The area of the region bounded by `x^(2)+y^(2)-2x-3=0 and y=|x|+1` is

Options

A. `(pi)/(2)-1` sq. units
B. `2pi` sq. units
C. `4pi` sq. units
D. `pi//2` sq. units

Answer

A

Explanation

`x^(2)+y^(2)-2x-3=0`
`"or "(x-1)^(2)+y^(2)=4`
`A=overset(0)underset(1-sqrt(2))int(sqrt(4-(x-1)^(2))-(-x+1))dx+overset(1)underset(0)int(sqrt(4-(x-1)^(2))-(x+1))dx`
`=(x-1)/(2)sqrt(4-(x-1)^(2))+(4)/(2)sin^(-1)""(x-1)/(2)+(x^(2))/(2)-x:|_(1-sqrt(2))^(0)+(x-1)/(2)sqrt(4-(x-1)^(2))+(4)/(2)sin^(-1)""(x-1)/(2)-(x^(2))/(2)-x:|_(0)^(1)`
`=(-(sqrt(3))/(2)-(pi)/(3))-((-sqrt(2))/(2)sqrt(2)-(pi)/(2)+(3-2sqrt(2))/(2)-1+sqrt(2))+(-(1)/(2)-1)-(-(sqrt(3))/(2)-(pi)/(3))`
`=-(-1-(pi)/(3)+(3)/(2)-sqrt(2)-1+sqrt(2))-(3)/(2)`
`=(pi)/(2)-1` sq. units.