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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.
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