Change of variable Theorem problem. What am I doing wrong? I posted a question here the other day, asking me to show that $\int_{\Bbb{R}^n}f=nV\int_{0}^{\infty}g(r)r^{n-1}dr$ where $V$ is the volume of the unit ball (with respect to the $p(>1)$ norm.), and $f(x)=g(||x||)$, $g:[0,\infty)\to [0,\infty)$ integrable. The intuitive thing for me to do was setting $x=rw$, where $r>0$ and $w\in S^1$ (the unit sphere).
With some differentiation with respect to $x\in S_1$, I could arrive at a Jacobian of $r^{n-1}$, and things would start to get close to the result I was trying to arrive at. The thing is: this is not a diffeomorphism. I actually can't evaluate what that is. I mean, I need a directed vector from $S^1$, that would be $n$ variables, and then a radius, that is another variable in addition. So I need $n+1$ variables to get $\Bbb{R}^n$. The Change of Variables Theorem will obviously won't apply here. How can I circumvent this problem, or this theorem? I am completely lost here. Any hints? 
 A: For $r<s$ define the shell $S(r,s):=\{x\in{\mathbb R}^n\,|\,r\leq\|x\|<s\}$. If $V$ is the volume of the unit ball then the MVT provides the estimate
$${\rm vol}(S(r,s))=V(s^n-r^n)=nV(s-r)\rho^{n-1}$$
for some $\rho\in\>]r,s[\>$. As $f(x)=g(\|x\|)$ the mean value theorem for definite integrals gives
$$\int_{S(r,s)} f(x)\>{\rm d}(x)=g(\rho')\,{\rm vol}(S(r,s))=n V\>g(\rho')\rho^{n-1}(s-r)\ ,$$
with both $\rho'$ and $\rho$ in the interval $\>]r,s[\>$.
Consider now partitions
$$0=r_0<r_1<r_2\ldots \to\infty\tag{1}$$
of the $r$-axis. Then
$$\int_{{\mathbb R}^n} f(x)\>{\rm d}(x)=\sum_{k=1}^\infty \int_{S(r_{k-1},r_k)} f(x)\>{\rm d}(x)=Vn\>\sum_{k=1}^\infty g(\rho_k')\,\rho_k^{n-1}\>(r_k-r_{k-1})\ .$$
Here the right hand side is a Riemann sum for the integral
$$Vn\>\int_0^\infty g(r)\,r^{n-1}\>dr\ .$$
In fact it is not difficult to provide error estimates which prove that for any given $\epsilon>0$ this sum is $<\epsilon$ away from the integral, if the partition $(1)$ is fine enough. I leave the details to you.
