How to determine the region of integration after switching the order of integration in higher dimensions?

Question

I ran into this problem when I was reading the proof of the Sobolev imbedding theorem in Adams's book: We let $\Omega_k$ denote the insection of $\Omega$ with a k-dimensional plane H, where $\Omega\subset\mathbb{R}^n$ is a domain,let $\Omega_{k,\rho}=\{x\in\mathbb{R}^n:dist(x,\Omega_k)<\rho\}$, and let u and all its derivativs be extended to be zero outside $\Omega$.Denoting by $dx^{'}$ the k-volume element in $H$,than why the following equality is right: $$\int_{\Omega_k}dx^{'}\int_{B_{\rho}(x)}|D^{\alpha}u(y)|^pdy=\int_{\Omega_{k,\rho}}|D^{\alpha}u(y)|^pdy\int_{H\cap B_{\rho}(y)}dx^{'}$$