# How do you find the volume of the solid generated by revolving this region about the y axis, x= y^2 and x= y+2?

##### 1 Answer
Sep 21, 2015

$V = \frac{72 \pi}{5}$

#### Explanation:

${x}_{1} = {y}^{2}$
${x}_{2} = y + 2$

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

y_1,y_2=?
${x}_{1} = {x}_{2}$
${y}^{2} = y + 2$
${y}^{2} - y - 2 = 0$
${y}^{2} - y - 2 = {y}^{2} - 2 y + y - 2 = y \left(y - 2\right) + y - 2 = \left(y - 2\right) \left(y + 1\right) = 0$

${y}_{1} = - 1 , {y}_{2} = 2$

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

$V = \pi {\int}_{- 1}^{2} \left({y}^{2} + 4 y + 4 - {y}^{4}\right) \mathrm{dy}$

$V = \pi \left({y}^{3} / 3 + 2 {y}^{2} + 4 y - {y}^{5} / 5\right) {|}_{-} {1}^{2}$

$V = \pi \left(\frac{8}{3} + 8 + 8 - \frac{32}{5} + \frac{1}{3} - 2 + 4 - \frac{1}{5}\right)$

$V = \frac{72 \pi}{5}$