Reference for polar Coordinates in $\mathbb R^n$!

Question

Let $f$ be a real-valued integrable function on $\mathbb R^n$. It is known from the real variables theory that there exists a unique Borel measure $d\sigma$ on the unit sphere $\mathbb S^{n-1}=\{x\in\mathbb{R}^n:|x|=1\}$ such that \begin{align}\tag{1}\label{eq:1} \int_{\mathbb{R}^n} f(x) \,dx = \int_0^\infty \int_{\mathbb S^{n-1}} f(r\omega) \, d\sigma(\omega) \, r^{n-1} dr. \end{align}

1. Is there a constant in $(\ref{eq:1})$ depends on $n$ and $\pi$?

2. I am looking for a reference for the formula $(\ref{eq:1})$.

Thank you in advance

Answer 1

It is a very special case of coarea formula. You can find an undergraduate proof of it in the Mathematical Analysis book of Giaquinta and Modica (volume 5).