Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line.
$y = 4 - x ^ { 2 } , y = 0 ;$ the line $y = 6$
A) $\frac { 704 } { 15 } \pi$
B) $\frac { 1408 } { 15 } \pi$
C) $\frac { 2816 } { 15 } \pi$
D) $\frac { 1408 } { 5 } \pi$

Question

Quizplus · Accepted Answer

The volume of the solid is found using the method of cylindrical shells. The region is revolved around the line $y = 6$, so the radius of a shell at $x$ is $6 - (4 - x^2) = 2 + x^2$. The height of the shell is $4 - x^2$. The volume is given by the integral $V = 2\pi \int_{-2}^{2} (2 + x^2)(4 - x^2) \, dx$, which evaluates to $\frac{1408}{15} \pi$.

Find the Volume of the Solid Generated by Revolving the Region