Find the area bounded by curves `(x-1)^2+y^2=1`and `x^2+y^2=1`.

Explanation

Curve `(x-1)^(2)+y^(2)=1 " " ` …(1)
` :. y=sqrt(1-(x-1)^(2))`
Represents a circle whose centre is (1, 0) and radius is 1.

and curve, `x^(2)+y^(2)=1 " " ` ...(2)
` :. y=sqrt(1-x^(2))`
Represents a circle whose centre is (0, 0) and radius is 1.
Two circles meet at `(x-1)^(2)=x^(2)`
`implies 2x=1impliesx=(1)/(2)`
` :.` Required area (shaded region)
`=2[int_(0)^(1//2)y_(1)dx+int_(1//2)^(1)y_(2)dx]`
`=2[int_(0)^(1//2)sqrt(1-(x-1)^(2))dx+int_(1//2)^(1)sqrt(1-x^(2))dx]`
`=2[(x-1)/(2)sqrt(1-(x-1)^(2))+(1)/(2)"sin"^(-1)(x-1)/(1)]_(0)^(1//2)+2[(x)/(2)sqrt(1-x^(2))+(1)/(2)"sin"^(-1)x]_(1//2)^(1)`
`=2[((1)/(2)-1)/(2)sqrt(1-(1)/(4))+(1)/(2)"sin"^(-1)(-(1)/(2))-((-1)/(2))0-(1)/(2)"sin"^(-1)(-1)]+2[0+(1)/(2)"sin"^(-1)(1)-(1)/(4)sqrt(1-(1)/(4))-(1)/(2)"sin"^(-1)(1)/(2)]`
`=2[-(1)/(4)*(sqrt(3))/(2)-(1)/(2)*(pi)/(6)+0+(1)/(2)*(pi)/(2)]+(pi)/(2)-(1)/(2)*(sqrt(3))/(2)-(pi)/(6)`
`=-(sqrt(3))/(4)-(pi)/(6)+(pi)/(2)+(pi)/(2)-(sqrt(3))/(4)-(pi)/(6)`
`=((2pi)/(3)-(sqrt(3))/(2))` sq. units.