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
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" When you do a \"definite\" integration with symbolic limits (instead of
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concrete numbers), the result you get may not be unconditionally true. The \
necessary conditions are not stated by ",
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". The ",
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integral, you will often not get the right answer for a definit integral if
\
you just subtract values of the limits at each point. ",
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supply the limits in numerical form. (Open to see an example)"
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The last answer is correct. It is not equal to -1 because the two limits are
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Now I want to determine the value of the definite integral as a function of
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In a thermodynamics course for chemistry students, I asked them to calculate \
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With the help of some theory, students could deduce for themselves that the
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Cv is the heat capacity at constant volume, R is the universal gas constant,
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It has absorbed 13.7 kiloJoules.
Now I want to see how the heat taken up depends on the process parameter t.\
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"...the function to be integrated had no poles or other
discontinuities.\nI \
told my students that ",
StyleBox["Mathematica",
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" 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, ",
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" 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",
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". 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.
"
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*)
(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)