Find the volume of the solid generated by revolving the region in the first quadrant bounded by the $x$-axis and the curve $y = \sin 8 x , 0 \leq x \leq \pi / 8$ about the line $x = \pi / 8$.
A) $\frac { \pi ^ { 2 } } { 64 }$
B) $\frac { 1 } { 32 } \pi ^ { 2 } - \pi$
C) $\frac { 1 } { 32 } \pi$
D) $\frac { 1 } { 32 } \pi ^ { 2 }$

Question

Quizplus · Accepted Answer

The volume of the solid is calculated using the method of cylindrical shells. The formula is $ V = 2\pi \int_{0}^{\pi/8} (x - \pi/8) \sin(8x) \, dx $. Solving this integral gives the volume as $\frac{1}{32} \pi^2$.

Find the Volume of the Solid Generated by Revolving the Region