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## 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.

$V = \pi \int_a^b [r(x)]^2 dx$
$V = \pi \int_0^h \frac{r^2}{h} x dx$
$V = \frac{\pi r^2}{h} \int_0^h x dx$
$V = \frac{\pi r^2 h}{2}$

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

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