In several papers by John Monahan and Alan Genz there's mention of a spherical-radial transformation: $$ \int_{\mathbb{R}^n} f(\mathbf{x}) ~\mathrm{d}\mathbf{x} = \int_0^\infty \int_{\mathbf{z}'\mathbf{z} = 1} f(r\mathbf{z}) ~r^{n-1} ~\mathrm{d}\mathbf{z}~\mathrm{d}r $$ where $r$ is a radius and $\mathbf{z}$ is a vector in $\mathbb{R}^n$. How does one show this?
I've been trying integration by substitution but I'm not sure how to handle the fact that the right side has $n+1$ variables plus a constraint and the left side has $n$ variables and no constraints.