Volume of the Paraboloid

Problem

Use integration to derive the volume of a paraboloid of radius $r$ and height $h$. Compare the volume of the paraboloid to the volume of the cylinder with equal base and height.

Solution

Consider half a parabola where the interval of $x$ is $[0, h]$.

$r(x)=a\sqrt{x}$

At $x = 0$ ,

$r(0) = a\sqrt{0} = 0$.

At $x=h$ ,

$r(h) = a\sqrt{h} = r$.

Therefore,

$a = \frac{r}{\sqrt{h}}$

and

$r(x) = \frac{r}{\sqrt{h}} \sqrt{x}$.

Use the disk method to calculate the volume.

$V=\pi \int _{a}^{b}[r(x)]^{2}dx$

$V=\pi \int _{0}^{h}{\frac {r^{2}}{h}}xdx$

$V={\frac {\pi r^{2}}{h}}\int _{0}^{h}xdx$

$V = \frac{\pi r^2 h}{2}$

Comparing with the volume the cylinder, ${V}_{cylinder} = \pi r^2 h$, the volume of the paraboloid is half the volume of the cylinder.