Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. $x = y ^ { 2 } , x = 8 y - y ^ { 2 }$
(i) $y$-axis; (ii) the line $x = 18$
A) $\frac { 1,024 } { 3 } \pi ; \frac { 1,280 } { 3 } \pi$
B) $\frac { 16 } { 3 } \pi ; \frac { 128 } { 3 } \pi$
C) $\frac { 64 } { 3 } \pi ; \frac { 128 } { 3 } \pi$
D) $\frac { 64 } { 3 } \pi ; \frac { 1,280 } { 3 } \pi$
E) $\frac { 16 } { 3 } \pi ; \frac { 1,280 } { 3 } \pi$

Question

Quizplus · Accepted Answer

The region is bounded by the graphs of $x = y ^ { 2 } $ and $x = 8 y - y ^ { 2 }$. The volume of the solid generated by revolving the region about the $y$-axis is given by$$V = \pi \int_0^4 (8y - y^2)^2 - (y^2)^2 dy = \pi \int_0^4 (64y^2 - 16y^3 + y^4) dy = \frac{1024}{3}\pi$$The volume of the solid generated by revolving the region about the line $x = 18$ is given by$$V = \pi \int_0^4 (18 - (8y - y^2))^2 - (18 - y^2)^2 dy = \frac{1280}{3}\pi$$

Find the Volume of the Solid Generated by Revolving the Region