Question

A triangle bounded by the line $y=0,y=x$ and $x=4$ is revolved about the $x$-axis. Find the volume of the solid of revolution.

Answer 1

Given, lines are $y=0,y=x$ and $x=4$ is revolved about $x$-axis.

Therefore, volume of the solid of revolution $={\int }_0^4\pi y^{2}dx$
$=\pi {\int }_0^4{x}^{2}dx$ ....$[\because y=x]$
$=\pi {\left(\frac{x^3}{3}\right)}_0^4$
$=\frac{64\pi }{3}$ cubic units.