Integral of the gradient of a semilinear function. Let $u:\mathbb{R}^n  \rightarrow \mathbb{R}$  be a semilinear map so that for any $k\in \mathbb{R}$ the surface $\partial C_k^+=\{x:k=u(x)\}$ is contained in the union of finitely many hyperplanes.
If $\mu_{n-1}$ is the standard measure in $\mathbb{R}^{n-1}$ then we can define 
for a subset $U\subset \mathbb{R}^n$ contained in the union of finitely many hyperplanes and hyperspherical surfaces, $\mu_{n-1}(U)$ to be the sum of the measures in the varius $n-1$ dimensional components.
What I don't understand is the following:
$$\int_{C_k^+\cap B(0,1)} |\nabla u| dx = \int_k^\infty \mu_{n-1}(B(0,1)\cap \partial C_t^+)dt$$  
Somehow the gradient disappears, I understand that $$C_k^+\cap B(0,1)=\cup_{t=k}^\infty (B(0,1)\cap \partial C_t^+) $$
but can't proceed.
Any help would be really appreciated!!

If $C_k^+$ means $\{x: u(x)\geq k\}$, then Coarea formula reads:
$$
\int_{C_k^+\cap B(0,1)} |\nabla u| dx = \int_{-\infty}^\infty \mu_{n-1}(\{x\in B(0,1): u(x)=t\})dt \\
= \int_{k}^\infty \mu_{n-1}(\{x\in B(0,1): u(x)=t\})dt = \int_k^\infty \mu_{n-1}(B(0,1)\cap \partial C_t^+)dt.
$$
To see why the gradient disappeared, I drew a little illustration here:

The reason $|\nabla u|$ disappear because we view it as the normal vector for the level curve $\{(x,u(x)): u(x) = t\}$.
