Finding $\mathrm{vol}(f^{-1}(y))$

Question

• Consider a many-to-one function $f:X\to Y$
• $X$ is a high-dimensional vector space and $Y$ a low-dimensional vector space
• The analytical form of $f$ is known
• You are probably free to make any 'nice' assumptions about $f$ (e.g. infinitely-differentiable), and the spaces (e.g. Hilbert).
• Let $\mathrm{vol}(f^{-1}(y))$ denote the volume of the subset of $X$ that maps to $y\in Y$.

Question: Are there general methods for computing $\mathrm{vol}(f^{-1}(y))$ (either exactly or to within some arbitrary precision)? Is there an area of mathematics concerned with such calculations?