Sketch the region bounded by the curves `y=sqrt(5-x^2)` and `y=|x-1|` and find its area.

Options

A. `((5pi)/(4)-2)` sq units
B. `((5pi-2))/(4)` sq units
C. `((5pi-2))/(2)` sq units
D. `((pi)/(2)-5)` sq units

Answer

B

Explanation

Given, `y=sqrt(5-x^(2))andy=|x-1|`
or `y^(2)+x^(2)=5 and y=|x-1|`

`therefore` Required area
`=int_(-1)^(2)sqrt(5-x^(2))dx-int_(-1)^(1)(1-x)dx-int_(1)^(2)(x-1)dx`
`=[(x)/(2)sqrt(5-x^(2))+(5)/(2)sin^(-1).(x)/(sqrt(5))]_(-1)^(2)-[x-(x^(2))/(2)]_(-1)^(1)-[(x^(2))/(2)-x]_(1)^(2)`
`=[1+(5)/(2)sin^(-1).(2)/(sqrt(5))+1+(5)/(2)sin^(-1).(1)/(sqrt(5))]-[1-(1)/(2)-(-1-(1)/(2))]-[2-2-((1)/(2)-1)]`
`=2+(5)/(2)sin^(-1).((2)/(sqrt(5))sqrt(1-(1)/(5))+(1)/(sqrt(5))sqrt(1-(4)/(5)))-(5)/(2)`
`=(5)/(2)sin^(-1)(1)-(1)/(2)=(5pi)/(4)-(1)/(2)=((5pi-2)/(4))` sq units