Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line
$y = 14$ $y = \sin x , y = 0,0 \leq x \leq \frac { \pi } { 2 }$
A) $\pi \left( 14 - \frac { \pi } { 2 } \right)$
B) $\pi \left( 14 - \frac { \pi } { 4 } \right)$
C) $\pi \left( 56 - \frac { \pi } { 4 } \right)$
D) $\pi \left( 28 - \frac { \pi } { 2 } \right)$
E) $\pi \left( 28 - \frac { \pi } { 4 } \right)$

Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line 
$y = 14$ $y = \sin x , y = 0,0 \leq x \leq \frac { \pi } { 2 }$
A) $\pi \left( 14 - \frac { \pi } { 2 } \right)$
B) $\pi \left( 14 - \frac { \pi } { 4 } \right)$
C) $\pi \left( 56 - \frac { \pi } { 4 } \right)$
D) $\pi \left( 28 - \frac { \pi } { 2 } \right)$
E) $\pi \left( 28 - \frac { \pi } { 4 } \right)$

Quizplus · Accepted Answer

The volume is calculated using the washer method: $ V = \pi \int_{0}^{\pi/2} [(14)^2 - (14 - \sin x)^2] \, dx $. Simplifying gives $ V = \pi \int_{0}^{\pi/2} [28\sin x - (\sin x)^2] \, dx $. Evaluating this integral results in $ \pi \left( 28 - \frac{\pi}{4} \right) $.

Find the Volume of the Solid Generated by Revolving the Region