# How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region y=8-x^2 y=x^2 x=0 rotated about the y axis?

##### 1 Answer
Sep 12, 2015

$16 \pi$

#### Explanation:

The volume of the solid generated would be calculated in two parts. The two functions are parabolas one opening up and the other opening down on the same axis of symmetry x=0. The figure is shown alongside. The point of intersection is (2,4) on the right side and (-2,4) on the left side. The elementary shells would be formed along the line y=4. The volume integrals would be

${\int}_{0}^{2} 2 \pi x \mathrm{dx} \left(4 - {x}^{2}\right) + {\int}_{0}^{2} 2 \pi x \mathrm{dx} \left(8 - {x}^{2} - 4\right)$
(volume of a shell on green) (volume of shell on red parabola)
parabola)

=$4 \pi {\int}_{0}^{2} \left(4 x - {x}^{3}\right) \mathrm{dx}$

=$4 \pi {\left[2 {x}^{2} - {x}^{4} / 4\right]}_{0}^{2}$

= $16 \pi$