What is the 2D shape that gives the maximum finite volume of a solid of revolution?
The following image (inspired by the one at Weisstein, Eric W. "Pappus's Centroid Theorem." From MathWorld--A Wolfram Web Resource.) shows a right triangle, a rectangle and a semicircle; all three of them have area equal to $\frac{\pi}{2}R^2$.
The horizontal coordinate of the centroid is $\frac{R}{3}$ for the right triangle, $\frac{R}{2}$ for the rectangle and $\frac{4}{3}\frac{R}{\pi}$ for the semicircle.
The volumes are $\frac{1}{3}\pi^2R^3$ for the cone, $\frac{1}{2}\pi^2R^3$ for the cylinder and $\frac{4}{3}\pi R^3$ for the sphere and so the cylinder has the maximum volume; so, is the rectangle the lamina that, when rotated, gives the maximum volume?
