What is the relation between the gaussian integral and the volume of the n-ball?

Question

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?

Answer 1

$$ \pi^{n/2} = \int \exp \left( -\sum_{i=1}^n x_i^2 \right) \mathrm d V = \int_0^\infty \exp \left( -r^2 \right) r^{n-1} \mathrm d r \int \mathrm d \Omega. $$ 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 of $n-1$ 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 $$ \int_{r<R} \mathrm d V = \int_0^R r^{n-1} \mathrm d r \int \mathrm d \Omega. $$