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Tuesday, March 26, 2013

Centroid - Area, 2

Category: Integral Calculus, Analytic Geometry, Algebra, Physics, Mechanics

"Published in Newark, California, USA"

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


Photo by Math Principles in Everyday Life

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


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