Definite Integration Question 216
Question: The value of $ \int _{0}^{\pi }{| {{\sin }^{3}}\theta |d\theta } $ is
[UPSEAT 2003]
Options:
A) 0
B) 3/8
C) 4/3
D) $ \pi $
Show Answer
Answer:
Correct Answer: C
Solution:
$ I=\int_0^{\pi }{|{{\sin }^{3}}\theta |}d\theta $
Since $ \sin \theta $ is positive in interval $ (0,\pi ) $
$ \therefore I=\int_0^{\pi }{{{\sin }^{3}}\theta d\theta =\int_0^{\pi }{\sin \theta (1-{{\cos }^{2}}\theta )d\theta }} $
$ =\int_0^{\pi }{\sin \theta d\theta +\int_0^{\pi }{(-\sin \theta ){{\cos }^{2}}\theta d\theta }} $
$ =[-\cos \theta ]_0^{\pi }+( \frac{{{\cos }^{3}}\theta }{3} )_0^{\pi }=\frac{4}{3} $ .
BETA
