# How do you find the volume of the region bounded b the line y = x-2, the x-axis, x=2, and x=4 is revolved about the line x = -1?

Oct 18, 2015

$V = \frac{52 \pi}{3}$

#### Explanation:

$V = \pi {\int}_{{y}_{1}}^{{y}_{2}} \left({R}^{2} - {r}^{2}\right) \mathrm{dy}$

$y = x - 2 , x = 2 \implies {y}_{1} = 2 - 2 = 0$
$y = x - 2 , x = 4 \implies {y}_{2} = 4 - 2 = 2$

$x = 4 \implies R = 4 - \left(- 1\right) = 5$
$y = x - 2 \implies x = y + 2 \implies r = y + 2 - \left(- 1\right) = y + 3$

$V = \pi {\int}_{0}^{2} \left({5}^{2} - {\left(y + 3\right)}^{2}\right) \mathrm{dy} = \pi {\int}_{0}^{2} \left(25 - {y}^{2} - 6 y - 9\right) \mathrm{dy}$

$V = \pi \left(16 y - 3 {y}^{2} - {y}^{3} / 3\right) {|}_{0}^{2} = \pi \left(32 - 12 - \frac{8}{3}\right)$

$V = \frac{52 \pi}{3}$