The area enclosed by the circle $x^2 + y^2 = 2$ excluding the area bounded by $y = x$ and $y^2 =x$, is

$\frac{1}{6}(12π-1)$

Clearly, required area A is given by

A = Area of the circle - Area of the shaded region.

$⇒A=2π-\int\limits_0^1(y_2-y_1)dx$

$⇒A=2π-\int\limits_0^1(\sqrt{x}-x)dx=2π-\left[\frac{2x^{3/2}}{3}-\frac{x^2}{2}\right]_0^1$

$=2π-\frac{1}{6}=\frac{1}{6}(12π-1)$