Let $f: \mathbb{R}^n \rightarrow \mathbb{\overline{R}} $ be non-negative and (Borel)-measurable and $p>0$.
Then:
$$ \int_{\mathbb{R}^n} f^p dx = p \int_{0}^{\infty} t^{p-1} \lambda^n(\{f>t\})dt $$
where $\lambda^n$ denotes the Lebesgue-Measure in $\mathbb{R}^n$
This is an exercise in a book on measure-theory. It's in the chapter where Fubini's theorem is proven - so it's likely that it can be proven with it. I think it's a quite fascinating identity, but I have no idea how to prove it.
Note first that, for every nonnegative $u$, $$u^p = p \int_{0}^u t^{p-1} \mathrm dt = p \int_{0}^{\infty} t^{p-1} \mathbf 1_{\{u>t\}}\mathrm dt.$$ Use this for $u=f(x)$ and integrate the identity with respect to the Lebesgue measure $\lambda^n$ on $\mathbb R^n$. Then the desired result stems from Tonelli theorem and from the fact that, for every nonnegative $t$, $$\int_{\mathbb R^n}\mathbf 1_{\{f(x)>t\}}\mathrm dx=\lambda^n(\{f\gt t\}).$$