Find the area under y=2sin(x) and above y=2cos(x) for π≤x≤2π. (Note that this area may not be defined over the entire interval.)

Question

Find the area under y=2sin(x) and above y=2cos(x) for π≤x≤2π. Note that this area may not be defined over the entire interval.

area =

(I am struggling to find the answer for where the portion of the integral it is defined...)

Answer 1

Since there is a 2 on both, think about the unit circle in terms of radians and an x = y line.

That would show you that y > x on [π, 2π] only on [π, 5/4 π)

Thus, we integrate 2 sin x - 2 cos x on [π, 5/4 π)

Using I[a,b] for the integral from a to b and E[a,b] for the evaluation from a to b

I[π, 5/4 π] 2 sin x - 2 cos x dx = -2 cos x - 2 sin x E[π, 5/4 π] = -2 * -√2/2 - 2 * -√2/2 - (-2 * -1 - 2 * 0) =

2√2 - 2