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 = -4

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 =

(The answer is NOT 4 or -4 -- the two answers I keep getting for the integral I set up)

Answer 1

Let find popint x- coordinate of intrsection point: 2cox = 2sinx; tanx = 1, x = 5π/ only intersection on interval

[π, 2π]. So, A = ∫_π^5π/4(2sinx - 2cosx)dx + ∫_5π/4^2π(2cosx - 2sinx)dx = (- 2cosx - 2sinx)_π^5π/4 + (2sinx + 2cosx)_5π/4^2π = (√2 + √2) - (2 - 0) + (0 + 2) - (- √2 - √2) = 4√2;

Area A = 4√2 ≈ 5.657