How to calculate varying moment of inertia in a slender beam given final curve of deflection?

I have no mechanics background so be gentle. Looking for advice on how to approach this problem:

I have a long, slender, straight beam that is then bent into a curve using wire rope as a tension element between either end. Very similar to an archery bow.

I want the final shape of the beam (specifically the surface away from the radius of curvature) to be a parabola whose vertex is in the center of the beam. The beam will be a constant thickness but varying width. Modulus of Elastisity is assumed constant and there is no yielding in the material (or a thickness that prevents the stress from rising too high is selected).

By varying the width and therefore the moment of inertia, I should be able to follow the parabola.  

How would you approach this problem?  Is this standard FEA stuff?

Thanks in advance for any ideas!