Find the area included between the parabolasy^{2} = 4ax and x^{2} = 4by.

To find area enclosed by

Y^{2} = 4ax

...(i)

And X^{2} = 4by

...(ii)

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

Or, x^{4} – 64ab^{2}x = 0

Or, x(x^{3} – 64ab^{2}) = 0

Or, x = 0 and x =

Then y = 0 and y =

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 x - axis,

Points of intersection of parabolas are O (0,0) and

These are shown in the graph below: -

The shaded region is required area, and it is sliced into rectangles of width and length (y^{1} – y^{2})ΔX.

This approximation rectangle slides from x = 0 to , so

Required area = Region OQCPO

The area included between the parabolasy^{2} = 4ax and x^{2} = 4by is

16