Although this question is marked as a Trigonometry question, the solution requires calculus to solve.
You are given two curves and want to find the space between them, as depicted by this graph.
Let f(x) = 4sin(x).
Let g(x)=4cos(x).
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The x-coordinate of the left hand side intersection is a=pi/4.
The x-coordinate of the right hand side intersection is b=(5pi)/4.
The positive area above the x-axis is equal to the negative area below the x-axis and would cancel each other out. So we must stick to finding the area above the x-axis and then double that area.
One strategy would be to find the area, A under f(x)=4sin(x) between x=pi/4 and x=pi and then subtract off the area, B, of g(x)=4cos(x) between x=pi/4 and x=pi/2. Finally, we would need to double that area to find the total area, A_"Total", both above and below the x-axis. In other words, we are finding 2(A-B).
Area A is
A=int_(pi/4)^pi4sin(x)dx=[-4cos(x)]_(pi/4)^pi
= 4+2sqrt(2)
Area B is
B=int_(pi/4)^(pi/2)4cos(x)dx=[4sin(x)]_(pi/4)^(pi/2)
B=4-2sqrt(2)
Therefore, the total area A_"Total" is
A_"Total"=2(A-B)
" "=2(4+2sqrt(2)-4+2sqrt(2))
" "=8sqrt(2)~~11.31
