What I have is length of the bottom line $L$ and area under parabolic curve $S$. How can I find this parabolic curve equation, depending on area under it? The following picture illustrates the problem.
Assume your equation is $y = ax^2$. Compute the area $S$ through integration to find the area under the curve from $x \in [0,L/2]$. You should be able to find $a$ in terms of $L,S$.