Find the area of the region bounded by the parabola y² = 2px, x² = 2py

In y² = 2px 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 = 2py 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 = 2px and x² = 2py simultaneously

Put in y² = 2px

⇒ x⁴ = 8p³x

⇒ x³ = 8p³

⇒ x = 2p

Put x = 2p in y² = 2px

⇒ y² = 2p(2p)

⇒ y = 2p

Hence the intersection point of two parabola is (2p, 2p)

We require the area between the two parabolas

⇒ area bounded by two parabolas given = area under parabola y² = 2px – area under parabola x² = 2py …(i)

Let us find area under parabola y² = 2px

⇒ y = √2p√x

Integrate from 0 to 2p

Now let us find area under parabola x² = 2py

⇒ x² = 2py

Integrate from 0 to 2p

Using (i)

⇒ area bounded by two parabolas given =

⇒ area bounded by two parabolas given =

Hence area is unit²