Question

Find the area of the region included between:

y² = x and the line y = 4x.

Answer 1

The vertex of the parabola y = x² is at the origin 0(0, 0)

To find the points of the intersection of a line and the parabola.

Equating the values of y from the two equations, we get

x² = 4x 

∴ x – 4x = 0 

∴ x(x – 4) = 0 

∴ x = 0, x = 4

When x = 0, y = 4(0) = 0 

When x = 4, y = 4(4) = 16

∴ the points of intersection are 0(0, 0) and B(4, 16)

Required area = area of the region OABCO = (area of the region ODBCO) – (area of the region ODBAO)

Now, area of the region ODBCO = area under the line y = 4x between x = 0 and x = 4

= 4\(\int_0^4xdx\)

= 2(16 – 0)

= 32

Area of the region ODBAO = area under the parabola y = x² between x = 0 and x = 4