Volume of Paraboloid

Theorem
The volume of paraboloid is half the volume of its circumscribing cylinder.

Proof


Consider a cylinder of radius $r$ and height $h$, circumscribing a paraboloid $y= h \paren {\dfrac x r}^2$ whose apex is at the center of the bottom base of the cylinder and whose base is the top base of the cylinder.

Also consider the paraboloid $y = h - h \paren {\dfrac x r}^2$, with equal dimensions but with its apex and base flipped.

For every height $0 \le y \le h$, the disk-shaped cross-sectional area $\pi \paren {\sqrt {1 - \dfrac y h} r}^2$ of the flipped paraboloid is equal to the ring-shaped cross-sectional area $\pi r^2 - \pi \paren {\sqrt {\dfrac y h} }^2$ of the cylinder part outside the inscribed paraboloid.

Therefore, the volume of the flipped paraboloid is equal to the volume of the cylinder part outside the inscribed paraboloid.

In other words, the volume of the paraboloid is $\dfrac \pi 2 r^2 h$, half the volume of its circumscribing cylinder.