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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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    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 $b$ack, if you **know** otherwise :-)2012-12-14

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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 $L,S$.