Question

Draw a rough sketch and find the area of the region bounded by the two parabolas y² = 4x and x² = 4y by using methods of integration.

Answer 1

To find the area bounded by

y² = 4x

y = \(2\sqrt{x}\) ....(i)

And x² = 4y

y = \(\frac{x^2}{4}\) ....(ii)

On solving the equation (i) and (ii),

\(\big(\frac{x^2}{4}\big)^2\) = 4x

Or, x⁴ – 64x = 0

Or, x(x³ – 64) = 0

Or, x = 0, 4

Then y = 0, 4

Equation (i) represents a parabola with vertex (0, 0) and axis as x – axis. Equation (ii) represents a parabola with vertex (0, 0) and axis as y - axis.

Points of intersection of the parabola are (0, 0) and (4, 4).

A rough sketch is given as: -

Now the bounded area is the required area to be calculated, Hence,

Bounded Area, A = [Area between the curve (i) and x axis from 0 to 4] – [Area between the curve (ii) and x axis from 0 to 4]

On integrating the above definite integration,

Area of the region bounded by the parabolas y² = 4x and x² = 4y is \(\frac{16}{3}\) sq. units.