Find the area of this region in the coordinate plane that satisfies both the inequalities 𝑥^2 + 𝑦^2 ≤ 8 and y ≤ |𝑥| ?

Answer 1

Make a sketch.
The area you want lies inside the circle, with centre at the origin and r^2 - 8, but below the lines y = x and y = -x
It is easy to see that this is simply 3/4 of the area of the circle, since the two
straight lines form a 90° angle.

so area = (3/4)π(8) = 6π units^2