JEE MAIN - Mathematics (2016 (Offline) - No. 7)
The area (in sq. units) of the region $$\left\{ {\left( {x,y} \right):{y^2} \ge 2x\,\,\,and\,\,\,{x^2} + {y^2} \le 4x,x \ge 0,y \ge 0} \right\}$$ is :
$$\pi - {{4\sqrt 2 } \over 3}$$
$${\pi \over 2} - {{2\sqrt 2 } \over 3}$$
$$\pi - {4 \over 3}$$
$$\pi - {8 \over 3}$$
Explanation
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Points of intersection of the two curves are $$\left( {0,0} \right),\left( {2,2} \right)$$ and $$\left( {2, - 2} \right)$$
Area $$=$$ Area $$(OAB)-$$ area under parabola ($$0$$ to $$2$$ )
$$ = {{\pi \times {{\left( 2 \right)}^2}} \over 4} - \int\limits_0^2 {\sqrt 2 \sqrt x } \,dx$$
$$ = \pi - {8 \over 3}$$
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