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From: Peter Pein <petsie@dordos.net>
Newsgroups: comp.soft-sys.math.maple
Subject: Re: Integration over a complicated region in Maple?
Date: Sun, 22 May 2011 23:53:00 +0200
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Am 22.05.2011 21:01, schrieb Ray Vickson:
> In a recent positing in "sci.math" the following Mathematica command
> was given as the solution to a problem:
> Timing[12 Integrate[If[x1 > x2 && y1 > y2 && y2 > y3 &&
>   x2 + y2 < x3 + y3 && x3 + y3 < x1 + y1 && x3 > x1, 8, 0],
>     {x1, 0, 1}, {y1, 0, 1 - x1}, {x2, 0, 1}, {y2, 0, 1 - x2},
>     {x3, 0, 1}, {y3, 0, 1 - x3}]]
> 
> {77.766, 1/5}
> 
> So, the integration is that of the characteristic function of the set
> {(x1,y1,x2,y2,x3,y3): x1 > x2, y1 > y2, y2 > y3, x2 + y2 < x3 + y3, x3
> + y3 < x1 + y1, x3 > x1}, and Mathematical got the answer as 1/5.
> 
> Can this also be done in Maple? So far, I have not managed to do it.
> 
> (For the origin of the problem, see the "sci.math" thread with the
> delightful title "Non-transitive vampire breakfast cereal
> probabilities".)
> 
> RGV
obviously Maple fails:
eval(convert("12 Integrate[If[x1 > x2 && y1 > y2 && y2 > y3 &&
>   x2 + y2 < x3 + y3 && x3 + y3 < x1 + y1 && x3 > x1, 8, 0],
>     {x1, 0, 1}, {y1, 0, 1 - x1}, {x2, 0, 1}, {y2, 0, 1 - x2},
>     {x3, 0, 1}, {y3, 0, 1 - x3}]",FromMma));
gives an unevaluated nested integral, while Mma gives on my machine:

In[1]:= Timing[12
Integrate[If[x1>x2&&y1>y2&&y2>y3&&x2+y2<x3+y3&&x3+y3<x1+y1&&x3>x1,8,0],{x1,0,1},{y1,0,1-x1},{x2,0,1},{y2,0,1-x2},{x3,0,1},{y3,0,1-x3}]]
Out[1]= {79.217,1/5}