Question

Find the area of the region enclosed between the two circles: 

x² + y² = 4 and (x – 2)² + y² = 4.

Answer 1

Equations of the given circles are

x² + y² = 4  ....(i)

and (x - 2)² + y² = 4 .....(ii)

Equation (i) is a circle with centre O at origin and radius 2. Equation (ii) is a circle with centre C(2, 0) and radius 2. Solving equation (i) and (ii), we have

Thus, the points of intersection of the given circle are A(1,√3) and A'(1, √3) as shown in the fig.

Required area of the enclosed region OACA'O between circles

= 2[area of the region ODCAO]

= 2[area of the region ODAO + area of the region DCAD]

= 2[\(\int_0^1y\,dx+\int_1^2y\,dx\)]

=2[\(\int_0^1\sqrt{4-(x-2)^2}dx+\int_1^2\sqrt{4-x^2}dx\) ]