Integration of radial functions?

Question

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?

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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.

Answer 1

Spherical coordinates in $\mathbb{R}^n$ is treated fully in Karl Stromberg's book "Introduction to Classical Real Analysis" p. 369-370. Note that on p. 370, it shows the Jacobian of the coordinate transformation as $$r^{n-1} \prod_{j=1}^{n-2} (\sin\theta_j)^{n-j-1}$$ So if your function is radial, then \begin{align*}\int_{\mathbb{R}^n}f(|x|)\mathrm{d}x &=\int_{-\pi}^\pi\int_0^\pi\int_0^\pi\cdots\int_0^\infty f(r)r^{n-1}\prod_{j=1}^{n-2} (\sin\theta_j)^{n-j-1}\,dr \, d\theta_1\cdots d\theta_{n-2}\,d\theta_{n-1}\\&= \omega_{n-1}\int_0^\infty f(r) r^{n-1} \mathrm{d}r,\end{align*} where $$\omega_{n-1}=\int_{-\pi}^\pi\int_0^\pi\int_0^\pi\cdots\int_0^\pi \prod_{j=1}^{n-2} (\sin\theta_j)^{n-j-1}\,d\theta_1\cdots d\theta_{n-2}\,d\theta_{n-1}$$

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Stromberg, Karl Robert, An introduction to classical real analysis, Providence, RI: AMS Chelsea Publishing (ISBN 978-1-4704-2544-9/hbk). xiv, 577 p. (2015). ZBL1331.00003.

Answer 2

Since $\mathbb{R}^n=\{0\}\cup (0,\infty)\times S_r^{n-1}$, its volume element $dV$ can be written as a the product of $dr$ times the volume element of the $n-1$ sphere of radius $r$. By dimensional grounds, the latter is equal to $r^{n-1}$ times the volume element of the unit $(n-1)$-sphere. For example, if $n=2$ one has $dV=r dr\wedge d\theta$, note that the integral of $d\theta$ over $S^1$ is $2\pi$, i.e. the "volume" of $S^1$; similarly for $n=2$, $dV=r^2 \sin\theta\, dr\wedge d\theta$ and $\int _{S^2} \sin\theta d\theta\wedge d\phi=4\pi$. Since your function is radial, you can factorize the integral into a radial and an angular part. The integral of the angular part, which you denote with $\omega_{n-1}$, is therefore the volume of the unit $(n-1)$-sphere. A nice way of calculating $\omega_{n-1}$ is presented here http://scipp.ucsc.edu/~haber/ph116A/volume_11.pdf