Find the area of the region bounded by the curve y² = 4x, x² = 4y.

In y² = 4x parabola it is not defined for negative values of x hence the parabola will be to the right of Y-axis passing through (0, 0)

Similarly for x² = 4y parabola it is not defined for negative values of y hence parabola will be above X-axis opening upwards and passing through (0, 0)

Let us find the point of intersection by solving the equations y² = 4x and x² = 4y simultaneously

Put in y² = 4x

⇒ x⁴ = 64x

⇒ x³ = 64

⇒ x = 4

Put x = 4 in y² = 4x

⇒ y² = 4(4)

⇒ y = 4

Hence the intersection point of two parabola is (4, 4)

We require the area between the two parabolas

⇒ area bounded by two parabolas given = area under parabola y² = 4x – area under parabola x² = 4y …(i)

Let us find area under parabola y² = 4x

⇒ y = 2√x

Integrate from 0 to 4

Now let us find area under parabola x² = 4y

⇒ x² = 4y

Integrate from 0 to 4

Using (i)

⇒ area bounded by two parabolas given =

⇒ area bounded by two parabolas given =

Hence area is unit²