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 $x$-axis.
A) $\frac { 8,748 \pi } { 7 }$
B) $\frac { 4,374 \pi } { 5 }$
C) $\frac { 162 \pi } { 5 }$
D) $\frac { 162 \pi } { 7 }$
E) $\frac { 54 \pi } { 7 }$

Question

Quizplus · Accepted Answer

Using the disk method, the volume $V$ of the solid of revolution about the $x$-axis is given by $$V = \pi \int_{a}^{b} [f(x)]^2 dx$$, where $f(x) = 2x^3$, $a = 0$, and $b = 3$. Thus, $$V = \pi \int_{0}^{3} (2x^3)^2 dx = \pi \int_{0}^{3} 4x^6 dx = \pi \left[\frac{4x^7}{7}\right]_{0}^{3} = \frac{4\pi}{7} [3^7] = \frac{4\pi}{7} [2187] = \frac{8748\pi}{7}.$$

Use the Disk or the Shell Method to Find the Volume