If the area (in sq. units) of the region
`{(x,y): y^(2) lt= 4x,x+y lt=1,xgt=0,ygt=0}` is `asqrt2+b`, then `a-b` is equal to :

Options

A. `(8)/(3)`
B. `(10)/(3)`
C. `-(2)/(3)`
D. 6

Answer

D

Explanation

`A=int_(0)^(2sqrt2-2)(1-y(y^(2))/(4))dy`
`A=[y-(y^(2))/(2)-(y^(3))/(12)]_(0)^(2sqrt2-2)=y[1-(y)/(2)-(y^(2))/(12)]`
`=2(sqrt2-1)[1-(sqrt2-1)-(4)/(12)(sqrt2-1)^(2)]`
`=2(sqrt2-1)[2-sqrt2-(1)/(3)(2+1-2sqrt2)]`
`=2(sqrt2-1)[1-(sqrt2)/(3)]=2(sqrt2-(2)/(3)=1+(sqrt3)/(3))`
`=2((4)/(3)sqrt2-(5)/(3))`