homework and exercises - How do I find work done by friction over a curve represented by a polynomial?

I am facing a problem in Physics.

Problem: What will be the work done by the frictional force over a polynomial curve if a body is sliding on this polynomial($a+bx+cx^2+dx^3+\ldots$) curve from rest from the height $h_1$ to height $h_2$ (where $h_1 > h_2$).

I tried to solve this as follows:

frictional force $F = k mg \cos\theta$, where $mg \cos\theta$ is normal force at that point. $k$ is coefficient of friction

Total work done=Line Integration over the polynomial(dot product of F and displacement).

But to go ahead from this point,i do not know.