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

Oct 14, 2015

See the explanation below.

#### Explanation:

Here is the region and a thin slice taken perpendicular to the axis of rotation. (So we'll be using disks.)

The representative disk (at $x$) has volume $\pi {r}^{2} \cdot \text{thickness}$

In this case, radius $r = 2 {x}^{2}$ and thickness = $\mathrm{dx}$.

$x$ varies from $0$ to $2$, so we need

${\int}_{0}^{2} \pi {\left(2 {x}^{2}\right)}^{2} \mathrm{dx} = 4 \pi {\int}_{0}^{2} {x}^{4} \mathrm{dx}$ which is straightforward to evaluate.