question_answer

The are bounded by the circle \[{{x}^{2}}+{{y}^{2}}=4,\] line \[x=\sqrt{3}y\] and x-axis lying in the first quadrant, is

A) \[\pi /2\]                       

B)   \[\pi /4\]    

C)   \[\pi /3\]                       

D)   \[\pi \]

Correct Answer: C

Solution :

   Required area \[=\int\limits_{0}^{\sqrt{3}}{\frac{x}{\sqrt{3}}}\,\,dx+\int_{\sqrt{3}}^{2}{\sqrt{4-{{x}^{2}}}dx}\] \[=\frac{1}{\sqrt{3}}\left[ \frac{{{x}^{2}}}{2} \right]_{0}^{\sqrt{3}}+\left[ \frac{x}{2}\sqrt{4-{{x}^{2}}}+\frac{4}{2}{{\sin }^{-1}}\frac{x}{2} \right]_{\sqrt{3}}^{2}\] \[=\frac{\sqrt{3}}{2}+\left[ \pi -\frac{\sqrt{3}}{2}-\frac{2\pi }{3} \right]=\frac{\pi }{3}.\]