The area of the region described by $A = \{(x, y): x^2 + y^2 <1\, and\, y^2 ≤1-x\}$, is

$\frac{π}{2}+\frac{4}{3}$

Clearly,

Required area = Area of the circle - Area of the shaded region

$=π (1)^2 -2\int\limits_0^1 (\sqrt{1 − x^2} -\sqrt{1−x}) dx$

$=π -2\left[\frac{1}{2}x\sqrt{1 − x^2}+\frac{1}{2}\sin^{-1}(\frac{x}{1})+\frac{2}{3}(1-x)^{3/2}\right]_0^1$

$=π -2\left\{\frac{π}{4}+\frac{4}{3}\right\}=\frac{π}{2}+\frac{4}{3}$