Archimedes' Method for Computing Areas and Volumes - Proposition 5 of The Method

We can use Archimedes' method to determine the center of gravity of the paraboloid. By symmetry, we know that the center of gravity lies at some point $X$ on its axis (not indicated on the applet). Archimedes' Proposition 5, illustrated below, shows a paraboloid and a cone inscribed inside a cylinder. 

The sketch illustrates that the paraboloid, left where it is, balances the cone, moved to $H$, where $|AH|=|AD|$. So, by the law of the lever,

$$(\mbox{Volume of paraboloid})\times |AX|=(\mbox{Volume of cone})\times |AH|.$$

Since the volume of the paraboloid is half the volume of the cylinder (by the previous exercise), and the volume of the cone is one-third the volume of the cylinder, this implies that

$$\frac12|AX|=\frac13|AH|,\,\,\,\,{\rm{hence}}\,\,\,{\rm{that}}\,\,\,\, |AX|=\frac23|AH|.$$ So the center of gravity of the paraboloid lies on its axis, twice as far from the vertex as from the base.