__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

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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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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 x

_{C}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 y

_{C}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