The area (in sq units) of the region described by `{(x,y):y^(2)le2x and yge4x-1}` is

Options

A. `(7)/(32)`
B. `(5)/(64)`
C. `(15)/(64)`
D. `(9)/(32)`

Answer

D

Explanation

Given region is `{(x,y):y^(2)le2x and yge4x-1}y^(2)le2x` represents a region inside the parabola
`y^(2)=2x" … (i)"`
and `yge4x-1` represents a region to the left of the line
`y = 4x - 1 " … (ii)"`
The point of intersection of the curve (i) and (ii) is
`(4x-1)^(2)=2x`
`rArr 16x^(2)+1-8x=2x`
`rArr16x^(2)-10x+1=0`
`rArrx=(1)/(2),(1)/(8)`
At `x = (1)/(2), y = 1`
At `x=(1)/(8), y=(-1)/(2)`
`therefore` The points where these curves intersect, are `((1)/(2),1)` and `((1)/(8),-(1)/(2))`

Hence, required area `=int_(-1//2)^(1)((y+1)/(4)-(y^(2))/(2))dy`
`=(1)/(4)((y^(2))/(2)+y)_(-1//2)^(1)-(1)/(6)(y^(3))_(-1//2)^(1)`
`=(1)/(4){((1)/(2)+1)-((1)/(8)-(1)/(2))}-(1)/(6){1+(1)/(8)}`
`=(1)/(4){(3)/(2)+(3)/(8)}-(1)/(6){(9)/(8)}=(1)/(4)xx(15)/(8)-(3)/(16)=(9)/(32)`