In my measure theory class, my teacher wrote the following theorem:
Theorem (Cavalieri's Principle): Let $f:\Omega\to [0,+\infty]$ be a measurable function. Then for every monotone, $\mathscr{C}^1$, function $\Phi:\mathbb{R}_+\to\mathbb{R}_+ $ which satisfies $\Phi(0)=0$, we have $$\int_\Omega \Phi(f(x))\:\mathrm{d}x=\int_{\mathbb{R}_+} \Phi'(\lambda)\rho_f(\lambda)\:\mathrm{d}\lambda.$$
Also, he has defined $\rho_f$ as the following function: $$\lambda\mapsto \mu(\{x\in\Omega\:|\:f(x)>\lambda\}),$$ where $\mu$ is Lebesgue's measure.
I (kind of) understand this theorem but I don't know why it is called Cavalieri's principle. It does not seem related to Cavalieri's principle to me. Am I missing some intuition here?