Newton polygons and local integrability of negative powers of smooth functions in the plane

Let $f(x,y)$ be any smooth real-valued function with $f(0,0)=0$. For a sufficiently small neighborhood $U$ of the origin, we study the number

It is known that sometimes this number can be expressed in a natural way using the Newton polygon of $f$. We provide necessary and sufficient conditions for this Newton polygon characterization to hold. The behavior of the integral at the supremal $\epsilon$ is also analyzed.