Exercise on Proposition 4:
a) Each crosssection of the cylinder, left where it is, balances the crosssection of the paraboloid, moved to H.
b) Thus the cylinder, left where it is, will balance the paraboloid moved so that its center of gravity is H. By the Law of the Lever we have

Now AH = AD so this implies that the volume of the paraboloid is 1/2 the volume of the cylinder.
c) If r is the radius of the cylinder, then the height is h = r^{2} (since the paraboloid is generated by the curve x = y^{2}). So the volume of the cylinder is \(\pi\)r^{2}h = \(\pi\)r^{4}, and so the volume of the paraboloid is \(\pi\) r^{4}/2.
Exercise on Proposition 6:
a) Each crosssection of the hemisphere together with the crosssection of the cone, left where they are, balance the crosssection of the cone moved to H.
b) Thus the hemisphere and cone, left where they are, balance a cone placed so that its center of gravity is at H. Let X be the location of the center of gravity of the hemisphere. By the Law of the Lever,

Now we know that
