Compare the areas under the curves y = cos^{2}x and y = sin^{2} x between x = 0 and x = π.

Given equations are:

y = cos^{2}x …..(i)

y = sin^{2}x …..(ii)

x = 0 ……(iii)

x = …..(iv)

A table for values of y = cos^{2}x and y = sin^{2}x is: -

A rough sketch of the curves is given below: -

The area under the curve y = cos^{2}x, x = 0 and x = is

A_{1} = (area of the region OABCO + area of the region CEFGC)

A_{1} = 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 = cos^{2}x, x = 0 and x = is

A_{2} = (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 A_{1} = A_{2}

Therefore the areas under the curves y = cos^{2}x and y = sin^{2} x between x = 0 and x = π are equal.

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