It is wellknown that the volume of the unit ball in nspace is $\pi^{n/2}\\!/\Gamma(n/2+1)$. Do you know of a proof which explains this formula? Any proof which does not treat the cases n even and n odd separately (like using an explicit expression for $\Gamma(n/2+1)$ in terms of factorials) should be fine.
It is easier to take the derivative, and consider the volume of the n1sphere (i.e., the "surface area" of the boundary of the ball). Start with the integral $\int\_{\mathbb{R}^n} e^{x\_1^2  ...  x\_n^2} dx\_1 \dots dx\_n$. Fubini's theorem lets you decompose this into a product of 1dimensional integrals, and you get $\pi^{n/2}$. Since the integrand is spherically symmetric, you can change to the integral $\int\_0^\infty vol(S^{n1}(r)) e^{r^2} dr$, where $S^{n1}(r)$ is the unit n1sphere of radius r. The volume of this sphere is $r^{n1}$ times the volume of the unit sphere, so solving for that, you get $\frac{\pi^{n/2}}{\int_0^\infty r^{n1} e^{r^2} dr}$. A change of coordinates (u = r^{2}) in the denominator yields the integral defining $\Gamma(n/2)$. 


I like to write it as $\omega_n = \frac{\pi^\frac{n}{2}}{\frac{n}{2}!}$ (I've seen $\omega_n$ used both for the measure of the unit ball and for that of its boundary, but eh, I had to attach some name to it for below). I guess using the factorial notation for nonintegers isn't too popular, though. Alternatively, induction. It's true for $n=1$ (since $\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$) and $n=2$. So then: $\omega_{n+2} = \int_{x_1^2 + \dots + x_{n+2}^2 \leq 1}dx = \int_{x_{n+1}^2+x_{n+2}^2 \leq 1}\int_{x_1^2 + \dots + x_n^2 \leq 1  (x_{n+1}^2+x_{n+2}^2)}d(x_1,\dots,x_n)d(x_1,x_2).$ Polar coordinates in the plane give us $\omega_{n+2} = \int_0^{2\pi}\int_0^1\sqrt{1r^2}^n\omega_n r dr d\phi = 2\pi\omega_n \int_0^1(1r^2)^{\frac{n}{2}}rdr = \pi\omega_n \int_0^1(1r^2)^{\frac{n}{2}}2rdr.$ Substitute $s=1r^2$ and get $\omega_{n+2} = \pi\omega_n\int_0^1s^\frac{n}{2}ds = \omega_n \frac{2\pi}{n+2} = \frac{\pi^\frac{n}{2}}{\frac{n}{2}!}\frac{2\pi}{n+2} = \frac{\pi^\frac{n+2}{2}}{\frac{n+2}{2}!}.$ 


softquestion
tag for stuff without formulas :) – Ilya Nikokoshev Nov 19 '09 at 20:26