TITLE: Integration of radial functions?
QUESTION [21 upvotes]: Let $f(|x|)$ be a integrable radial function in $\mathbb{R}^n$ ($|\cdot|$ denotes the euclidean norm as in convention). The following identity is used to simplify computations
$$\int_{\mathbb{R}^n}f(|x|)\mathrm{d}x = \omega_{n-1}\int_0^\infty f(r) r^{n-1} \mathrm{d}r,$$
where $\omega_{n-1}$ denotes the surface area of the $(n-1)$-sphere of radius 1.
But I have never found any full proofs on such topic. Does anyone have some references in mind, better some source which I can actually refer to? 

updates
Recently I found this subject has been mentioned by Stein in the appendix from his book "Fourier Analysis", and it further refers to Buck's "Advanced Calculus", Folland's "Advanced Calculus" and Lang's "Undergraduate Analysis". 
I have gone through the later two. It seems to me that Lang's treatment is pretty self-contained, though it is given in exercise hence details omitted.
Mentioned by TCL, Evan's book on measure theory is decent but I'm not sure if it's too heavy. It let me feel like shooting mosquito with bazooka.

REPLY [2 votes]: Proof using coarea formula for general $\ell_p$-norm radial functions
Let $p \in [1,\infty]$ and let $\|\cdot\|_p$ denote the $\ell_p$-norm on $\mathbb R^n$. In particular, $p=2$ corresponds to the euclidean case in the OP's question. I will handle the general case.
Let $B_{n,p}(t) := \{x \in \mathbb R^p \mid \|x\|_p \le t\}$ be the $\ell_p$-ball of radius $t \ge 0$ in $\mathbb R^n$. Let $ :[0, \infty) \rightarrow \mathbb R$ be a measurable function, and  define measurable functions $g,u:\mathbb R^n \rightarrow \mathbb R$ by $g(y) := f(\|y\|_p)$, $r(y) := \|y\|_p$. The $\omega_{n,p}(t)$ be the surface area of the level-set $r^{-1}(t) := \{y \in \mathbb R^n \mid \|y\|_p = t\}=:S_{n,p}(0,t) = \partial B_{n,p}(t)$. Note that thanks to the triangle inequality for $\ell_p$-norms, $y \mapsto r(y)$ is a Lipschitz function on $\mathbb R^n$ and the level-set $r^{-1}(t)$ is a.e smooth for all $t \ge 0$. Finally, note that $\partial_j r(y) = \dfrac{y_j|y_j|^{p-2}}{\|y\|_p^{p-1}}$ and so for every $y \in \mathbb R^n\setminus\{0\}$, we have
$$
\begin{split}
\|\nabla r(y)\|_2 &= \frac{1}{\|y\|_p^{p-1}}\sqrt{\sum_{j=1}^n y_j^2y_j^{2(p-2)}}=\frac{1}{\|y\|_p^{p-1}}\sqrt{\sum_{j=1}^n y_j^{2(p-1)}}\\
&=a_{n,p}(y) := \begin{cases}1, &\mbox{ if }p \in \{2,\infty\},\\
\text{ some explicit expression},&\mbox{ else.}\end{cases}
\end{split}
$$
For $m \in \mathbb N$, let $\mathcal L^m$ be the $m$-dimensional Lebesgue measure (aka $m$-dimensional volume) and let $\mathcal H^m$ be the $m$-dimensional Hausdorff measure (aka $m$-dimensional surface area). Then by the coarea formula, we have
$$
\begin{split}
\int_{\mathbb R^n}a_{n,p}(y)f(\|y\|_p)dy &= \int_{\mathbb R^n}g(y)\|\nabla r(y)\|_2dy = \int_{\mathbb R}\left(\int_{r^{-1}(t)}g(y)d\mathcal H^{n-1}(y)\right)dt\\
&= \int_{0}^\infty f(t)\mathcal H^{n-1}(S_{n,p}(0,t))dt = \omega_{n,p}(1)\int_{0}^\infty f(t)t^{n-1}dt,
\end{split}
$$
Thus, we get the identity
$$
\int_{\mathbb R^n} a_{n,p}(y) f(\|y\|_p)dy = \omega_{n,p}(1)\int_{0}^\infty f(t)t^{n-1}dt.
$$