How does one compute the integral
$ \int_{\Omega_{\epsilon}} f( d(x))\ dx = ? $
where $d(x)$ is the distance to the boundary and $\Omega_\epsilon := \{ x\in \Omega: d(x)<\epsilon\}$, supposing that the boundary is smooth and $\epsilon$ is taken sufficiently small to ensure that there exists a point $p(x)\in \partial \Omega$ such that $d(x) = |x-p(x)|$ and $\nabla d(x) = -\hat{n}(p(x))$ for all $x\in \Omega_\epsilon$.
I imagine such an integral can be computed using some form of generalized polar coordinates tailored to the boundary, but I don't immediately see how. Can anyone provide a reference where this is done?
You could use the tubular neighborhood, which has a nice description for small $\varepsilon>0$.