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Given the length L of a curve going two given point $(a,\alpha)$, $(b,\beta)$ find the equation of the curve so that the curve together with the interval $[a,b]$ encloses the largest area. Am I correct in thinking this is Dido's problem? Is it possible to use Green's theorem to find the equation?

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Yes I think this is dido's problem or some variation (sorry for the pun). But think about what you're trying to extremize and with what constraint. You are trying to extremize $\int^b_a y(x) \text{d}x$ together with the constraint that $\int^b_a \sqrt{1+y'(x)^2}dx = L$ and you are also given that the endpoints are fixed i.e, $y(a) = \alpha$ and $y(b) = \beta$. What method does one usually employ if they want to extremize a functional subject to a constraint and given that the endpoints are fixed? You use Lagrange multipliers together with the Euler-Lagrange equation.

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