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Find the area included between the parabolasy2 = 4ax and x2 = 4by.
To find area enclosed by
Y2 = 4ax
And X2 = 4by
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