The ploygon width parallel to the x axis as a function of the y ordonate?

Considering a polygon with n vertices as input.

I need to calculate the integral of the form
[int_A p(y) dA ] where $p(y)$ is a piecewise polynomial function of $y$.

May be if I could find the expression $b(y)$ then the integral can be calculated by:
$$int_{0}^{y_max} p(y) b(y) dy $$

The question is : "is there a straightforward method to find $b(y)$ given the coordinates of the polygon vertices? Or is there a simpler manner to calculate the integral above?"