Elaboration on an integral expression of $L^P$ norm of a function

Question

I was trying to understand this example in my functional analysis book, but have problem to understand the last integral equality.( the steps before I derived and are clear, just I do not get the last equality.

Answer 1

Consider the change of variable $x = r w$ where $w$ lies on the unit sphere $S^{n-1}$ and $r \in [0, 1]$. Then, the theorem of integration by substitution for multiple variables lets us write that $$ \int_{x\in \mathbb{R}^n, |x| < 1} f(x) dx = \int_0^1 r^{n-1} \left( \int_{S^{n-1}} f(rw) dw \right) dr, $$ where the $r^{n-1}$ factor is the determinant of the Jacobian matrix of the change of variables.