How do you find the volume of the region bounded y = x² and y =1 is revolved about the line y = -2?

Oct 7, 2015

See the explanation.

Explanation:

Here is the region (in blue) with the line $y = - 2$ in red.

A representative slice in black and the two radii (vertical red lines).

The volume of a washer is $\left(\pi {R}^{2} - \pi {r}^{2}\right) \mathrm{dx}$
Where $R$ is the greater and $r$ the lesser radius.

In this question, $R = 1 - \left(- 2\right) = 3$ and $r = {x}^{2} - \left(- 2\right) = {x}^{2} + 2$

$x$ goes from $- 1$ to $1$, so we need

${\int}_{-} {1}^{1} \pi \left({R}^{2} - {r}^{2}\right) \mathrm{dx} = \pi {\int}_{-} {1}^{1} \left({3}^{2} - {\left({x}^{2} + 2\right)}^{2}\right) \mathrm{dx}$

Expand the polynomial integrand and evaluate to get: $\frac{104}{15} \pi$