n-dimensional integral of radial function

Question

Suppose $f:\mathbb{R}^n\to\mathbb{R}$ only depends on the distance to $x$, $f(x_1,\ldots,x_n)=f(\sum_i x_i^2)$. I want to know the volume integral of $f$ over the region $\{x\in\mathbb{R}^n:x_i\ge 0, i=1,\ldots,n,\sum_ix_i^2\le R\}$. The n-dimensional version of the region in the plane bounded by a circle at the origin of radius R, and the coordinate axes in the first quadrant. Is there a n-dimensional version of polar coodinates I can use to compute this integral, maybe in terms of the volume of the ball in $\mathbb{R}^n$, and in terms the integral of $f$ as a function of the radius $\sum x_i^2$?

Answer 1

Suppose that you have $f(x_1, \dots, x_n) = g(r)$, where $r = \sqrt{\sum x_i^2}$ and $g:\Bbb R_{\geq 0} \rightarrow \Bbb R$ is an integrable function.

Then the integral $\int_{\Bbb R^n} f(x_1, \dots, x_n)dx_1\dots dx_n$ can be evaluated as $\int_0^\infty g(r) \cdot S_{n - 1}r^{n - 1} dr$, where $S_{n - 1}$ is the volumn of the $(n - 1)$-dimensional unit sphere.

The value of $S_{n - 1}$ is discussed in the above wiki page. The simplest closed form is $S_{n - 1} = \frac{2\pi^{\frac n 2}}{\Gamma(\frac n 2)}$.