Find the area included between the parabolasy2 = 4ax and x2 = 4by.

To find area enclosed by

Y2 = 4ax


...(i)


And X2 = 4by


...(ii)


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



Or, x4 – 64ab2x = 0


Or, x(x3 – 64ab2) = 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 (y1 – y2)ΔX.


This approximation rectangle slides from x = 0 to , so


Required area = Region OQCPO








The area included between the parabolasy2 = 4ax and x2 = 4by is


16