The area (in square units) of the region enclosed by the two circles x^2+y^2=1 and (x-1)^2+y^2=1 is

Step-by-step solution

Intersection point of two circles x^2+y^2=1 (i) (x-1)^2+y^2=1 (ii) is given by (x-1)^2+ (1-x^2 )=1 x^2+1-2 x-x^2=0 x= 1 2 From Eq. (i), 1 4 +y^2=1y^2=1- 1 4 y= 3 2 Point A ( 1 2 , 3 2 ) and C ( 1 2 , - 3 2 ) So, Area of region O A B C O=2 Area of region O A B D O Area of O A B D O= Area of O A D O+ Area of A B D A= ₀^ 1 / 2 1-(x-1)^2 d x+ _ 1 / 2 ^1 1-x^2 d x= [ 1 2 (x-1) 1-(x-1)^2 + 1 2 ⁻¹ ( x-1 1 ) ]₀^ 1 / 2 + [ 1 2 x 1-x^2 + 1 2 ⁻¹ ( x 1 ) ]_ 1 / 2 ^1= [- 1 2 1 2 3 2 + 1 2 ⁻¹ ( -1 2 )- 1 2 0- 1 2 ⁻¹(-1) ]+ [ 1 2