## Book: RD Sharma - Mathematics (Volume 2)

### Chapter: 21. Areas of Bounded Regions

#### Subject: Maths - Class 12th

##### Q. No. 16 of Exercise 21.3

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16
##### 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

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