• To: mathgroup at smc.vnet.net
• Subject: [mg49262] Re: question about Integrate
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Sat, 10 Jul 2004 02:48:33 -0400 (EDT)
• Organization: Universitaet Leipzig
• References: <cclelo\$k9m\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,

a) you should always tell your students
"Don't use a CAS for a symbolic computation
that you can't do by yourself."
b) this is a bug in Integrate[] because

In[]:=2*Integrate[x[t]*Derivative[1][y][t], {t, Pi, 0}]
Out[]=2 Pi (BesselI[1, 1] - StruveL[1, 1])

and the StruveL[1,1] term is false

Regards
Jens

Florian Jaccard wrote:
>
> 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

```

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