Let $b > a > 0$. Fixing $y(a) = y(b) = 0$ and the total length $L = \int_a^b \sqrt{1+y'^2} dx$, I want to find the curve $y(x)$ that maximizes the volume of the (roughly toroidal) volume between the $x-z$ plane and the surface of revolution of $y(x)$ about the $y$ axis; that is, maximize $\int_a^b x y\, dx$. Can this be solved analytically?
This question is similar to Calculus of variations: find $y(a/2)$ if $y(x)$ maximizes the volume of rotation, but I want to rotate the curve about the other axis. (I also didn't see a way to enforce $y(b) = 0$ in that approach.) Following the solution approach there blindly and using WolframAlpha on the final differential equation results in a monstrous solution that doesn't show any signs of passing through $0$ at $x=b$.
The only sanity check I can think of is that when $a, b \gg L$, the problem is very close to the standard isoperimetric inequality, so the resulting curve should be close to a circular arc passing through $a$ and $b$ and having total length $L$. I am interested in the regime when $a$, $b$, and $L$ are all comparable.
