$1/|x|^\alpha$ is integrable on the unit ball in $\mathbb R^n$ iff $\alpha < n$

Question

I've always remember the statement that $1/|x|^\alpha$ is integrable on the unit ball in $\mathbb R^n$ if and only if $\alpha < n$. But I don't know how to prove it. Can anyone show me how to prove it or link a proof of it.

Any help is appreciated.

Answer 1

Let $A_n(R)$ be the surface area of $\|x\|=R$ in $\mathbb{R}^n$. Then obviously $A_n(R)= C\cdot R^{n-1}$ for some positive constant $C$, and:

$$ \int_{\|x\|\leq 1}\frac{d\mu}{\|x\|^{\alpha}} = \int_{0}^{1}\frac{A_n(z)}{z^\alpha}\,dz = C\cdot \int_{0}^{1}z^{n-1-\alpha}=\frac{C}{n-\alpha}$$ provided that $\alpha\color{red}{<}n$.