I want to prove that identity: $$\int _{B^n(0,R)} f(x)dx = \int _0^R r^{n-1} \int _{S^{n-1}} f(ry)dS(y)dr$$

where $B^n(0,R)$ is $n$-dimensional ball and $S^{n-1}$ is the $(n-1)$-dimensional sphere (of radius $1$)

I start by using the co-area formula with the function $\phi(x) = |x|$ and since $\phi (x) = c$ is exactly $cS^{n-1}$, I get: $$\int _{B^n(0,R)} f(x)dx = \int _0^R \int_{rS^{n-1}} f(x)dS(x)dr$$

Afterwards I thought about the change of variables $x_i = ry_i$ which leads me to:

$$\int_0^R \int _{rS^{n-1}} f(x)dS(x)dr = \int_0^R r^n \int_{S^{n-1}} f(ry)dS(y)dr$$ which is almost what I need but the power of $r$ is $n$ instead of $n-1$. I can't understand what is the problem, because I get the power $n$ from the jacobian of the variable change ($J = r^n$)

Help would be appreciated


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