Question

Find the area of the region $\left\{\right(x,y):x^2+y^2\le 4,x+y\ge 2\}.$

Answer 1

$x^2+y^2\le 4$ denotes the area inside the circle, whose center is $(0,0)$ and radius is $2$ units.

The inequality $x+y\ge 2$ denotes the region on the right hand side of the line $x+y=2$, and the line intersects the coordinate axes at $(0,2)$ and $(2,0)$

This can be obtained by subtracting the area of a quarter circle and the right angled triangle which forms.

$\therefore \frac{\pi \times r^2}{4}-\frac{1}{2}\times r\times r$ is the required area, where r is the radius of the circle.

$=\frac{\pi \times 4}{4}-\frac{4}{2}$

$=\pi -2$

$\approx 1.142$ square units.