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