Question

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

• A. $\frac{1}{3}(12\pi -1)$
• B. $\frac{1}{6}(12\pi -1)$
• C. $\frac{1}{3}(6\pi -1)$
• D. $\frac{1}{6}(24\pi -1)$

Answer 1

The correct option is B $\frac{1}{6}(12\pi -1)$

Required area
$=$ Area of the circle $–$ Area bounded by given line and parabola

Required area $=\pi (\sqrt{2}{)}^2-\underset{0}{\overset{1}{\int }}(y-y^2)dy$
$=2\pi -{[\frac{y^2}{2}-\frac{y^3}{3}]}_0^1=2\pi -\frac{1}{6}$
$=\frac{1}{6}(12\pi -1)$ sq. units