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

Question

Quizplus · Accepted Answer

The volume of the solid generated by revolving the region bounded by $y = e^x$, $x = \ln 2$, and the coordinate axes about the line $x = \ln 2$ can be found using the method of cylindrical shells. The radius of a typical shell is $\ln 2 - x$, and the height is $e^x$. The volume of a thin shell is approximately $2\pi(\ln 2 - x)e^x \Delta x$, and integrating from $0$ to $\ln 2$ gives the total volume: $$V = \int_{0}^{\ln 2} 2\pi(\ln 2 - x)e^x dx = 2\pi [(\ln 2)e^x - e^x]_{0}^{\ln 2} = 2\pi[(\ln 2)2 - 2 - (\ln 2) + 1] = 2\pi(1 - \ln 2).$$

Find the Volume of the Solid Generated by Revolving the Region