f(x) = 6*sin(2x)
g(x) = 6*cos(2x)
Here is the graph of f(x) and g(x). They intersect at point A. The x coordinate of point A is pi/8 = 0.39 approximately
The region between the curves, from x = 0 to x = pi/8, is shown by the light blue shading
Note: The fact that f(pi/8) = g(pi/8) = 3*sqrt(2) indicates that we would have a fully enclosed region without the need for the right boundary of x = pi/8, so it's a bit redundant.
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The formulas we'll use can be found here. Scroll down til you reach the "Center of Mass Coordinates" section. The formulas in the blue box below are
which represent the coordinates of the centroid. The value of A is the area between the two curves, so,
Because the red g(x) curve is above the green f(x) curve all throughout the interval 0 < x < pi/8, this means that we must swap the locations and f(x) and g(x) when we subtract. So we should have these three formulas instead
This is to ensure A is positive and the centroid coordinates end up in the right spot.
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We need to find the area A. Using numerical integration, I get
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Now use this to find the x coordinate of the centroid (xbar =
Again I'll use numerical integration to make things go quicker and more efficient
Do the same for the y coordinate of the centroid (ybar =
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We found that
Therefore, the centroid's location is approximately (0.13365, 3.62132)
Here is an updated graph with the centroid point C added in
