Compare the areas under the curves y=cos2x and y=sin2x between x=0 and x=π.
Given
• First curve y = cos2 x
• Second curve y = sin2 x
• x= 0
• x= π
Consider the curve y = cos2 x
Now consider the y values for some random x values between 0 and π for the function y = cos2 x.
From the table we can clearly draw the graph for y = cos2 x
The required area under the curve is given by:
[using the property cos 2x = 2 cos2 x - 1]
[using the formula, ]
[as sin 2π = 0, sin 0 = 0]
Hence the required area of the curve y = cos2 x from x = 0 to x=π is sq. units. ------ (1)
Consider the curve y = sin2 x
Now consider the y values for some random x values between 0 and π for the function y = sin2 x.
From the table we can clearly draw the graph for y = sin2 x
The required area under the curve is given by:
[using the property cos 2x = 1- 2 sin2 x]
[using the formula, ]
[as sin 2π = 0, sin 0 = 0]
Hence the required area of the curve y = sin2 x from x = 0 to x=π is sq. units. ----- (2)
From (1) and (2), we can clearly state that, the areas under
y = cos2 x and y = sin2 x are similar which is sq. units.