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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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