# How do you find the volume bounded by x = 1, x = 2, y = 0, and y = x^2  revolved about the x-axis?

Nov 29, 2017

Volume bounded by $x = 1$, $x = 2$, $y = 0$ and $y = {x}^{2}$ is $\frac{31 \pi}{5}$ units.

#### Explanation:

The area bounded by $x = 1$, $x = 2$, $y = 0$ and $y = {x}^{2}$ is shown below (shaded in grey(

As it revolves around $x$-axis, its volume is given by

${\int}_{1}^{2} \left(\pi {y}^{2}\right) \mathrm{dx}$

= ${\int}_{1}^{2} \left(\pi {\left({x}^{2}\right)}^{2}\right) \mathrm{dx}$

= $\pi {\int}_{1}^{2} {x}^{4} \mathrm{dx}$

= $\pi {\left[{x}^{5} / 5\right]}_{1}^{2}$

= $\pi \left({2}^{5} / 5 - {1}^{5} / 5\right)$

= $\pi \times \frac{31}{5}$

= $\frac{31 \pi}{5}$