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} 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.