Given two functions $f, g: \mathbb{R}^2 \to \mathbb{R}$, how to simplify the derivative
$$S'(r) = \frac{d}{dr} \int_{f(x,y) \leq r} g(x,y) \,d\mathbb{R}^2.$$
The answer is to take convolution for $f \equiv x+y$, and wouldn't be too hard for $f \equiv x^2 + y^2$ as we have a nice coordinate change. But how to deal with the general cases, provided that $f$ is a polynomial in $x$ and $y$?
More generally, the question is:
Let $n, l$ be positive integers, $g: \mathbb{R}^n \to \mathbb{R}$ be a smooth funciton, and $f_1, \ldots f_l: \mathbb{R}^n \to \mathbb{R}$ be polynomials. In general, is there a formula for
$$S(r_1, \ldots, r_l) = \int_{f_i(x) \leq r_i} g(x) \,d\mathbb{R}^n$$
and its derivatives?