The area of the region is

We need to determine what region is being talked about.

x^{2} + y^{2} = 1 is a circle with centre at origin and unit radius.

So, x^{2} + y^{2} ≤ 1 represents the region inside that circle.

x + y = 1 is a line that intersects the 2 axes at unit distances from the origin, in the positive direction.

So x + y ≥ 1 represents all the points, i.e., the region above the line x + y = 1.

So, the region {(x, y): x^{2} + y^{2} ≤ 1 ≤ x + y} is –

For each point (x, y) on circle in first quadrant, y = √(1 – x^{2})

For each point (x, y) on line, y = 1 – x

So, area A of the region described is –

A =

=

= π/4 – 1/2 (Ans)

1