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Find the centroid of the area bounded by two curves for
Solution:
The first thing that we have to do is to draw or sketch the two given curves using the principles of Analytic Geometry as follows
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Next, we need to get their points of intersection by solving the two equations, two unknowns as follows
but
The above equation becomes
Equate each factor to zero and solve for the value of y. Therefore, y = 2 and y = -1.
Substitute the values of y either of the two equation in order to solve for the value of x, we have
If y = 2, then
If y = -1, then
Their points of intersection are (1, -1) and (4, 2).
Label further the figure and draw the horizontal strip, we have
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Photo by Math Principles in Everyday Life |
The area bounded by the two curves is
The x value of the centroid for the figure bounded by two curves is given by the formula
If the length of a strip is x, then xC is ½ x. The above equation becomes
Therefore,
The y value of the centroid for the figure bounded by two curves is given by the formula
If the length of a strip is x, then yC is also equal to y which is the distance of a strip from x axis. Since dy is a very small measurement, then dy is negligible. The above equation becomes
Therefore,
Therefore, the coordinates of the centroid are