Calculus solution to Proposition 11 of Archimedes' method?

Use calculus to prove this proposition. If in a rectangular parallelepiped with square bases a cylinder is inscribed having its bases in opposite square faces and touching with its surface the remaining four parallelogrammic faces, and if through the center of the circle which is the base of the cylinder and one side of the opposite square face a plane be drawn, the figure cut off by the plane so drawn is one-sixth the volume of the whole parallelepiped