Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve $y = e ^ { - 2 x }$, and the line $x = 2$ about the $y$-axis.
A) $\frac { 1 } { 2 } \pi \left( 1 - 5 \mathrm { e } ^ { - 4 } \right)$
B) $\frac { 1 } { 2 } \pi \left( 1 - 3 \mathrm { e } ^ { - 4 } \right)$
C) $\frac { 1 } { 2 } \pi \left( 1 - 4 \mathrm { e } ^ { - 4 } \right)$
D - $\frac { 1 } { 2 } \pi \left( 1 + 5 e ^ { - 4 } \right)$

Question

Quizplus · Accepted Answer

The volume of the solid generated by revolving the region around the y-axis can be found using the method of disks or shells. Here, using the shell method is appropriate since we're revolving around the y-axis. The volume $V$ is given by the integral $$V = 2\pi \int_{0}^{2} x e^{-2x} dx.$$ Solving this integral gives $\frac { 1 } { 2 } \pi \left( 1 - 5 \mathrm { e } ^ { - 4 } \right)$, which matches option A.

Find the Volume of the Solid Generated by Revolving the Region