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volume integration
Posted:
Jul 11, 1996 4:18 AM


Hello,
in advanced integral calculus lectures, students are tought how to derive the formula for the (hyper)volume of a kdimensional hyper sphere by solving the integral
r r 2 2 2 / / / 1 if a + .... + a < r    1 k V(r) =  ....  < da .... da    1 k / / \ 0 else 0 0
The solution is, of course, V(r) = \pi^(k/2) / GAMMA (k/2+1) * r^k
Now, I have a related problem: I do not want to compute the whole hypervolume, but the volume of the hypersphere, cut with a hypercube.
Thus, a point from the unithypercube [0...1]^k is randomly selected. I want to determine the volume of the hypersphere with radius r around this point, lying inside the unithypercube on average. I am not interested in the value of V_avg(r) for a given choice of the random point, but in the overall average.
This approach can be used to model a probability distribution function for the distance between two randomly chosen points.
I think, a proper integral for this problem would be:
r r 2 2 2 / / / 1 if (a b ) + .. + (a b ) < r    1 1 k k V (r) =  ..  < da db .. da db avg    1 1 k k / / \ 0 else 0 0
Is it possible to solve this integral symbolically and to give a simple formula like the formula for V(r). I would assume that it is a piecewise defined polynom in terms of the radius r with degree 2k but this is only my feeling.
Maybe some math program is able to produce the solution. Maple V did not. I also have tried to solve the integral numerically (with montecarlo integration) but the result is not exact enough since I have to evaluate d/dr (1V_{avg}(r))^N, N being a large number, maybe 1,000,000.
Maybe I have only to have a look into the right book. Which one?
Thank you for any hint and best regards
Christian
 Christian Boehm phone ++49/89/21782228 University of Munich fax ++49/89/21782192
boehm@dbs.informatik.unimuenchen.de



