Circular-arc area maximizer with a fixed graph length
= Circular-arc area maximizer with a fixed graph length
{title2=$y=\sqrt{R^2-(x-a)^2}-\sqrt{R^2-a^2}$}
For a nonnegative graph joining $(0,0)$ and $(2a,0)$ with length between $2a$ and $\pi a$, the area-maximizing smooth graph is a circular arch. The <Euler-Lagrange equation> integrates to the displayed <circle> formula. Its radius satisfies $l=2R\arcsin(a/R)$ with $R>a$. <Concavity> of $y-R\sqrt{1+(y^{\prime})^2}$ gives global maximality on the fixed-length class.