Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
$y = \ln x , y = 1 , y = 5 , x = 0$; about the $y$ - axis
A) $\frac { \pi } { 4 } \left( e ^ { 10 } - e \right)$
B) $\frac { \pi } { 2 } \left( 1 - e ^ { 10 } \right)$
C) $\frac { \pi } { 2 } \left( e ^ { 10 } - e ^ { 2 } \right)$
D) $\frac { \pi e ^ { 10 } } { 2 }$
E) $2 \pi \left( e ^ { 10 } - e ^ { 2 } \right)$

Question

Quizplus · Accepted Answer

The volume is calculated using the method of cylindrical shells, integrating from $y = 1$ to $y = 5$, with the formula $V = 2\pi \int_{1}^{5} x(y) \, dy$, where $x = e^y$. This results in $\frac{\pi}{2} (e^{10} - e^2)$.

Find the Volume of the Solid Obtained by Rotating the Region