It is easier to take the derivative, and consider the volume of the n-1-sphere (i.e., the "surface area" of the boundary of the ball).
Start with the integral of e-x_1^2 $\int_{\mathbb{R}^n} e^{-x_1^2 - ... - x_n^2 over Rn} dx_1 \dots dx_n$. Fubini's theorem lets you decompose this into a product of 1-dimensional integrals, and you get pin/2. $\pi^{n/2}$. Since the integrand is spherically symmetric, you can change to an integral of the volume of integral $\int_0^\infty vol(S^{n-1}(r)) e^{-r^2} dr$, where $S^{n-1}(r)$ is the unit n-1-sphere of radius rtimes e-r^2 dr. Solving for The volume of this sphere is $r^{n-1}$ times the volume of the unit sphere, so solving for that, you get a power of pi divided by the integral of rn-1 e-r^2 dr from 0 to infinity$\frac{\pi^{n/2}}{\int_0^\infty r^{n-1} e^{-r^2} dr}$. A change of coordinates (u = r2) in the denominator yields the integral defining Gamma(n/2) .$\Gamma(n/2)$.
