Definite Integration Question 280
Question: $ \int_0^{a}{x^{2}{{(a^{2}-x^{2})}^{3/2}}dx=} $
Options:
A) $ \frac{\pi a^{6}}{32} $
B) $ \frac{2a^{5}}{15} $
C) $ \frac{a^{6}}{32} $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
$ I=\int_0^{a}{x^{2}{{(a^{2}-x^{2})}^{3/2}}dx} $
Put $ x=a\sin \theta \Rightarrow dx=a\cos \theta d\theta $
$ I=\int_0^{\pi /2}{a^{2}{{\sin }^{2}}\theta .a^{3}{{\cos }^{3}}\theta .a\cos \theta d\theta } $
$ =a^{6}\int_0^{\pi /2}{{{\sin }^{2}}\theta {{\cos }^{4}}\theta d\theta =a^{6}\frac{\Gamma \frac{3}{2}.\Gamma \frac{5}{2}}{2.\Gamma \frac{8}{2}}} $
$ =a^{6}\frac{\frac{1}{2}.\sqrt{\pi }.\frac{3}{2}.\frac{1}{2}.\sqrt{\pi }}{2.3.2.1}=\frac{\pi a^{6}}{32} $ .
BETA
