Re: False result with Integrate ?

*To*: mathgroup at smc.vnet.net*Subject*: [mg8756] Re: [mg8711] False result with Integrate ?*From*: seanross at worldnet.att.net*Date*: Sat, 20 Sep 1997 22:28:21 -0400*Sender*: owner-wri-mathgroup at wolfram.com

> Gilles BARBIER wrote: > > > > Help ! > > > > Why : Integrate[ Sqrt[(x-y)^2],{x,0,1}],{y,0,1}] > > gives 0 with math2.2 or math3.0. > > > > The exact result is 1/3 !! > > > > Gilles > > EDF/DER > > Are you sure? Sqrt[(x-y)^2]=x-y and > Integrate[x,{x,0,1}]-Integrate[y,{y,0,1}]=0. Gilles BARBIER wrote: > > No, Sqrt[(x-y)^2]=Abs[x-y] > > Moreover, I do not understand how the integral of an > always positive function can be 0. > > In mathemetics, we can then demonstrate that, in this case, > the function has to be null "nearly everywhere". This is > obviously not the case here. > > Gilles. > EDF/DER. > Upon looking at the problem further, I don't think that the symbolic integrator is smart enough to split up the limits. In order to symbolically do this integral correctly, Integrate[Sqrt[(x-y)^2],{x,0,1},{y,0,1}]= Integrate[Sqrt[(x-y)^2],{x,y,1},{y,0,1}]- Integrate[Sqrt[(x-y)^2],{x,0,y},{y,0,1}]. That is a pretty subtle trick for a symbolic processor to do. Apparently, Mma isn't up to the task. Another side note to this is that Sqrt[] is a multivalued function being both positive and negative, hence any integral of Sqrt of anything is zero. The symbolic processor was not up to the task of integrating Abs[Sqrt[]]. By the way, NIntegrate[Sqrt[(x-y)^2],{x,0,1},{y,0,1}] returns 0.3333333.