Using integration, find the area of the region
`{(x,y) : x^(2) +y^(2) le 1, x+y ge 1, x ge 0, y ge 0}`

Explanation

`X^(2) + y^(2) = 1 ……………..(1)`
`x+y = 1…………………(2)`
Sloving (1) and (2)
`x^(2) + (1-x)^(2) = 1`
` x^(2) +x^(2) - 2x +1=1 `
` 2x^(2) - 2x = 0`
`2x(x-1) =0`
` x =0 or x =1`

Required area= shaded area ACBDA = area (OACBO) - area (OADBO)
` underset(0)overset(1)int(y_("circle") - y_("line"))dx`
` underset(0)overset(1)intsqrt(1-x^(2)dx)- underset(0)overset(1)int(1-x)dx`
`=[(xsqrt(1-x)^(2))/(2)+(1)/(2)sin^(-1)x]_(0)^(1)`
`[(0+(1)/(2).(pi)/(2))-0]-[(1-(1)/(2))]`
`((pi)/(4)-(1)/(2))` square units