Distance Along a Curve

Question

Given $f(x)=10^3( \sin^2(\frac{33\pi}{x})+\sin^2(x\pi))-10$, what is the distance along the curve between $[a,b]$ in the form of $h(a,b)$? If in some interval $[m,n]$ a subster of $[a,b]$ $f$ is all negative, I want the distance in that interval to be negative and subtracted from the entire $h(a,b)$. I also want the same thing applied to $g(x)=|f(x)|$.