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Re: question about Integrate
*To*: mathgroup at smc.vnet.net
*Subject*: [mg49269] Re: [mg49241] question about Integrate
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Sat, 10 Jul 2004 02:48:39 -0400 (EDT)
*References*: <200407090626.CAA20492@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
On 9 Jul 2004, at 15:26, Florian Jaccard wrote:
> *This message was transferred with a trial version of CommuniGate(tm)
> Pro*
> Dear specialists,
>
> I asked my students to compute the area of the following closed curve :
>
> x[t_] := Sin[t]; y[t_] := E^Cos[t];
>
> ParametricPlot[{x[t], y[t]},
> {t, 0, 2*Pi}];
>
> I expected the following input :
>
> 2*NIntegrate[x[t]* Derivative[1][y][t],{t, Pi, 0}]
>
> which gives the following output :
>
> 3.5509993784243483
>
> (They also could have done it like this :
> 2*NIntegrate[y[t]*Derivative[1][x][t],{t, 0, Pi}]
> and it gives the same output)
>
> As the help browser says "N[Integrate[ . ]] calls NIntegrate for
> integrals
> that cannot be done symbolically" , I never told my students to avoid
> that
> way :
>
> 2*N[Integrate[x[t]*Derivative[1][y][t],{t, Pi, 0}]]
>
> But, surprise, it gives an other answer :
>
> 2.1262
>
> Checking carefully, I could see that NIntegrate did it fine, but
> N[Integrate...] is wrong. It seems that the special functions BesselI
> and
> StruveL (used by Mathematica if you type Integrate[...]) are making the
> mistake...
>
>
> Now, what should I tell my students ? To never believe Integrate if not
> checked with NIntegrate ?
> Must I recognize that there is a bug in Integrate, or what did I
> understand
> wrong ?
>
> Thanks for your help !
>
> Florian JACCARD
>
>
>
It seems you have just encountered the same bug in Integrate in
Mathematica 5.0 that two other person pointed out in recent postings.
In Matheamtica 4.2 the answer given by Integrate is correct.
What should you tell your students? Well, a bug is a bug, and they
should already know that computer programs have bugs and that this is
not likely to change anytime soon. Also, I think it is worth telling
them that NIntegrate is more reliable than Integrate and explaining
why. It is very instructive to understand why there are so many
problems with Integrate and not with NIntegrate - the reason is not
that Integrate has been programmed by less careful programmers. Finally
there is the question, of how it is possible that bugs that were not
present in version 4 now appear in version 5. Many people seem to find
this hard to accept, but it is indeed an unfortunate side effect of
many improvements which have been made in Integrate in version 5. It
is instructive to understand how this can come about. You can try to do
it yourself: unprotected Integrate and try to re-define it to avoid the
above error for the largest class of integrals of this kind you can
think of. It is very easy to do this using Mathematica's pattern
matching but there is a high chance that you will not notice that
something matches your pattern that shouldn't and suddenly you have
fixed one bug but created another. Then there is another way in which
improvements can make things (temporarily) worse. I don't know how many
people still remember that before Mathematica 5, this group used to
receive many postings complaining that Integrate could not manage some
particular not very hard integral. Someone would then point out that if
you applied Simplify or FullSimplify to the integrand first, Integrate
would then be able to give an answer. I think, there have been no such
messages since version 5. Obviously Integrate is now making much better
use of Simplify or FullSimplify. But, probably, that also means that
any bugs in these functions are being passed on to Integrate, which was
not the case in version 4. In fact I am only speculating that this
might be happening, but I think it illustrates quite convincingly how
making things better in general can make them worse in some particular
cases.
Andrzej Kozlowski
Chiba, Japan
http://www.mimuw.edu.pl/~akoz/
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