We know that under some "well-behaving" conditions, we have the coarea formula
$$ \int_{\mathbb{R}^d} f(x)dx = \int_\mathbb{R} d\epsilon \int_{u^{-1}(\epsilon)} f(x) \frac{d\sigma}{|\nabla u|} $$ where $\sigma$ is the surface measure of $u^{-1}(\epsilon)$. Now suppose that the function $f$ is "concentrated" (or compactly supported) in a small open neighborhood of $u^{-1} (0)$, that is, $f(x) = 0$ for $|u(x)|>\delta$ where $\delta$ is small, and $f$ varies slowly along the surfaces $u^{-1}(\epsilon) $. In this case, it feels like you can approximate the integral to leading order of $\delta$ by first integrating $d\epsilon$ and then integrating $u^{-1}(0)$, but I'm not sure exactly the details.