RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 28 of Exercise 21.1

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28

Find the area of the region bounded by the curve x = at2, y = 2at between the ordinates corresponding t = 1 and t = 2

Given equations are:

x = at2 ...... (1)


y = 2at ..... (2)


t = 1 ..... (3)


t = 2 ..... (4)


Equation (1) and (2) represents the parametric equation of the parabola.


Eliminating the parameter t, we get



This represents the Cartesian equation of the parabola opening towards the positive x - axis with focus at (a,0).


A rough sketch of the circle is given below: -


28.PNG


When t = 1, x = a


When t = 2, x = 4a


We have to find the area of shaded region.


Required area


= (shaded region ABCDEF)


= 2(shaded region BCDEB)


(the area can be found by taking a small slice in each region of width Δx, then the area of that sliced part will be yΔx as it is a rectangle and then integrating it to get the area of the whole region)


(As x is between and the value of y varies, here y is Cartesian equation of the parabola)


(as )



On integrating we get,


(by applying power rule)


On applying the limits we get,





Hence the area of the region bounded by the curve x = at2, y = 2at between the ordinates corresponding t = 1 and t = 2 is equal to square units.


Chapter Exercises

More Exercise Questions

7

Sketch the graph of in [0,4] and determine the area of the region enclosed by the curve, the x - axis and the lines x = 0, x = 4