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=π4.
The x-coordinate of the right hand side intersection is b=5π4.
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=π4 and x=π and then subtract off the area, B, of g(x)=4cos(x) between x=π4 and x=π2. Finally, we would need to double that area to find the total area, ATotal, both above and below the x-axis. In other words, we are finding 2(A−B).
Area A is
A=∫ππ44sin(x)dx=[−4cos(x)]ππ4
=4+2√2
Area B is
B=∫π2π44cos(x)dx=[4sin(x)]π2π4
B=4−2√2
Therefore, the total area ATotal is
ATotal=2(A−B)
=2(4+2√2−4+2√2)
=8√2≈11.31
