Question

Find the area of the region included between:

y² = 4x and y = x.

Answer 1

The vertex of the parabola y² = 4x is at the origin O = (0, 0).

To find the points of intersection of the line and the parabola, equaling the values of 4x from both the equations we get,

∴ y² = y 

∴ y – y = 0 

∴ y(y – 1) = 0 

∴ y = 0 or y = 1

When y = 0, x = 0/2 =0

When y = 1, x = 1/2

∴ the points of intersection are O(0, 0) and B(1/2,1)

Required area = area of the region OABCO = area of the region OABDO – area of the region OCBDO

Now, area of the region OABDO = area under the parabola y² = 4x between x = 0 and x = 1/2

Area of the region OCBDO = area under the line y = 2x between x = 0 and x = 1/2

\(\therefore\) required area