Area of region bounded by the curve `y=(4-x^(2))/(4+x^(2)), 25y^(2)=9x and y=(3)/(5)|x|-(6)/(5)` which contains (1, 0) point in its interior is

Options

A. `{pi- 4 tan^(-1).(1)/(2)+(1)/(10)}` sq. units
B. `{pi-2 tan^(-1).(1)/(2)-(1)/(5)}` sq. units
C. `{pi+4 tan^(-1).(1)/(2)-(1)/(5)}` sq. units
D. none of these

Answer

A

Explanation

Graph of the functions is shown in the following figure

`"Required area"=2int_(0)^(1)(3)/(5)sqrtxdx+int_(1)^(2){(4-x^(2))/(4+x^(2))-(3x-6)/(5)}dx`
`=(4)/(5)+int_(1)^(2){(8)/(4+x^(2))-(3x-1)/(5)}dx`
`=(4)/(5)+[4 tan^(-1).(x)/(2)]_(1)^(2)-(1)/(5)[(3x^(2))/(2)-x]_(1)^(2)`
`=(4)/(5)+pi-4 tan^(-1).(1)/(2)-(7)/(10)`
`={pi-4 tan^(-1).(1)/(2)+(1)/(10)}"sq. units"`