How to proof the co-area formula on surfaces:
Let for each $t \in [0,T]$, $\phi( t,\cdot) : {\bar \Omega} \rightarrow R$ be Lipschitz continuous and assume that for each $r \in ( \textrm{inf}_{\Omega} \phi, \textrm{sup}_{\Omega} \phi)$ the level set $\Gamma_r = \{x|\phi(x,\cdot)=r\}$ is a smooth $d$-dimensional hypersurface in $R^{d+1}$. Suppose $u : {\bar \Omega} \rightarrow R$ is continuous and integrable. Then \begin{equation} \int^{\textrm{sup}_{\Omega}}_{\textrm{inf}_{\Omega}} \left( \int_{\Gamma_r} u \right) d r = \int_{\Omega} u |\nabla \phi|. \end{equation}
This formula can be found in G. Dziuk and C.M. Elliott. An Eulerian approach to transport and diffusion on evolving implicit surfaces. Computing and Visualization in Science, 13(1):17–28, 2010.