The area of the region containing the points (x, y) satisfying $4\le x^2+y^2\le 2(\left|x\right|+\left|y\right|)$

The points in the required region satisfy $4\le x^2+y^2\le 2(\left|x\right|+\left|y\right|)$

Since the curve (1) is symmetrical about both the axes, the required area is 4 times the area of the region in the first quadrant. Therefore, it is sufficient to sketch the region and to find the area in the first quadrant.

In the first quadrant, the curve (1) consist of two curves $x^2+y^2\ge 4$ and $x^2+y^2-2x-2y\ge 0$

$\therefore$ Required area = (area ABCDA)

= 4 (area of semi-circle ABCDA) – (area of sector ADCA)

= 4 (area of semi – circle ABCA) – (area of sector OADCO – area of 

 triangle OAC)