A) \[\pi {{a}^{2}}\]
B) \[\frac{1}{2}\pi {{a}^{2}}\]
C) \[\frac{1}{3}\pi {{a}^{2}}\]
D) \[\frac{1}{4}\pi {{a}^{2}}\]
Correct Answer: D
Solution :
\[\int_{0}^{a}{\sqrt{{{a}^{2}}-{{x}^{2}}}dx}\] \[=\left[ \frac{x}{2}\sqrt{{{a}^{2}}-{{x}^{2}}}+\frac{{{a}^{2}}}{2}\,{{\sin }^{-1}}\left( \frac{x}{a} \right) \right]_{0}^{a}\] \[=\left[ 0+\frac{{{a}^{2}}}{2}\,{{\sin }^{-1}}(1)-0-\frac{{{a}^{2}}}{2}{{\sin }^{-1}}(0) \right]\] \[=\frac{{{a}^{2}}}{2}.\frac{\pi }{2}-0=\frac{{{a}^{2}}\pi }{4}\]
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