TITLE: Explanation for gamma function in formula for $n$-ball volume
QUESTION [14 upvotes]: It is well-known that the volume of the unit ball in n-space 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.

REPLY [4 votes]: Here is an argument similar to that of S. Carnahan, but more direct (without sphere surface area.) 
Denote the volume of a unit ball in $\mathbb{R}^n$ by $c_n$, then the volume of a ball of radius $r$ equals $c_nr^n$. 
We use the (Lebesgue-Stieltjes?) formula $\int fd\mu=\int_0^\infty \mu\{x:f(x)>t\}dt$ for a positive function $f$ on a measure space $(X,\mu)$. 
Let $(X,\mu)$ be $\mathbb{R}^n$ with Lebesgue measure and $f(x)=e^{-\sum x_i^2}$. Then $$\pi^{n/2}=\left(\int_{-\infty}^\infty e^{-x^2}dx\right)^n=\int fd\mu=\int_0^1 c_n(-\log t)^{n/2}dt=\\=c_n\int_0^\infty s^{n/2}e^{-s}ds=c_n\Gamma(n/2+1),$$
where we substitute $t=e^{-s}$.