Suppose we have a parabola-shaped cross-section of a riverbed, filled with water to a height y = h in the vertical direction, with the surface of the water extending between x = a and x = b in the horizontal direction. How could one express the perimeter of the riverbed wetted by water as a function of the area of water A contained in the cross-section, which is a function of time.
I imagine a good starting point is to apply the integral expression for the arc-length of a parabolic function $\int_a^{b} dx\sqrt{1 + (f'(x))^2}$, with $f(x) = cx^{2}$ as the input function, but I'm not sure how to proceed from there. Any help would be appreciated.