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

Indeed:

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.