RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 24 of Exercise 21.1

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24

Show that the areas under the curves y = sin x and y = sin 2x between x = 0 and are in the ration 2:3.

Given equations are:

y = sin x …..(i)


y = sin 2x …..(ii)


x = 0 ……(iii)


x = …..(iv)


A table for values of y = sin x and y = sin 2x is: -



A rough sketch of the curves is given below: -


24.PNG


The area under the curve y = sin x , x = 0 and is


A1 = (area of the region OPBCA)


(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)


(as y = sin x)


On integrating we get,



On applying the limits we get




The area under the curve y = sin 2x , x = 0 and is


A2 = (area of the region OABCO)


(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)


(as y = sin 2x)


On integrating we get,



On applying the limits we get




So the ratio of the areas under the curves y = sin x and y = sin 2x between x = 0 and are



Hence showed


24

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