A) \[\frac{\pi {{a}^{6}}}{32}\]
B) \[\frac{2{{a}^{5}}}{15}\]
C) \[\frac{{{a}^{6}}}{32}\]
D) None of these
Correct Answer: A
Solution :
\[I=\int_{0}^{a}{{{x}^{2}}{{({{a}^{2}}-{{x}^{2}})}^{3/2}}dx}\] Put \[x=a\sin \theta \Rightarrow dx=a\cos \theta \,d\theta \] \[I=\int_{0}^{\pi /2}{{{a}^{2}}{{\sin }^{2}}\theta .{{a}^{3}}{{\cos }^{3}}\theta .a\cos \theta \,d\theta }\] \[={{a}^{6}}\int_{0}^{\pi /2}{{{\sin }^{2}}\theta {{\cos }^{4}}\theta \,d\theta ={{a}^{6}}\frac{\Gamma \frac{3}{2}.\,\Gamma \frac{5}{2}}{2.\Gamma \frac{8}{2}}}\] \[={{a}^{6}}\frac{\frac{1}{2}.\sqrt{\pi }.\frac{3}{2}.\frac{1}{2}.\sqrt{\pi }}{2.3.2.1}=\frac{\pi {{a}^{6}}}{32}\].
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