Let $y''+fy=0$ be a second-order linear ode on $y$, where $f(x)>0$, and $I=\left[ a,b \right)$ be an interval. Suppose we want to estimate the number of zeros of a (not identically zero) solution of the ode on the interval $I$. Note that this question is well defined, because by the Sturm separation theorem, choosing a different solution can change the number of zers by at most 1.

It follows from the Sturm–Picone comparison theorem that if $0<m \leq f(x) \leq M$ on the interval ($m,M$ are numbers), and $N$ is the number of zeros of a solution on $I$, then
$$\left \lfloor \frac{(b-a)\sqrt{m}}{\pi} \right \rfloor \leq N \leq \left \lceil \frac{(b-a)\sqrt{M}}{\pi} \right \rceil$$
Now, we can break $I$ into subintervals, and hopefully we get better estimates. A hopeful thought lead to hoping that the number of zeros on $I$ can be approximated by the integral $S=\int_{a}^{b} \frac{\sqrt{f(x)}}{\pi} dx$.

My question is, how good is this approximation? What can be said about the error $\left| N-S \right|$?