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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.

enter image description here

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You should take into account that since you didn't provide the origin, equation is not unique with respect to vertical and horizontal translation. – Kaster Dec 14 '12 at 9:28
I edited the tags. The question certainly has nothing to do with elliptic curves, and I'm fairly certain that it has nothing to do with integral equations either. Do roll back, if you know otherwise :-) – Jyrki Lahtonen Dec 14 '12 at 10:03

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 $L,S$.

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