# How do you find the volume of the solid generated by revolving the region bounded by the curves y=x^2, y=0 x=2 rotated about the x-axis?

##### 1 Answer
Aug 30, 2015

See the explanation.

#### Explanation:

A at a value of $x$ in $\left[0 , 1\right\}$, a representative disk has radius $\text{top "y - "bottom } y$ which will be ${x}^{2} - 0 = {x}^{2}$

Th volume of a disk is $\pi {r}^{2} \cdot \text{thickness}$.

The thickness in this case is $\mathrm{dx}$

Volume of slice: $\pi {\left({x}^{2}\right)}^{2} \mathrm{dx}$

Volume of the solid ${\int}_{0}^{1} \pi {\left({x}^{2}\right)}^{2} \mathrm{dx}$

We need:

$\pi {\int}_{0}^{1} {x}^{4} \mathrm{dx} = \pi {\left[{x}^{5} / 5\right]}_{0}^{1}$

$= \pi \left[{\left(1\right)}^{5} / 5 - \frac{{0}^{5}}{5}\right]$

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

The volume is $\frac{\pi}{5}$