# Max an integral with endpoints varying with the extremal function.

Hello all,

Usually the calculus of variations' take on not variable endpoints means to choose a point in a fixed function, but what I need is for that point to be defined by the function I am maximizing; that is, my problem is to find y(x) to maximize:

$\int^{k_1}_{k_0}{\int_{y(x)}^{t_1}F(x,y,z)\,dz\,dx}$

If it makes it clearer, the geometrical interpretation (if I imagine it correctly) is to choose y(x) which maximizes the volume of the 3-dimensional area (inside a 3-dimensional compact cube) of F(x,y,z) ($F_y$>0), given that the width of the area is in itself given by the choice of y(x).

My problem is more complicated (since it involves probability measures), but if someone help me solve this (or point me into the right direction) I believe I can work out the rest.

sorry if this question is obvious for some (most), but I'm not a mathematician (economist) and I spent the last week reading several books on calculus of variations and I didn't find anything really helpful to solve my research problem.

Thank all, Pedro.

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Is the $y$ in $F(x,y,z)$ the same as $y(x)$? In any case, there seems to be no connection between different $x$ values: for each $x \in [k_0, k_1]$ you want to choose $y$ to maximize $\int_y^{t_1} F(x,y,z)\ dz$. –  Robert Israel Mar 25 '12 at 7:46