# Find isovalue such that a volume integral under the isosurface has a specific value

Lets say i have a continiously differentialbe function $$f(x, y, z)$$ that is nonnegative everywhere and has a finite volume integral over all space that yields 1, $$\int^\infty_{-\infty }dx\int^\infty_{-\infty }dy\int^\infty_{-\infty }dz f(x,y,z) = 1.$$ Now i'd like to find the levelsurface defined by an isovalue $$k$$, such that the integral inside the volume bounded by that surface yields a specific value $$0.

My thought is that an isovalue should fix an isosurface which fixes a volume which fixes a volume integral value and thus there should be a direct mapping from isovalue to volume integral value. Can i find this map in a closed form or is this in general impossible ? What i would like to have would be a function of the form $$U(k) = l$$ where $$l = \int_{V_k}fdV$$ and $$V_k$$ is the volume inside the isosurface defined by isovalue $$k$$.

The functions $$f(x,y,z)$$ could be the magnitude square of atomic orbitals for example. The question stems from the attempt to plot an isovalue contur that contains exactly 95% of the probability to find an electron. I could find such contours/isovalues by trial and error but i wondered if there is a more systematic approach to find the isovalue.