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MathGroup Archive 2004

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Re: question about Integrate

  • 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>
  • Reply-to: kuska at informatik.uni-leipzig.de
  • 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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