Question

The value of $\int _ { 0 } ^ { \pi } \left| \sin ^ { 3 } \theta \right| d \theta$ is

• A. 0
• B. 3/8
• C. 4/3
• D. $\pi$

Answer 1

Answer: (C)

4/3

Answer 2

$I = \int _ { 0 } ^ { \pi } \left| \sin ^ { 3 } \theta \right| d \theta$

Since $( 0 , \pi )$

$\therefore I = \int _ { 0 } ^ { \pi } \sin ^ { 3 } \theta d \theta = \int _ { 0 } ^ { \pi } \sin \theta \left( 1 - \cos ^ { 2 } \theta \right) d \theta$

$= \int _ { 0 } ^ { \pi } \sin \theta d \theta + \int _ { 0 } ^ { \pi } ( - \sin \theta ) \cos ^ { 2 } \theta d \theta$

$= [ - \cos \theta ] _ { 0 } ^ { \pi } + \left( \frac { \cos ^ { 3 } \theta } { 3 } \right) _ { 0 } ^ { \pi } = \frac { 4 } { 3 }$.