Re: question about Integrate

• To: mathgroup at smc.vnet.net
• Subject: [mg49261] Re: question about Integrate
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Sat, 10 Jul 2004 02:48:32 -0400 (EDT)
• Organization: The University of Western Australia
• References: <cclelo\$k9m\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <cclelo\$k9m\$1 at smc.vnet.net>,
"Florian Jaccard" <florian.jaccard at eiaj.ch> wrote:

> 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...

Indeed this bug was recently reported on MathGroup (and to WRI). The
StruveL should not be there. The correct answer is

2 Pi BesselI[1,1]

> Now, what should I tell my students ? To never believe Integrate if not
> checked with NIntegrate ?

That is _always_ good advice.

Cheers,
Paul

--
Paul Abbott                                   Phone: +61 8 9380 2734
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```

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