# How do you find the volume of the solid obtained by rotating the region bounded by y=x and y=x^2 about the x-axis?

Apr 24, 2018

$V = \frac{2 \pi}{15}$

#### Explanation:

First we need the points where $x$ and ${x}^{2}$ meet.

$x = {x}^{2}$

${x}^{x} - x = 0$

$x \left(x - 1\right) = 0$

$x = 0 \mathmr{and} 1$

So our bounds are $0$ and $1$.

When we have two function for the volume, we use:
$V = \pi {\int}_{a}^{b} \left(f {\left(x\right)}^{2} - g {\left(x\right)}^{2}\right) \mathrm{dx}$

$V = \pi {\int}_{0}^{1} \left({x}^{2} - {x}^{4}\right) \mathrm{dx}$

$V = \pi {\left[{x}^{3} / 3 - {x}^{5} / 5\right]}_{0}^{1}$

$V = \pi \left(\frac{1}{3} - \frac{1}{5}\right) = \frac{2 \pi}{15}$