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.