 Circular-arc area maximizer with a fixed graph length

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.