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MathGroup Archive 2004

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question about Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49241] question about Integrate
  • From: "Florian Jaccard" <florian.jaccard at eiaj.ch>
  • Date: Fri, 9 Jul 2004 02:26:11 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

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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