Question

Find the area of the region included between parabolas y^2 = 4ax

Answer 1

√4ax is the area of the region included between parabolas y^2 = 4ax

Answer 2

We have, y2 = 4ax --------------------------- (1)

x2 = 4ay ---------------------------- (2)

(1) and (2) intersects hence

x = y2/4a (a > 0)

=> (y2/4a)2 = 4ay

=> y4 = 64a3y

=> y4 – 64a3y = 0

=> y[y3 – (4a)3] = 0

=> y = 0, 4a

When y = 0, x = 0 and when y = 4a, x = 4a.

The points of intersection of (1) and (2) are O(0, 0) and A(4a, 4a).

The area of the region between the two curves

= Area of the shaded region

= 0∫4a(y1 – y2)dx

= 0∫4a[√(4ax) – x2/4a]dx

= [2√a.(x3/2)/(3/2) – (1/4a)(x3/3)]04a

= 4/3√a(4a)3/2 – (1/12a)(4a)3 – 0

= 32/3a2 – 16/3a2

= 16/3a2 sq. units.