Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method.
-The region bounded by $y = 7 x - x ^ { 2 }$ and $y = x$ about the line $x = 6$
A) $162 \pi$
B) $108 \pi$
C) $324 \pi$
D) $216 \pi$

Question

Quizplus · Accepted Answer

The volume is calculated using the shell method. The intersection points of $y = 7x - x^2$ and $y = x$ are $x = 0$ and $x = 6$. The shell radius is $6 - x$ and the height is $(7x - x^2) - x$. Integrating from 0 to 6 gives the volume $V = 2\pi \int_0^6 (6-x)(6x-x^2) \, dx = 216\pi$.

Find the Volume of the Solid Generated by Revolving the Region