Chapter 9.6, Problem 5P

A curve $y=y(x),$ joining two points $x_1$ and $x_2$ on the $x$ axis, is revolved around the $x$ axis to produce a surface and a volume of revolution. Given the surface area, find the shape of the curve $y=y(x)$ to maximize the volume. Hint: You should find a first integral of the Euler equation of the form $yf\left(y,x^{\prime },\lambda \right)=C.$ since $y=0$ at the endpoints, $C=0.$ Then either $y=0$ for all x, or $f=0.$ But $y\equiv 0$ gives zero volume of the solid of revolution, so for maximum volume you want to solve $f=0$.