How do you prove that the volume of any paraboloid is always half the volume of the circumscribed cylinder?
Simply use the generic formulas to show that whatever specific values are used, the ratio is always 1:2.
The calculation of a general parabolic solid volume is described here:
Using the same variable dimensions for the base radius and the height of both the parabola and the cylinder, the two equations can be compared as a ratio. The algebraic solution proves that the 1:2 ratio of Parabolic:Cylinder volumes is correct.