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MathGroup Archive 2004

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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
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
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Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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