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: -


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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.


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