Book: RD Sharma - Mathematics (Volume 2)
Chapter: 21. Areas of Bounded Regions
Subject: Maths - Class 12th
Q. No. 25 of Exercise 21.1
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Compare the areas under the curves y = cos2x and y = sin2 x between x = 0 and x = π.
Given equations are:
y = cos2x …..(i)
y = sin2x …..(ii)
x = 0 ……(iii)
x = 
…..(iv)
A table for values of y = cos2x and y = sin2x is: -
A rough sketch of the curves is given below: -
The area under the curve y = cos2x, x = 0 and x = 
is
A1 = (area of the region OABCO + area of the region CEFGC)
A1 = 2(area of the region CEFGC)
(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)
Apply reduction formula:
On integrating we get,
On applying the limits we get
The area under the curve y = cos2x, x = 0 and x = 
is
A2 = (area of the region OBDGEO)
(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)
Apply reduction formula:
On integrating we get,
On applying the limits we get
Hence A1 = A2
Therefore the areas under the curves y = cos2x and y = sin2 x between x = 0 and x = π are equal.
