Integral formula for polar coordinates The polar coordinates of point $x \in \mathbb{R} \setminus \{0\}$ are pairs $(r,\gamma)$, where $0 < r < \infty$ and $\gamma \in S^{d-1} = \{x \in \mathbb{R}^{d}\mid |x| = 1\}$. These are determined by
$$r = |x|, \quad\gamma = x/|x|,$$ and reciprocally by $x = r\gamma$.
Then we have:
$$\int_{\mathbb{R}^{d}}f(x)dx = \int_{S^{d-1}} \left( \int_{0}^{\infty}f(r\gamma)r^{d-1}dr \right) d\sigma(\gamma).$$
The proof of this formula using the Fubini's theorem. However, I can't understand that the relationship between those measure spaces. And I was so confused by this equation:
$$\mu_{1}(E) = \int_{E}r^{d-1}dr.$$
Besides, I also want to know about the integral formula for the general case. Is there a universal steps to construct the integral formula with respect to the Jacobi in the Reimann integral. 
 A: I recently discovered a very nice treatment of spherical integration in the paper Integration over spheres and the Divergence Theorem by John A. Baker (American mathematical monthly 104 (1997), 36-47). The story goes as follows. Let $n \geq 2$ and $g \colon S^{n-1} \to \mathbb{R}$ be continuous. 
Define $\hat{g} \colon \mathbb{R}^n \to \mathbb{R}$ by 
$$
\hat{g}(x)=
\begin{cases}
g(|x|^{-1}x) &\hbox{if $x \neq 0$} \\
0 &\hbox{if $x=0$}.
\end{cases}
$$
Now define
$$
\int_{S^{n-1}} g\, d\sigma_{n-1} = n\int_{B(0,n)} \hat{g}(x)\, dx.
$$
The following result can be proved ($B(a,b)$ is the spherical shell with radii $a$ and $b$).
Theorem. Suppose $0 \leq a < b$ and $f \colon B(a,b) \to \mathbb{R}$ is continuous. Then
$$
\int_{a \leq |x| \leq b} f(x)\, dx = \int_a^b r^{n-1} \left( \int_{S^{n-1}} f(rs)\, d\sigma_{n-1}(s)\right)dr.
$$
The proof is very nice, and uses the differentiability properties of the map
$$
\varphi(r) = \int_{a \leq |x| \leq r} f(x)\, dx,
$$
since it turns out that
$$
\frac{d\varphi}{dr} = r^{n-1} \int_{S^{n-1}} f(rs)\, d\sigma_{n-1}(s).
$$
