question_answer

The area of the region, enclosed by the circle \[{{x}^{2}}+{{y}^{2}}=2\] which is not common to the region bounded by the parabola \[{{y}^{2}}=x\] and the straight line \[y=x\], is                                [JEE MAIN Held on 07-01-2020 Morning]

A) \[\frac{1}{6}(24\pi -1)\]  

B) \[\frac{1}{6}(12\pi -1)\]

C) \[\frac{1}{3}(12\pi -1)\]  

D) \[\frac{1}{3}(6\pi -1)\]

Correct Answer: B

Solution :

[b] Area \[=2\pi -\int\limits_{0}^{1}{\left( \sqrt{x}-x \right)dx}\] \[=2\pi -\left[ \frac{2{{x}^{\frac{3}{2}}}}{3}-\frac{{{x}^{2}}}{2} \right]_{0}^{1}\] \[=2\pi -\left( \frac{1}{6} \right)\] \[=\frac{12\pi -1}{6}\] square units