Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = 2 x ^ { 3 } , y = 0 , x = 3$ about the line $x = 8$.
A) $\frac { 2,268 \pi } { 5 }$
B) $\frac { 2,754 \pi } { 5 }$
C) $\frac { 1,863 \pi } { 5 }$
D) $\frac { 9,477 \pi } { 5 }$
E) $\frac { 2,484 \pi } { 5 }$

Question

Quizplus · Accepted Answer

The shell method is more suitable for this problem because we are revolving around a vertical line (x=8) and dealing with functions of x. The volume of the solid can be found by integrating the volume of cylindrical shells from x=0 to x=3. The radius of each shell is $8 - x$, and the height is given by $y = 2x^3$. The volume of a shell is $2\pi \times \text{radius} \times \text{height} \times \text{thickness}$, leading to the integral $\int_{0}^{3} 2\pi (8-x)(2x^3) dx$. Solving this integral gives the volume as $\frac { 2,268 \pi } { 5 }$.

Use the Disk or the Shell Method to Find the Volume