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Question: The value of \(\int _ { 0 } ^ { 2 \pi } \left| \sin ^ { 3 } \theta \right| d \theta\) is...

The value of ∫02π∣sin⁡3θ∣dθ\int _ { 0 } ^ { 2 \pi } \left| \sin ^ { 3 } \theta \right| d \theta is

A

0

B

C

8/38 / 3

D

π\pi

Answer

8/38 / 3

Explanation

Solution

We have 2∫0πsin⁡3θdθ=2∫0π(3sin⁡θ−sin⁡3θ)4dθ2 \int _ { 0 } ^ { \pi } \sin ^ { 3 } \theta d \theta = 2 \int _ { 0 } ^ { \pi } \frac { ( 3 \sin \theta - \sin 3 \theta ) } { 4 } d \theta

=12[−3cos⁡θ+cos⁡3θ3]0π=12[−3(−1−1)+(−1−1)3]=83= \frac { 1 } { 2 } \left[ - 3 \cos \theta + \frac { \cos 3 \theta } { 3 } \right] _ { 0 } ^ { \pi } = \frac { 1 } { 2 } \left[ - 3 ( - 1 - 1 ) + \frac { ( - 1 - 1 ) } { 3 } \right] = \frac { 8 } { 3 }.