Even if I've red other threads treating this question, it's still obscure to me what deeply relates the multiple Guassian integral $\int e^{-x^2} = \sqrt \pi$ and the area of a $n$-ball.

Someone could offer an insight?


$$ \pi^{n/2} = \int \exp \left( -\sum_{i=1}^n x_i^2 \right) \mathrm d V = \int_0^\infty \exp \left( -r^2 \right) r^{n-1} \mathrm d r \int \mathrm d \Omega. $$ The second equality is change of cartesian coordinates into spherical and application of Fubini theorem. $\int \mathrm d \Omega$, which is also the surface of a sphere in $n$ dimensions of unitary radius, can be computed from this relation. This is because $\int_0^\infty \exp \left( -r^2 \right) r^{n-1} \mathrm d r = \frac 12\Gamma \left(\frac n 2 \right)$ and we know now to deal with Gamma function.

Knowing how to compute $\int\mathrm d \Omega$ you can compute the volume easily. Once again using spherical coordinates and Fubini theorem $$ \int_{r<R} \mathrm d V = \int_0^R r^{n-1} \mathrm d r \int \mathrm d \Omega. $$

  • $\begingroup$ Thank you. Just a question: $\int \mathrm d \Omega$ gives the surface area of a $(n-1)-$sphere of unitary $r$? $\endgroup$ – Lo Scrondo Mar 5 '16 at 13:40
  • $\begingroup$ @LoScrondo Yes, you are right. I'll edit the answer to fix this. $\endgroup$ – Korf Mar 5 '16 at 13:43

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