JAMB 1999 · UME · Q38

Find the volume of the solid generated when the area enclosed by $y = 0$ , $y = 2x$ and $x = 3$ is rotated about the $x$ -axis.

Worked solution (try it first)

The volume of revolution about the $x$ -axis is $\pi\int y^2\,dx$ .
Here $y = 2x$ , so $y^2 = 4x^2$ , from $x = 0$ to $x = 3$ .
$\pi\int_0^3 4x^2\,dx = \pi\left[\frac43x^3\right]_0^3$
$= \pi \times \frac43 \times 27$
$= 36\pi$ .
So the volume is $36\pi$ cubic units, option B.