Find the area between y=cosx and y=sinx from 0 to 2pi.
To find the zeros, I combined the equations cosx-sinx=0
What's next?
Watch out.
Between 0 and 45 degrees, the cos is bigger than the sin.
Between 45 and 90, the sin is bigger than the cos.
In both sectors, there is area between them. The same in fact.
Therefore for quadrant 1, do 0 to pi/4 and double the result.
Then look at the other three quadrants.
integral cos x dx from 0 to pi/4 = sin pi/4 - sin 0
= sqrt(2)/2
integral sin x dx from 0 to pi/4 = -cos pi/4 + 0 = -sqrt(2) /2
difference = sqrt 2
then integral from 0 to pi/2 = 2 sqrt 2
etc
Oh, forgot you do not know where sin = cos
sure cos x - sin x = 0
1 -tan x = 0
tan x = 1
x = pi/4 (x = y at 45 deg)
To find the area between the curves y = cos(x) and y = sin(x) from 0 to 2π, you need to evaluate the integral of the absolute difference between the two functions over that interval.
Since you have already combined the equations cos(x) - sin(x) = 0, you are looking for the points where cos(x) = sin(x). To find those points, you can rearrange the equation as cos(x) - sin(x) = 0, or sin(x) = cos(x).
To find the points where sin(x) = cos(x), you can use the trigonometric identity tan(x) = sin(x) / cos(x). If tan(x) = 1, then sin(x) = cos(x).
The solutions to tan(x) = 1 are x = π/4 and x = 5π/4, which are the points where the curves y = cos(x) and y = sin(x) intersect between 0 and 2π.
The next step is to evaluate the integral of the absolute difference between the two functions, which gives the area between them. The integral can be set up as follows:
∫[0,2π] |cos(x) - sin(x)| dx,
where the limits of integration are from 0 to 2π. This integral calculates the area between the curves from x = 0 to x = 2π.
To evaluate this integral, you can break it into two separate integrals:
∫[0,π/4] (cos(x) - sin(x)) dx + ∫[π/4 , 2π] (sin(x) - cos(x)) dx.
Calculate each integral separately using standard integration techniques, and then add the two results together to find the total area between the curves from x = 0 to x = 2π.