Let $A_{n-1}$ be the area of the sphere in $\Bbb R^n$ given by $S_{n-1}=\{ (x_1,\dots,x_n) \in \Bbb R^n \lvert \sum_{i=1}^n x_i^2=1 \}.$ We have that the area of a sphere of radius $r$ is given by $r^{n-1}A_{n-1}$.
A step in a computation says that $$\int_{\Bbb R^n}e^{-\sum_{i=1}^n x_i^2}dx_1 \dots dx_n = A_{n-1} \int_0^{+ \infty}e^{-r^2}r^{n-1}dr $$
There is obviously a change of variables with $r$ but I don't see why exactly this equality holds, can someone explain ?