Volume of n-sphere? What is wrong with my version?

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Steve
Steve 2013-11-11
编辑： the cyclist 2013-11-11
Hello Experts,
I have to calculate the volume of 10-dim sphere.
Here what I did:
N = 100000;
RandVars = -1 + 2*rand(10,N);
V = zeros(N,1);
for i = 1:N
if (sum(RandVars(:,i).^2) < 1)
V(i) = 1;
end
end
% Mean values using the Monte-Carlo method:
M = (1/N)*sum(V,1);
% Error estimation:
C = V - M*ones(N,1);
Error = sqrt((1/(N-1))*sum(C.^2,2));
Please have a look what is wrong as I get M = 0.002. Thanks a lot in advance.

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the cyclist
the cyclist 2013-11-11
编辑：the cyclist 2013-11-11
You forgot to take into account that your bounding hypercube has volume 2^N, not 1, where N is the number of dimensions. (N=10 in your specific case.)

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