If $\int_n^m{f(x,y)}dy=g(x)$, is there a way to find, or approximate $f(x,y)$ given $g(x)$

Question

If I'm given $f(x,y)$, when $$\int_n^m{f(x,y)}dy=g(x)$$ then I know how that I can at least approximate $g(x)$ using a rieman sum, however if I am instead given $g(x)$ I don't know how to even approximate $f(x,y)$, other than by trying out every possible function I think that $f(x,y)$ might be. Is there a way to find or approximate $f(x,y)$ that's better than guess and check given $g(x)$?

Answer 1

This equation is far from uniquely determining $f$, and there are many solutions.

For example, if $\chi:[n,m]\to\mathbb R$ is a continuous function whose integral is $1$ (and there are many of those), then $f(x,y) = g(x) \chi(y)$ is a solution.