I have a background in math, but I think the following construction appears often in physics.
Is there a term for a functional which maps one smooth function $f: R^n \rightarrow R$ to another $L(f):R \rightarrow R$ which represents the integral along each level set? That is, $L(f)(a) = \int_{f^{-1}(a)} d\mu $ where the measure $d\mu$ might be the induced measure from the metric on $R^n$, or it might also depend on the original $f$ in some intelligent way.
For instance, if the initial function is $f:R^2 \rightarrow R$ is defined $f(x)=x^2+y^2$ then $L(f)(a) = 2 \pi \sqrt{a}$.
If the original function is $f:R \rightarrow R$ is defined $f(x)=x^2$, then we'd have $L(f)(a)=0$ for $a<0$, $1$ for $a=0$, and $2$ for $a>0$. In both cases it might be more natural to integrate with a term derived from the value of $f'$ normal to the level set.
I suppose one could generalize to any smooth map $f:M \rightarrow M'$ from a Riemannian manifold to a another smooth manifold and get a new function that represents a computed volume of the preimage for each point: $L(f):M' \rightarrow R$.
My question: Is there a good way to compute $L(f)$ for some class of function $f$?
Any thoughts would be appreciated. It is reminiscent of (but I hope simpler than) stationary phase approximations and path integral calculations.