We can start by graphing the two functions:
The shaded region represents the area we are looking for. To find the bounds of integration, we need to find the intersection points of the two functions:
4sin(x) = 4cos(x)
sin(x) = cos(x)
tan(x) = 1
x = π/4 or 5π/4
Since we are only interested in the area between the two functions for π/2 ≤ x ≤ 3π/2, we need to consider only the intersection point x=5π/4.
So, we can set up the integral as follows:
Area = ∫π/2 5π/4 4sin(x) - 4cos(x) dx
= 4∫π/2 5π/4 (sin(x) - cos(x)) dx
= 4[cos(x) + sin(x)]π/2 5π/4
= 4[(0 + 1) - (1/√2 + 1/√2)]
= 4(1 - √2)/2
= 2(1 - √2)
Therefore, the area under y=4sin(x) and above y=4cos(x) for π/2 ≤ x ≤ 3π/2 is 2(1 - √2).
Find the area under y=4sin(x) and above y=4cos(x) for π2≤x≤3π2.
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