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Topic: volume integration
Replies: 0

 Christian Boehm Posts: 2 Registered: 12/12/04
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 k-dimensional 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 unit-hypercube [0...1]^k is randomly selected.
I want to determine the volume of the hypersphere with radius r around
this point, lying inside the unit-hypercube 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 (1-V_{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

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Christian Boehm phone ++49/89/2178-2228
University of Munich fax ++49/89/2178-2192

boehm@dbs.informatik.uni-muenchen.de