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:

The requireed area is  the area of the region bounded by the circle minus the area of the region bounded by $y^2=x$ and the line $y=x$

Required area =Total area – enclosed area =  $2\pi -\underset{0}{\overset{1}{\int }}\sqrt{x}-xdx$

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

                                                                    $$\begin{gathered} =2\pi -\left(\frac{2}{3}-\frac{1}{2}\right) \\ =2\pi -\left(\frac{1}{6}\right) \\ =\frac{12\pi -1}{6} \end{gathered}$$