Smaller area enclosed by the circle `x^2+y^2=4`and the line `x + y = 2`is(A) `2(pi-2)` (B) `pi-2` (C) `2pi-1` (D) `2(pi+2)`

Options

A. `2(pi - 2)`
B. `pi-2`
C. `2pi-1`
D. `2(pi+2)`

Answer

B

Explanation

The smaller bounded by the circle `x^(2)+y^(2)=4` and the line `x+y=2` is shown by the shaded region ACBA.

The points of intersection of the circle and line are A(2, 0) and B(0, 2).
` :.` Required area (shaded region).
`=ar(OACBO)-ar(triangle OAB)`
`=int_(0)^(2)sqrt(4-x^(2))dx-int_(0)^(2)(2-x)dx`
`=[(x)/(2)sqrt(4-x^(2))+(4)/(2)"sin"^(-1)(x)/(2)]_(0)^(2)-[2x-(x^(2))/(2)]_(0)^(2)`
`=[(2)/(2)sqrt(4-4)+(4)/(2)"sin"^(-1)(1)-0-(4)/(2)"sin"^(-1)(0)]-[4-2]`
`=[2*(pi)/(2)]-[4-2]=(pi-2)` sq. units.