The area of the region (in sq. units), enclosed by the circle x 2 + y 2 = 2 which is not common to the…

1. $\frac{1}{6}\left(24\pi -1\right)$
2. $\frac{1}{3}\left(6\pi -1\right)$
3. $\frac{1}{3}\left(12\pi -1\right)$
4. $\frac{1}{6}\left(12\pi -1\right)$

Solution

Required area

$=2\pi -{\int }_0^1\left(\sqrt{x}-x\right)dx$

$=2\pi -{\left(\frac{2x^{\frac{3}{2}}}{3}-\frac{x^2}{2}\right)}_0^1$

$=2\pi -\left(\frac{2}{3}-\frac{1}{2}\right)$

$=2\pi -\left(\frac{1}{6}\right)$

$=\frac{12\pi -1}{6} \mathrm{sq}.\mathrm{units}$

Asked in: JEE Main 2020 (07 Jan Shift 1)

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