A) \[\frac{1}{\sqrt{3}}+\frac{4\pi }{3}\]
B) \[\frac{1}{\sqrt{3}}+\frac{2\pi }{3}\]
C) \[\frac{1}{2\sqrt{3}}-\frac{\pi }{3}\]
D) \[\frac{1}{2\sqrt{3}}+\frac{2\pi }{3}\]
Correct Answer: A
Solution :
\[{{x}^{2}}+3x-4=0\] \[(x+4)(x-1)=0\] \[x=-4,x=1\] Area\[=\left( \int\limits_{0}^{1}{\sqrt{3}.\sqrt{x}dx} \right)+\int\limits_{0}^{1}{\left( \int\limits_{0}^{1}{\sqrt{3}.\sqrt{x}dx}+\int\limits_{1}^{2}{\sqrt{4-{{x}^{2}}}dx} \right)\times 2}\]\[=\left( \sqrt{3}\left( \frac{{{x}^{3/2}}}{3/2} \right)_{0}^{1}+\left( \frac{x}{2}\sqrt{4-{{x}^{2}}}+2{{\sin }^{-1}}\frac{x}{2} \right)_{1}^{2} \right)\times 2\] \[=\left( \sqrt{3}\left( \frac{2}{3} \right)+\left\{ 2.\frac{\pi }{2}-\frac{\sqrt{3}}{2}+\frac{\pi }{3} \right\} \right)\times 2\] \[\left( \frac{2}{\sqrt{3}}-\frac{\sqrt{3}}{2}+\frac{2\pi }{3} \right)\times 2\]\[=\left( \frac{2}{2\sqrt{3}}+\frac{2\pi }{3} \right)\times 2=\frac{1}{\sqrt{3}}+\frac{4\pi }{3}\]dYou need to login to perform this action.
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