Find the area included between the line `y=x` and the parabola `x^2=4y`.

Options

A. `(8)/(3)` sq units
B. `(4)/(3)` sq units
C. `(9)/(4)` sq units
D. `(7)/(3)` sq units

Answer

A

Explanation

Equation of parabola is `x^(2)=4y` and equation of line is y = x.
Solving, we get `x^(2)=x`
or `x(x-4)=0`
`therefore x = 0, 4`
`therefore` Line y = x cuts parabola at two points O and

B, x-coordinate of O is 0 and x-coordinate of B is 4.
Required area = area OCBAO
`=int_(0)^(4)(y_(1)-y_(2))dx=int_(0)^(4)(x-(x^(2))/(4))dx`
`=[(x^(2))/(2)-(x^(3))/(12)]_(0)^(4)=[(16)/(2)-(64)/(12)]=(8)/(3)` sq units