Definite-Integration Question 542
Question: $ \int_0^{a}{x^{4}\sqrt{a^{2}-x^{2}}},dx= $
Options:
A) $ \frac{\pi }{32} $
B) $ \frac{\pi }{32}a^{6} $
C) $ \frac{\pi }{16}a^{6} $
D) $ \frac{\pi }{8}a^{6} $
Show Answer
Answer:
Correct Answer: B
Solution:
Put $ x=a\sin \theta \Rightarrow dx=a\cos \theta d\theta $ Now $ \int_0^{a}{x^{4}\sqrt{a^{2}-x^{2}}}dx=a^{6}\int_0^{\pi /2}{{{\sin }^{4}}\theta \cos \theta \cos \theta ,d\theta } $ $ =a^{6}\int_0^{\pi /2}{{{\sin }^{4}}\theta {{\cos }^{2}}\theta ,d\theta } $ $ =a^{6}\frac{\Gamma ( \frac{5}{2} ).\Gamma ( \frac{3}{2} )}{2\Gamma 4}=\frac{\pi }{32}a^{6} $ , (Using gamma function).
BETA
