RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 23 of Exercise 21.1

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23

Find the area bounded by the curve y = cosx, x - axis and the ordinates x = 0 and x = 2π.

Given equations are:

y = cos x …..(i)


x - axis …..(ii)


x = 0 ……(iii)


x = …..(iv)


A table for values of y = cos x is: -



A rough sketch of the curves is given below: -


23.PNG


We have to find the area of shaded region.


Required area


= (shaded region ABOA + shaded region BCDB + shaded region DEFD)


(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 (0,) and the value of y varies)


(as y = cos x)


On integrating we get,



On applying the limits we get



= 1 - 0 + | - 1 - 1| + 0 - ( - 1) = 4


Hence the area bounded by the curve y = cosx, x - axis and the ordinates x = 0 and x = 2π is equal to 4 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