The region under the curve $y = 3 \sin ^ { 2 } x , 0 \leq x \leq \pi$ is rotated about the $x$-axis. Find the volume of the resulting solid. Select the correct answer.
A) $\frac { 9 } { \pi ^ { 2 } }$
B) $\frac { 27 } { 8 } \pi ^ { 2 }$
C) $\frac { 3 } { \pi }$
D) $\frac { \pi ^ { 2 } } { 9 }$
E) $\frac { \pi } { 9 }$

Question

Quizplus · Accepted Answer

The volume of the solid obtained by the rotation of the region under the curve is given by: 
\begin{align*}
	ext{Volume} &= \int_{0}^{\pi} \pi y^2 dx \
&= \int_{0}^{\pi} \pi (3\sin^2 x)^2 dx \
&= 9\pi \int_{0}^{\pi} \sin^4 x dx \
&= 9\pi \cdot \frac{3\pi}{8} &&	ext{(using  reduction formula)}\
&= \frac{27}{8} \pi^2
\end{align*}

The Region Under the Curve y=3sin⁡2x,0≤x≤πy = 3 \sin ^ { 2 } x , 0 \leq x \leq \piy=3sin2x,0≤x≤π

The Region Under the Curve $y = 3 \sin ^ { 2 } x , 0 \leq x \leq \pi$