01ioo`03001oo`0002=oo`03001oogoo02Aoo`03001oogoo011oo`009goo00<007ooOol0
0goo
00<007ooOol077oo0P009Goo0P0097oo0P0017oo00<007ooOol07goo0P008goo0`0097oo0`00
47oo
000WOol20003Ool00`00Oomoo`1`Ool00`00Oomoo`1nOol002Moo`05001oogooOol0001cOol0
0`00
Oomoo`1nOol002Moo`04001oogoo001dOol00`00Oomoo`1nOol002Moo`03001oo`0007Eoo`03
001o
ogoo07ioo`009goo0P00Mgoo00<007ooOol0OGoo000QOooo000002Moo`03001oogoo00Aoo`03
001o
ogoo00Eoo`03001oogoo00Eoo`03001oogoo00Aoo`03001oogoo00Eoo`03001oogoo00Eoo`03
001o
ogoo00Eoo`03001oogoo00Aoo`03001oogoo00Eoo`03001oogoo00Eoo`03001oogoo00Aoo`03
001o
ogoo00Eoo`03001oogoo00Eoo`03001oogoo00Aoo`03001oogoo00Eoo`03001oogoo00=oo`03
001o
o`0000Moo`03001oogoo00Aoo`03001oogoo00Eoo`03001oogoo00Eoo`03001oogoo00Aoo`03
001o
ogoo00Eoo`03001oogoo00Eoo`03001oogoo00Aoo`03001oogoo00Eoo`03001oogoo00Eoo`03
001o
ogoo00Aoo`03001oogoo00Eoo`03001oogoo00Eoo`03001oogoo00Eoo`03001oogoo00Aoo`03
001o
ogoo00Eoo`40005oo`5oo`009goo00<007ooOol08goo00<007ooOol097oo00<007ooOol08goo
00<0
07ooOol00goo00<007ooOol07Woo00<007ooOol08goo00<007ooOol097oo00<007ooOol03goo
000W
Ool00`00Oomoo`1fOol00`00Oomoo`1mOol002Moo`03001oogoo07Ioo`03001oogoo07eoo`00
9goo
00<007ooOol0MWoo00<007ooOol0OGoo000WOol2001gOol00`00Oomoo`1mOol002Moo`03001o
ogoo
07Ioo`03001oogoo07eoo`009goo00<007ooOol0MWoo00<007ooOol0OGoo000WOol00`00Oomo
o`1f
Ool00`00Oomoo`1mOol002Moo`03001oogoo07Ioo`03001oogoo07eoo`009goo0P00Mgoo00<0
07oo
Ool0OGoo000WOol00`00Oomoo`1fOol00`00Oomoo`1mOol002Moo`03001oogoo07Ioo`03001o
ogoo
07eoo`009goo00<007ooOol0MWoo00<007ooOol0OGoo000WOol00`00Oomoo`1fOol00`00Oomo
o`1m
Ool002Moo`03001oogoo07Ioo`03001oogoo07eoo`009goo0P00Mgoo00<007ooOol0OGoo000W
Ool0
0`00Oomoo`1fOol00`00Oomoo`1mOol002Moo`03001oogoo07Ioo`03001oogoo07eoo`009goo
00<0
07ooOol0MWoo00<007ooOol0OGoo000WOol00`00Oomoo`1gOol00`00Oomoo`1lOol002Moo`80
07Uo
o`03001oogoo07]oo`009goo00<007ooOol0N7oo00<007ooOol0Ngoo000WOol00`00Oomoo`1i
Ool0
0`00Oomoo`1jOol002Moo`03001oogoo07Yoo`03001oogoo07Uoo`001goo1 at 000Woo0P0017oo
0P00
17oo0P0017oo0P001Goo00<007ooOol0Ngoo00<007ooOol0N7oo0009Ool01 at 00Oomoogoo0000
0Woo
00 at 007ooOol0009oo`04001oogoo0002Ool01000Oomoo`000Woo00<007ooOol00Woo00<007oo
Ool0
Ngoo00<007ooOol0N7oo0009Ool01 at 00Oomoogoo00000Woo00@007ooOol0009oo`04001oogoo
0002
Ool01000Oomoo`000Woo00<007ooOol00Woo0P00OGoo00<007ooOol0Mgoo0009Ool01 at 00Oomo
ogoo
00000Woo00 at 007ooOol0009oo`04001oogoo0002Ool01000Oomoo`000Woo00<007ooOol00Woo
00<0
07ooOol0OGoo00<007ooOol0MWoo0009Ool01 at 00Oomoogoo00000Woo00@007ooOol0009oo`04
001o
ogoo0002Ool01000Oomoo`000Woo00<007ooOol00Woo00<007ooOol0OWoo00<007ooOol0MGoo
0008
Ool20004Ool20004Ool20004Ool20004Ool20005Ool00`00Oomoo`1nOol00`00Oomoo`1eOol0
02Mo
o`03001oogoo07moo`03001oogoo07Aoo`009goo0P00PGoo00<007ooOol0Lgoo000WOol00`00
Oomo
o`21Ool00`00Oomoo`1bOol002Moo`03001oogoo085oo`03001oogoo079oo`009goo00<007oo
Ool0
PWoo00<007ooOol0LGoo000WOol00`00Oomoo`22Ool00`00Oomoo`1aOol002Moo`03001oogoo
08=o
o`03001oogoo071oo`009goo0P00QGoo00<007ooOol0Kgoo000WOol00`00Oomoo`24Ool00`00
Oomo
o`1_Ool002Moo`03001oogoo08Eoo`03001oogoo06ioo`009goo00<007ooOol0QGoo00<007oo
Ool0
KWoo000WOol00`00Oomoo`26Ool00`00Oomoo`1]Ool002Moo`8008Moo`03001oogoo06eoo`00
9goo
00<007ooOol0Qgoo00<007ooOol0K7oo000WOol00`00Oomoo`28Ool00`00Oomoo`1[Ool002Mo
o`03
001oogoo08Qoo`03001oogoo06]oo`009goo00<007ooOol0RGoo00<007ooOol0JWoo000WOol0
0`00
Oomoo`29Ool00`00Oomoo`1ZOol002Moo`8008]oo`03001oogoo06Uoo`009goo00<007ooOol0
Rgoo
00<007ooOol0J7oo000WOol00`00Oomoo`2;Ool00`00Oomoo`1XOol000Moo` at 000=oo`8000Ao
o`80
00Aoo`8000Aoo`8000Eoo`03001oogoo08aoo`03001oogoo06Moo`001goo00<007ooOol00goo
00 at 0
07ooOol0009oo`04001oogoo0002Ool01000Oomoo`000Woo00 at 007ooOol000Aoo`03001oogoo
08ao
o`03001oogoo06Moo`0027oo00<007ooOol00Woo00 at 007ooOol0009oo`04001oogoo0002Ool0
1000
Oomoo`000Woo00 at 007ooOol000Aoo`8008ioo`03001oogoo06Ioo`002Goo00D007ooOomoo`00
009o
o`04001oogoo0002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`03001oogoo009oo`03001o
ogoo
08ioo`03001oogoo06Eoo`001goo00 at 007ooOol0009oo`04001oogoo0002Ool01000Oomoo`00
0Woo
00 at 007ooOol0009oo`04001oogoo0004Ool00`00Oomoo`2>Ool00`00Oomoo`1UOol000Qoo`80
00Ao
o`8000Aoo`8000Aoo`8000Aoo`8000Eoo`03001oogoo08moo`03001oogoo06Aoo`009goo00<0
07oo
Ool0Sgoo00<007ooOol0I7oo000WOol00`00Oomoo`2 at Ool00`00Oomoo`1SOol002Moo`80095o
o`03
001oogoo06=oo`009goo00<007ooOol0TGoo00<007ooOol0HWoo000WOol00`00Oomoo`2BOol0
0`00
Oomoo`1QOol002Moo`03001oogoo099oo`03001oogoo065oo`009goo00<007ooOol0Tgoo00<0
07oo
Ool0H7oo000WOol2002DOol00`00Oomoo`1POol002Moo`03001oogoo09Aoo`03001oogoo05mo
o`00
9goo00<007ooOol0UGoo00<007ooOol0GWoo000WOol00`00Oomoo`2EOol00`00Oomoo`1NOol0
02Mo
o`03001oogoo09Ioo`03001oogoo05eoo`009goo0P00V7oo00<007ooOol0G7oo000WOol00`00
Oomo
o`2GOol00`00Oomoo`1LOol002Moo`03001oogoo09Qoo`03001oogoo05]oo`009goo00<007oo
Ool0
VGoo00<007ooOol0FWoo000WOol00`00Oomoo`2IOol00`00Oomoo`1JOol002Moo`03001oogoo
09Yo
o`03001oogoo05Uoo`009goo0P00W7oo00<007ooOol0F7oo000WOol00`00Oomoo`2KOol00`00
Oomo
o`1HOol002Moo`03001oogoo09aoo`03001oogoo05Moo`0027oo0P0017oo0P0017oo0P0017oo
0P00
17oo0P001Goo00<007ooOol0WGoo00<007ooOol0EWoo0007Ool01000Oomoo`000Woo00 at 007oo
Ool0
009oo`04001oogoo0002Ool01000Oomoo`000Woo00 at 007ooOol000Aoo`03001oogoo09eoo`03
001o
ogoo04ioo`03001oogoo00Eoo`002Goo00D007ooOomoo`00009oo`04001oogoo0002Ool01000
Oomo
o`000Woo00 at 007ooOol0009oo`03001oogoo009oo`8009moo`03001oogoo04aoo`03001oogoo
00Io
o`002Woo00 at 007ooOol0009oo`04001oogoo0002Ool01000Oomoo`000Woo00@007ooOol0009o
o`03
001oogoo009oo`03001oogoo09moo`03001oogoo04Yoo`03001oogoo00Moo`001goo00 at 007oo
Ool0
009oo`04001oogoo0002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`04001oogoo0004Ool0
0`00
Oomoo`2OOol00`00Oomoo`18Ool2000:Ool000Qoo`8000Aoo`8000Aoo`8000Aoo`8000Aoo`80
00Eo
o`03001oogoo0:1oo`03001oogoo04Ioo`03001oogoo00Yoo`009goo00<007ooOol0XGoo00<0
07oo
Ool0A7oo00<007ooOol02goo000WOol00`00Oomoo`2ROol00`00Oomoo`12Ool00`00Oomoo`0<
Ool0
02Moo`800:=oo`03001oogoo045oo`03001oogoo00eoo`009goo00<007ooOol0Xgoo00<007oo
Ool0
?goo00<007ooOol03Woo000WOol00`00Oomoo`2TOol00`00Oomoo`0mOol00`00Oomoo`0?Ool0
02Mo
o`03001oogoo0:Eoo`03001oogoo03]oo`03001oogoo011oo`009goo00<007ooOol0YWoo00<0
07oo
Ool0>7oo0P004goo000WOol2002XOol00`00Oomoo`0fOol00`00Oomoo`0COol002Moo`03001o
ogoo
0:Qoo`03001oogoo03Aoo`03001oogoo01Aoo`009goo00<007ooOol0ZGoo00<007ooOol0<Woo
00<0
07ooOol05Goo000WOol00`00Oomoo`2ZOol00`00Oomoo`0`Ool00`00Oomoo`0FOol002Moo`03
001o
ogoo0:]oo`03001oogoo02ioo`03001oogoo01Moo`009goo00<007ooOol0[7oo00<007ooOol0
:goo
0P006Woo000WOol2002^Ool00`00Oomoo`0XOol2000LOol002Moo`03001oogoo0:ioo`8002Mo
o`03
001oogoo01aoo`009goo00<007ooOol0/7oo00<007ooOol08Woo0P007goo000WOol00`00Oomo
o`2a
Ool2000POol2000QOol002Moo`03001oogoo0;=oo`03001oogoo01]oo`8002=oo`009goo0P00
]Goo
0`005goo0`009Goo000WOol00`00Oomoo`2gOol2000BOol3000XOol002Moo`03001oogoo0;Uo
o` at 0
00aoo`8002]oo`009goo00<007ooOol0_Goo3000;Goo0009Ool30002Ool20004Ool20004Ool2
0004
Ool20005Ool00`00Oomoo`3fOol000Yoo`04001oogoo0002Ool01000Oomoo`000Woo00 at 007oo
Ool0
009oo`04001oogoo0002Ool00`00Oomoo`02Ool00`00Oomoo`3fOol000Moo`D00005Ool007oo
Ool0
0002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`04001oogoo0004Ool2003gOol000Moo`04
001o
ogoo0002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`04001oogoo0002Ool01000Oomoo`00
17oo
00<007ooOol0mWoo0008Ool00`00Ool00002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`04
001o
ogoo0002Ool01000Oomoo`00oGoo0009Ool20003Ool20004Ool20004Ool20004Ool2003nOol0
0?mo
ob5oo`00ogoo8Goo003oOolQOol00?moob5oo`00ogoo8Goo003oOolQOol00?moob5oo`00ogoo
8Goo
003oOolQOol00001\
\>"],
  ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.01804, -20070.2, \
0.0259872, 365.902}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell["I don't believe my eyes!", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(qt\  /. \ tend \[Rule] 2*\[Pi]\)], "Input"],

Cell[BoxData[
    \(\(\(29325.540655806682`\)\(\[InvisibleSpace]\)\) +
      0.`\ \[ImaginaryI]\)], "Output"]
}, Open  ]],

Cell["\<\
Now it seems the engine has taken up more than 29 kiloJoules at the end!
The discontinuity that showed up at t = \[Pi] was totally unexpected \
because...\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Plot[dqdt, \ {t, 0, 2*Pi}]\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: .61803
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Courier
%%IncludeFont: Courier
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.0238095 0.151576 0.22602 1.51827e-005 [
[.17539 .21352 -3 -9 ]
[.17539 .21352 3 0 ]
[.32696 .21352 -3 -9 ]
[.32696 .21352 3 0 ]
[.47854 .21352 -3 -9 ]
[.47854 .21352 3 0 ]
[.63011 .21352 -3 -9 ]
[.63011 .21352 3 0 ]
[.78169 .21352 -3 -9 ]
[.78169 .21352 3 0 ]
[.93327 .21352 -3 -9 ]
[.93327 .21352 3 0 ]
[.01131 .07419 -36 -4.5 ]
[.01131 .07419 0 4.5 ]
[.01131 .37785 -30 -4.5 ]
[.01131 .37785 0 4.5 ]
[.01131 .52967 -30 -4.5 ]
[.01131 .52967 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.17539 .22602 m
.17539 .23227 L
s
[(1)] .17539 .21352 0 1 Mshowa
.32696 .22602 m
.32696 .23227 L
s
[(2)] .32696 .21352 0 1 Mshowa
.47854 .22602 m
.47854 .23227 L
s
[(3)] .47854 .21352 0 1 Mshowa
.63011 .22602 m
.63011 .23227 L
s
[(4)] .63011 .21352 0 1 Mshowa
.78169 .22602 m
.78169 .23227 L
s
[(5)] .78169 .21352 0 1 Mshowa
.93327 .22602 m
.93327 .23227 L
s
[(6)] .93327 .21352 0 1 Mshowa
.125 Mabswid
.05412 .22602 m
.05412 .22977 L
s
.08444 .22602 m
.08444 .22977 L
s
.11476 .22602 m
.11476 .22977 L
s
.14507 .22602 m
.14507 .22977 L
s
.2057 .22602 m
.2057 .22977 L
s
.23602 .22602 m
.23602 .22977 L
s
.26633 .22602 m
.26633 .22977 L
s
.29665 .22602 m
.29665 .22977 L
s
.35728 .22602 m
.35728 .22977 L
s
.38759 .22602 m
.38759 .22977 L
s
.41791 .22602 m
.41791 .22977 L
s
.44822 .22602 m
.44822 .22977 L
s
.50885 .22602 m
.50885 .22977 L
s
.53917 .22602 m
.53917 .22977 L
s
.56948 .22602 m
.56948 .22977 L
s
.5998 .22602 m
.5998 .22977 L
s
.66043 .22602 m
.66043 .22977 L
s
.69074 .22602 m
.69074 .22977 L
s
.72106 .22602 m
.72106 .22977 L
s
.75138 .22602 m
.75138 .22977 L
s
.81201 .22602 m
.81201 .22977 L
s
.84232 .22602 m
.84232 .22977 L
s
.87264 .22602 m
.87264 .22977 L
s
.90295 .22602 m
.90295 .22977 L
s
.96358 .22602 m
.96358 .22977 L
s
.9939 .22602 m
.9939 .22977 L
s
.25 Mabswid
0 .22602 m
1 .22602 L
s
.02381 .07419 m
.03006 .07419 L
s
[(-10000)] .01131 .07419 1 0 Mshowa
.02381 .37785 m
.03006 .37785 L
s
[(10000)] .01131 .37785 1 0 Mshowa
.02381 .52967 m
.03006 .52967 L
s
[(20000)] .01131 .52967 1 0 Mshowa
.125 Mabswid
.02381 .10456 m
.02756 .10456 L
s
.02381 .13492 m
.02756 .13492 L
s
.02381 .16529 m
.02756 .16529 L
s
.02381 .19565 m
.02756 .19565 L
s
.02381 .25639 m
.02756 .25639 L
s
.02381 .28675 m
.02756 .28675 L
s
.02381 .31712 m
.02756 .31712 L
s
.02381 .34748 m
.02756 .34748 L
s
.02381 .40821 m
.02756 .40821 L
s
.02381 .43858 m
.02756 .43858 L
s
.02381 .46894 m
.02756 .46894 L
s
.02381 .49931 m
.02756 .49931 L
s
.02381 .04383 m
.02756 .04383 L
s
.02381 .01346 m
.02756 .01346 L
s
.02381 .56004 m
.02756 .56004 L
s
.02381 .5904 m
.02756 .5904 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
.02381 .01832 m
.02846 .01711 L
.03279 .01621 L
.03754 .01546 L
.0402 .01516 L
.04262 .01495 L
.04385 .01487 L
.04516 .0148 L
.04627 .01476 L
.0475 .01473 L
.04882 .01472 L
.04957 .01472 L
.05024 .01472 L
.05156 .01475 L
.05281 .01479 L
.05397 .01485 L
.0552 .01492 L
.05742 .0151 L
.06005 .01538 L
.06244 .0157 L
.06787 .01666 L
.07287 .01781 L
.08414 .02139 L
.09408 .02563 L
.10458 .03115 L
.14255 .05931 L
.18298 .10065 L
.22188 .1472 L
.26325 .19826 L
.30309 .24283 L
.32131 .25999 L
.34142 .27567 L
.36119 .28728 L
.37203 .29198 L
.3822 .29538 L
.39203 .29787 L
.40238 .29991 L
.41208 .30165 L
.42089 .30351 L
.4258 .30485 L
.43043 .30645 L
.43533 .30858 L
.44063 .31157 L
.44527 .3149 L
.45019 .31928 L
.459 .32982 L
.46843 .34559 L
.47857 .36829 L
.49689 .42313 L
.51663 .49152 L
Mistroke
.52706 .52504 L
.53816 .55519 L
.54374 .56765 L
.54899 .57761 L
.5541 .58566 L
.55879 .59163 L
.56327 .59612 L
.56799 .59961 L
.57044 .60096 L
.57273 .60193 L
.57402 .60236 L
.57524 .60269 L
.57591 .60284 L
.57664 .60298 L
.57792 .60317 L
.57915 .60328 L
.58027 .60332 L
.58149 .6033 L
.58279 .60321 L
.58392 .60307 L
.58497 .60289 L
.58735 .60233 L
.58998 .60144 L
.59237 .60042 L
.59774 .59739 L
.60293 .59357 L
.60856 .58856 L
.61876 .57744 L
.65973 .51431 L
.69918 .43791 L
.73712 .35892 L
.77752 .27511 L
.81639 .19963 L
.85772 .12958 L
.89753 .07576 L
.91573 .05641 L
.93583 .03918 L
.95539 .02675 L
.96612 .02179 L
.97619 .01832 L
Mfstroke
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  ImageSize->{288, 177.938},
  ImageMargins->{{43, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40004P0000/A000`40O003h00Oogoo8Goo003o
OolQ
Ool00?moob5oo`00ogoo8Goo003oOolQOol00?moob5oo`00ogoo8Goo003oOolQOol00?moob5o
o`00
ogoo8Goo000WOol00`00Oomoo`3fOol002Moo`03001oogoo0?Ioo`009goo00<007ooOol0mWoo
000W
Ool20002Ool6003_Ool002Moo` at 000Eoo`D00>9oo`8000Moo`009goo00<007ooOol02goo0`00
gGoo
0P002Goo000WOol00`00Oomoo`0>Ool2003IOol2000;Ool002Moo`03001oogoo011oo`800=Io
o`03
001oogoo00]oo`009goo00<007ooOol04Woo00<007ooOol0dGoo0P003Woo000WOol00`00Oomo
o`0C
Ool2003 at Ool00`00Oomoo`0>Ool002Moo`03001oogoo01Eoo`03001oogoo0<aoo`03001oogoo
00mo
o`009goo0P005goo00<007ooOol0bWoo00<007ooOol047oo000WOol00`00Oomoo`0GOol00`00
Oomo
o`37Ool2000COol002Moo`03001oogoo01Qoo`800<Ioo`03001oogoo01=oo`009goo00<007oo
Ool0
6Woo00<007ooOol0`Woo00<007ooOol057oo000WOol00`00Oomoo`0KOol00`00Oomoo`30Ool0
0`00
Oomoo`0EOol000Moo`D0009oo`8000Aoo`8000Aoo`8000Aoo`8000Eoo`03001oogoo01aoo`03
001o
ogoo0;ioo`03001oogoo01Ioo`002Goo00D007ooOomoo`00009oo`04001oogoo0002Ool01000
Oomo
o`000Woo00 at 007ooOol0009oo`03001oogoo009oo`03001oogoo01eoo`03001oogoo0;aoo`03
001o
ogoo01Moo`002Goo00D007ooOomoo`00009oo`04001oogoo0002Ool01000Oomoo`000Woo00 at 0
07oo
Ool0009oo`03001oogoo009oo`8001moo`03001oogoo0;Yoo`03001oogoo01Qoo`0000=oo`00
0000
0P0017oo00D007ooOomoo`00009oo`04001oogoo0002Ool01000Oomoo`000Woo00 at 007ooOol0
009o
o`03001oogoo009oo`03001oogoo01moo`03001oogoo0;Qoo`03001oogoo01Uoo`002Goo00D0
07oo
Oomoo`00009oo`04001oogoo0002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`03001oogoo
009o
o`03001oogoo021oo`03001oogoo0;Ioo`03001oogoo01Yoo`0027oo0P0017oo0P0017oo0P00
17oo
0P0017oo0P001Goo00<007ooOol08Goo00<007ooOol0]7oo00<007ooOol06goo000WOol00`00
Oomo
o`0ROol00`00Oomoo`2cOol00`00Oomoo`0KOol002Moo`03001oogoo02=oo`03001oogoo0;5o
o`03
001oogoo01aoo`009goo00<007ooOol097oo00<007ooOol0[goo00<007ooOol07Goo000WOol0
0`00
Oomoo`0UOol00`00Oomoo`2]Ool00`00Oomoo`0NOol002Moo`8002Moo`03001oogoo0:aoo`03
001o
ogoo01ioo`009goo00<007ooOol09goo00<007ooOol0ZWoo00<007ooOol07goo000WOol00`00
Oomo
o`0XOol00`00Oomoo`2XOol00`00Oomoo`0POol002Moo`03001oogoo02Qoo`03001oogoo0:Mo
o`03
001oogoo025oo`009goo00<007ooOol0:Goo00<007ooOol0YWoo00<007ooOol08Goo000WOol0
0`00
Oomoo`0ZOol00`00Oomoo`2TOol00`00Oomoo`0ROol002Moo`03001oogoo02]oo`03001oogoo
0:9o
o`03001oogoo02=oo`009goo00<007ooOol0;7oo00<007ooOol0X7oo00<007ooOol097oo000W
Ool2
000^Ool00`00Oomoo`2OOol00`00Oomoo`0TOol002Moo`03001oogoo02eoo`03001oogoo09io
o`03
001oogoo02Eoo`009goo00<007ooOol0;Woo00<007ooOol0WGoo00<007ooOol09Goo000WOol0
0`00
Oomoo`0_Ool00`00Oomoo`2KOol00`00Oomoo`0VOol002Moo`03001oogoo031oo`03001oogoo
09Yo
o`03001oogoo02Ioo`009goo00<007ooOol0<Goo00<007ooOol0V7oo00<007ooOol09goo000W
Ool0
0`00Oomoo`0aOol00`00Oomoo`2HOol00`00Oomoo`0WOol002Moo`03001oogoo039oo`03001o
ogoo
09Ioo`03001oogoo02Qoo`009goo0P00=7oo00<007ooOol0U7oo00<007ooOol0:Goo000WOol0
0`00
Oomoo`0dOol00`00Oomoo`2COol00`00Oomoo`0YOol002Moo`03001oogoo03Aoo`03001oogoo
099o
o`03001oogoo02Yoo`009goo00<007ooOol0=Goo00<007ooOol0TGoo00<007ooOol0:Woo000W
Ool0
0`00Oomoo`0QOol5000 at Ool00`00Oomoo`0?Ool4000SOol2000VOol3000ROol2000:Ool00`00
Oomo
o`0GOol3000AOol002Moo`03001oogoo02=oo`03001oogoo015oo`03001oogoo00ioo`03001o
ogoo
02=oo`04001oogoo000VOol00`00Oomoo`0POol01000Oomoo`002Goo00<007ooOol05goo00 at 0
07oo
Ool0011oo`009goo00<007ooOol08goo00<007ooOol04Goo00<007ooOol03goo00<007ooOol0
97oo
00<007ooOol08Woo1 at 0097oo00<007ooOol01Woo00<007ooOol067oo00 at 007ooOol0011oo`00
9goo
0P0097oo00<007ooOol04Woo00<007ooOol03goo00<007ooOol097oo00<007ooOol08Goo00 at 0
07oo
Ool002=oo`8000Uoo`03001oogoo01Qoo`<0015oo`009goo00<007ooOol08goo00<007ooOol0
4goo
00<007ooOol037oo00 at 007ooOol0029oo`04001oogoo000TOol00`00Ool0000SOol00`00Oomo
o`07
Ool00`00Oomoo`0JOol00`00Oomoo`0 at Ool002Moo`03001oogoo029oo`8001Ioo`03001oogoo
00ao
o`8002Aoo`8002Ioo`8002=oo`<000Ioo`03001oogoo01]oo`<0011oo`009goo00<007ooOol0
>goo
00<007ooOol0Qgoo00<007ooOol0;Woo000WOol00`00Oomoo`0lOol00`00Oomoo`25Ool00`00
Oomo
o`0_Ool002Moo`03001oogoo03eoo`03001oogoo08Aoo`03001oogoo02moo`009goo00<007oo
Ool0
?Woo00<007ooOol0PWoo00<007ooOol0<7oo000WOol00`00Oomoo`0oOol00`00Oomoo`21Ool0
0`00
Oomoo`0`Ool0025oool000009goo00<007ooOol017oo00<007ooOol01Goo00<007ooOol01Goo
00<0
07ooOol017oo00<007ooOol01Goo00<007ooOol01Goo00<007ooOol01Goo00<007ooOol017oo
00<0
07ooOol017oo0P001goo00<007ooOol017oo00<007ooOol01Goo00<007ooOol01Goo00<007oo
Ool0
17oo00<007ooOol01Goo00<007ooOol01Goo00<007ooOol01Goo00<007ooOol017oo00<007oo
Ool0
1Goo00<007ooOol01Goo00<007ooOol017oo00<007ooOol01Goo00<007ooOol01Goo00<007oo
Ool0
17oo00<007ooOol01Goo00<007ooOol00Woo00 at 007ooOol000Ioo`03001oogoo00Eoo`03001o
ogoo
00Eoo`03001oogoo00Eoo`03001oogoo00Aoo`03001oogoo00Eoo`40005oo`5oo`009goo00<0
07oo
Ool08goo00<007ooOol077oo00<007ooOol01Goo00<007ooOol08goo00<007ooOol097oo00<0
07oo
Ool08goo00D007ooOomoo`00029oo`03001oogoo00moo`009goo00<007ooOol0 at goo00<007oo
Ool0
Ngoo00<007ooOol0<Woo000WOol00`00Oomoo`14Ool00`00Oomoo`1iOol00`00Oomoo`0cOol0
02Mo
o`03001oogoo04Eoo`03001oogoo07Qoo`03001oogoo03=oo`009goo00<007ooOol0AWoo00<0
07oo
Ool0MWoo00<007ooOol0=7oo000WOol00`00Oomoo`16Ool00`00Oomoo`1fOol00`00Oomoo`0d
Ool0
02Moo`8004Qoo`03001oogoo07Aoo`03001oogoo03Eoo`009goo00<007ooOol0B7oo00<007oo
Ool0
Lgoo00<007ooOol0=Goo000WOol00`00Oomoo`19Ool00`00Oomoo`1aOol00`00Oomoo`0fOol0
02Mo
o`03001oogoo04Yoo`80075oo`03001oogoo03Ioo`009goo00<007ooOol0C7oo00<007ooOol0
KGoo
00<007ooOol0=goo000WOol00`00Oomoo`1=Ool00`00Oomoo`1/Ool00`00Oomoo`0gOol002Mo
o`03
001oogoo04ioo`8006]oo`03001oogoo03Qoo`009goo0P00DGoo0P00JGoo00<007ooOol0>7oo
000W
Ool00`00Oomoo`1BOol2001VOol00`00Oomoo`0iOol002Moo`03001oogoo05Aoo`<006=oo`03
001o
ogoo03Uoo`009goo00<007ooOol0Egoo1 at 00GGoo00<007ooOol0>Woo000WOol00`00Oomoo`1L
Ool4
001IOol00`00Oomoo`0jOol002Moo`03001oogoo061oo`D005=oo`03001oogoo03]oo`009goo
00<0
07ooOol0IGoo00<007ooOol0D7oo00<007ooOol0>goo000WOol00`00Oomoo`1VOol00`00Oomo
o`1>
Ool00`00Oomoo`0lOol002Moo`8006Qoo`8004ioo`03001oogoo03aoo`009goo00<007ooOol0
JGoo
00<007ooOol0Bgoo00<007ooOol0?7oo000WOol00`00Oomoo`1ZOol00`00Oomoo`19Ool00`00
Oomo
o`0mOol002Moo`03001oogoo06]oo`03001oogoo04Qoo`03001oogoo03eoo`009goo00<007oo
Ool0
K7oo00<007ooOol0AWoo00<007ooOol0?Woo000WOol00`00Oomoo`1]Ool00`00Oomoo`15Ool0
0`00
Oomoo`0nOol002Moo`03001oogoo06eoo`03001oogoo04Aoo`03001oogoo03moo`009goo00<0
07oo
Ool0KWoo00<007ooOol0 at goo00<007ooOol0?goo000WOol2001_Ool00`00Oomoo`12Ool00`00
Oomo
o`10Ool002Moo`03001oogoo06moo`03001oogoo045oo`03001oogoo041oo`009goo00<007oo
Ool0
Kgoo00<007ooOol0 at 7oo00<007ooOol0 at Goo000WOol00`00Oomoo`1_Ool00`00Oomoo`10Ool0
0`00
Oomoo`11Ool002Moo`03001oogoo071oo`03001oogoo03moo`03001oogoo045oo`009goo00<0
07oo
Ool0L7oo00<007ooOol0?Woo00<007ooOol0 at Woo0007Ool50002Ool20004Ool20004Ool20004
Ool2
0005Ool00`00Oomoo`1`Ool00`00Oomoo`0nOol00`00Oomoo`12Ool000Uoo`05001oogooOol0
0002
Ool01000Oomoo`000Woo00 at 007ooOol0009oo`04001oogoo0002Ool00`00Oomoo`02Ool00`00
Oomo
o`1aOol00`00Oomoo`0lOol00`00Oomoo`13Ool000Uoo`05001oogooOol00002Ool01000Oomo
o`00
0Woo00 at 007ooOol0009oo`04001oogoo0002Ool00`00Oomoo`02Ool2001bOol00`00Oomoo`0l
Ool0
0`00Oomoo`13Ool000Uoo`05001oogooOol00002Ool01000Oomoo`000Woo00 at 007ooOol0009o
o`04
001oogoo0002Ool00`00Oomoo`02Ool00`00Oomoo`1aOol00`00Oomoo`0kOol00`00Oomoo`14
Ool0
00Uoo`05001oogooOol00002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`04001oogoo0002
Ool0
0`00Oomoo`02Ool00`00Oomoo`1bOol00`00Oomoo`0jOol00`00Oomoo`14Ool000Qoo`8000Ao
o`80
00Aoo`8000Aoo`8000Aoo`8000Eoo`03001oogoo079oo`03001oogoo03Uoo`03001oogoo04Eo
o`00
9goo00<007ooOol0Lgoo00<007ooOol0>7oo00<007ooOol0AGoo000WOol00`00Oomoo`1cOol0
0`00
Oomoo`0gOol00`00Oomoo`16Ool002Moo`03001oogoo07=oo`03001oogoo03Moo`03001oogoo
04Io
o`009goo0P00MGoo00<007ooOol0=Goo00<007ooOol0Agoo000WOol00`00Oomoo`1dOol00`00
Oomo
o`0eOol00`00Oomoo`17Ool002Moo`03001oogoo07Aoo`03001oogoo03Aoo`03001oogoo04Qo
o`00
9goo00<007ooOol0MGoo00<007ooOol0<goo00<007ooOol0B7oo000WOol00`00Oomoo`1eOol0
0`00
Oomoo`0bOol00`00Oomoo`19Ool002Moo`03001oogoo07Eoo`03001oogoo039oo`03001oogoo
04Uo
o`009goo00<007ooOol0MWoo00<007ooOol0<7oo00<007ooOol0BWoo000WOol00`00Oomoo`1f
Ool0
0`00Oomoo`0`Ool00`00Oomoo`1:Ool002Moo`8007Moo`03001oogoo02moo`03001oogoo04]o
o`00
9goo00<007ooOol0MWoo00<007ooOol0;Woo00<007ooOol0C7oo000WOol00`00Oomoo`1gOol0
0`00
Oomoo`0]Ool00`00Oomoo`1<Ool002Moo`03001oogoo07Moo`03001oogoo02aoo`03001oogoo
04eo
o`009goo00<007ooOol0Mgoo00<007ooOol0;7oo00<007ooOol0CGoo000WOol00`00Oomoo`1h
Ool0
0`00Oomoo`0ZOol00`00Oomoo`1>Ool002Moo`03001oogoo07Qoo`03001oogoo02Yoo`03001o
ogoo
04ioo`009goo00<007ooOol0N7oo00<007ooOol0:Goo00<007ooOol0Cgoo000WOol2001jOol0
0`00
Oomoo`0XOol00`00Oomoo`1?Ool002Moo`03001oogoo07Uoo`03001oogoo02Moo`03001oogoo
051o
o`009goo00<007ooOol0NGoo00<007ooOol09goo00<007ooOol0D7oo000WOol00`00Oomoo`1i
Ool0
0`00Oomoo`0VOol00`00Oomoo`1AOol002Moo`03001oogoo07Yoo`03001oogoo02Eoo`03001o
ogoo
055oo`009goo00<007ooOol0NWoo00<007ooOol097oo00<007ooOol0DWoo000WOol00`00Oomo
o`1j
Ool00`00Oomoo`0TOol00`00Oomoo`1BOol002Moo`8007]oo`03001oogoo02=oo`03001oogoo
05=o
o`009goo00<007ooOol0Ngoo00<007ooOol08Woo00<007ooOol0Dgoo000WOol00`00Oomoo`1k
Ool0
0`00Oomoo`0QOol00`00Oomoo`1DOol002Moo`03001oogoo07]oo`03001oogoo025oo`03001o
ogoo
05Aoo`009goo00<007ooOol0Ngoo00<007ooOol087oo00<007ooOol0EGoo000WOol00`00Oomo
o`1l
Ool00`00Oomoo`0NOol00`00Oomoo`1FOol000Moo` at 000=oo`8000Aoo`8000Aoo`8000Aoo`80
00Eo
o`03001oogoo07aoo`03001oogoo01ioo`03001oogoo05Ioo`001goo00<007ooOol00goo00 at 0
07oo
Ool0009oo`04001oogoo0002Ool01000Oomoo`000Woo00 at 007ooOol000Aoo`03001oogoo07ao
o`03
001oogoo01eoo`03001oogoo05Moo`0027oo00<007ooOol00Woo00 at 007ooOol0009oo`04001o
ogoo
0002Ool01000Oomoo`000Woo00 at 007ooOol000Aoo`8007eoo`03001oogoo01eoo`03001oogoo
05Mo
o`002Goo00D007ooOomoo`00009oo`04001oogoo0002Ool01000Oomoo`000Woo00 at 007ooOol0
009o
o`03001oogoo009oo`03001oogoo07eoo`03001oogoo01]oo`03001oogoo05Qoo`001goo00 at 0
07oo
Ool0009oo`04001oogoo0002Ool01000Oomoo`000Woo00 at 007ooOol0009oo`04001oogoo0004
Ool0
0`00Oomoo`1mOol00`00Oomoo`0JOol00`00Oomoo`1IOol000Qoo`8000Aoo`8000Aoo`8000Ao
o`80
00Aoo`8000Eoo`03001oogoo07ioo`03001oogoo01Uoo`03001oogoo05Uoo`009goo00<007oo
Ool0
OWoo00<007ooOol067oo00<007ooOol0FWoo000WOol00`00Oomoo`1nOol00`00Oomoo`0GOol0
0`00
Oomoo`1KOol002Moo`03001oogoo07moo`03001oogoo01Ioo`03001oogoo05]oo`009goo00<0
07oo
Ool0Ogoo00<007ooOol05Goo00<007ooOol0G7oo000WOol20021Ool00`00Oomoo`0DOol00`00
Oomo
o`1LOol002Moo`03001oogoo081oo`03001oogoo01=oo`03001oogoo05eoo`009goo00<007oo
Ool0
PGoo00<007ooOol04Goo00<007ooOol0GWoo000WOol00`00Oomoo`22Ool00`00Oomoo`0 at Ool0
0`00
Oomoo`1NOol002Moo`03001oogoo089oo`03001oogoo00moo`03001oogoo05moo`009goo00<0
07oo
Ool0Pgoo00<007ooOol03Goo00<007ooOol0H7oo000WOol00`00Oomoo`23Ool00`00Oomoo`0<
Ool0
0`00Oomoo`1QOol002Moo`8008Eoo`03001oogoo00Yoo`03001oogoo069oo`009goo00<007oo
Ool0
Q7oo00<007ooOol027oo0P00IGoo000WOol00`00Oomoo`25Ool20007Ool00`00Oomoo`1UOol0
02Mo
o`03001oogoo08Moo`8000=oo`8006Qoo`009goo00<007ooOol0R7oo1000JWoo000WOol00`00
Oomo
o`3fOol002Moo`03001oogoo0?Ioo`009goo00<007ooOol0mWoo003oOolQOol00?moob5oo`00
ogoo
8Goo003oOolQOol00?moob5oo`00ogoo8Goo003oOolQOol00?moob5oo`00ogoo8Goo003oOolQ
Ool0
0001\
\>"],
  ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.01804, -17486.2, \
0.0259872, 259.443}}],

Cell[BoxData[
    TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
      False,
      Editable->False]], "Output"]
}, Open  ]],

Cell[TextData[{
  "...the function to be integrated had no poles or other
discontinuities.\nI \
told my students that ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " had determined a faulty indefinite integral and that they should use \
NIntegrate to obtain correct results. Only some months later I understood
the \
real reason:\n\nAlthough the function to be integrated is continuous, ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " has determined an indefinite integral that contains (unexpected, to me)
\
discontinuities. This indefinite integral is allright, ",
  StyleBox["but you should be careful with definite integration across the \
discontinuities",
    FontSlant->"Italic"],
  ". I think the mechanism of \"GenerateConditions\" should be used to warn
\
users, because they can unexpectedly get results that may seem (and indeed,
\
are) contradictory.\n\nWe like it when students can experiment freely with \
the program and understand its results. In this case, they could not, and it
\
took me some time to find out what's wrong.\nThe useability of the program \
would increase if it gave users a clear warning in cases like these. The \
\"GenerateConditions\" mechanism seems designed to do so. And so it should.
"
}], "Text"]
}, Closed]]
},
FrontEndVersion->"4.0 for Microsoft Windows",
ScreenRectangle->{{0, 800}, {0, 527}},
CellGrouping->Manual,
WindowSize->{626, 369},
WindowMargins->{{0, Automatic}, {Automatic, 0}}
]


(***********************************************************************
Cached data follows.  If you edit this Notebook file directly, not using
Mathematica, you must remove the line containing CacheID at the top of
the file.  The cache data will then be recreated when you save this file
from within Mathematica.
***********************************************************************)

(*CellTagsOutline
CellTagsIndex->{
  "Example"->{
    Cell[2737, 79, 47, 1, 33, "Text",
      CellTags->"Example"]}
  }
*)

(*CellTagsIndex
CellTagsIndex->{
  {"Example", 56729, 1796}
  }
*)

(*NotebookFileOutline
Notebook[{
Cell[1717, 49, 52, 0, 150, "Title"],
Cell[1772, 51, 378, 9, 71, "Text"],

Cell[CellGroupData[{
Cell[2175, 64, 559, 13, 90, "Text"],
Cell[2737, 79, 47, 1, 33, "Text",
  CellTags->"Example"],

Cell[CellGroupData[{
Cell[2809, 84, 81, 1, 42, "Input"],
Cell[2893, 87, 47, 1, 42, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[2977, 93, 49, 1, 30, "Input"],
Cell[3029, 96, 40, 1, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[3106, 102, 85, 1, 42, "Input"],
Cell[3194, 105, 90, 2, 29, "Output"]
}, Open  ]],
Cell[3299, 110, 151, 3, 33, "Text"]
}, Closed]],
Cell[3465, 116, 116, 3, 30, "Text"],

Cell[CellGroupData[{
Cell[3606, 123, 125, 3, 42, "Input"],
Cell[3734, 128, 57, 1, 42, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[3828, 134, 134, 3, 42, "Input"],
Cell[3965, 139, 68, 1, 42, "Output"]
}, Open  ]],
Cell[4048, 143, 155, 4, 52, "Text"],

Cell[CellGroupData[{
Cell[4228, 151, 39, 1, 30, "Input"],
Cell[4270, 154, 40, 1, 29, "Output"]
}, Open  ]],
Cell[4325, 158, 673, 17, 169, "Text"],

Cell[CellGroupData[{
Cell[5023, 179, 50, 0, 47, "Subsection"],
Cell[5076, 181, 334, 6, 71, "Text"],

Cell[CellGroupData[{
Cell[5435, 191, 82, 2, 30, "Input"],
Cell[5520, 195, 51, 1, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[5608, 201, 82, 2, 30, "Input"],
Cell[5693, 205, 82, 1, 29, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[5812, 211, 95, 2, 30, "Input"],
Cell[5910, 215, 14916, 474, 186, 5087, 348, "GraphicsData", "PostScript", \
"Graphics"],
Cell[20829, 691, 130, 3, 29, "Output"]
}, Open  ]],
Cell[20974, 697, 177, 3, 52, "Text"],

Cell[CellGroupData[{
Cell[21176, 704, 199, 4, 42, "Input"],
Cell[21378, 710, 188, 4, 44, "Output"]
}, Open  ]],
Cell[21581, 717, 251, 5, 71, "Text"],

Cell[CellGroupData[{
Cell[21857, 726, 82, 1, 42, "Input"],
Cell[21942, 729, 110, 2, 29, "Output"]
}, Open  ]],
Cell[22067, 734, 132, 3, 52, "Text"],

Cell[CellGroupData[{
Cell[22224, 741, 81, 1, 42, "Input"],
Cell[22308, 744, 756, 12, 113, "Output"]
}, Open  ]],

Cell[CellGroupData[{
Cell[23101, 761, 67, 1, 30, "Input"],
Cell[23171, 764, 15287, 489, 186, 5068, 358, "GraphicsData", "PostScript", \
"Graphics"],
Cell[38461, 1255, 130, 3, 29, "Output"]
}, Open  ]],
Cell[38606, 1261, 40, 0, 33, "Text"],

Cell[CellGroupData[{
Cell[38671, 1265, 63, 1, 30, "Input"],
Cell[38737, 1268, 110, 2, 29, "Output"]
}, Open  ]],
Cell[38862, 1273, 178, 4, 52, "Text"],

Cell[CellGroupData[{
Cell[39065, 1281, 59, 1, 30, "Input"],
Cell[39127, 1284, 15553, 462, 186, 4617, 322, "GraphicsData", "PostScript",
\
"Graphics"],
Cell[54683, 1748, 130, 3, 29, "Output"]
}, Open  ]],
Cell[54828, 1754, 1263, 22, 261, "Text"]
}, Closed]]
}
]
*)




(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)




  • Prev by Date: Re: Mathematica and Apple Events (Q:)
  • Previous by thread: Re: Mathematica and Apple Events (Q:)