The area of the region bounded by parabola `y^(2)=x` and the straight line `2y = x` is

Options

A. `4/3"sq units"`
B. `1" sq unit "`
C. `2/3 " sq unit"`
D. `1/3" sq units"`

Answer

A

Explanation

We have to find the area enclosed by parabola`y^(2)=x` and the straight line `2y=x`.

`:. (x-2)^(2)=x`
`rArr x^(2)=4xrArrx(x-4)=0`
`rArr x=4rArry=2" and "x=0rArry=0`
So, the intersection points are (0,0) and (4,2).
Area enclosed by shaded region,
`A=int_(0)^(4)[sqrtx-x/2]dx`
`=[(c^(1/2+1))/(1/2+1)-1/2 . x^(2)/2]_(0)^(4)=[2. x^(3//2)/3-x^(2)/4]_(0)^(4)`
`=2/3 4^(3//2)-16/4-2/3 .0+1/4 .0`
`=16/3-16/4=(64-48)/12=16/12=4/3 " sq units"`