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# How do you find the center of mass with calculus?

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To determine the center of mass using calculus, the area and moments about x and y must be determined. The moments are divided by the area to give the coordinates of the center of mass.

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1. Determine the area of the given curve

First, find the area of the given curve. This can be done by evaluating the integral: A = |_(a..b) f(x) dx. If the area is between two curves, the area is found by evaluating the integral: A = |_(a..b) (f(x) - g(x)) dx.

2. Find the moments about x and y

To find the moment about x, use the integral: M_x = |_(a..b) ((1/2) * (f(x))^2) dx. The equation for finding the moment about y is: M_y = |_(a..b) (x * f(x)) dx. If the area is between two curves, the moments about x and y are found with the following equations: M_x = |_(a..b) ((1/2) * ((f(x))^2 - (g(x))^2)) dx, and M_y = |_(a..b) (x * (f(x) - g(x))) dx.

3. Find the coordinates for the center of mass

The coordinates for the center of mass are found by dividing the moments by the area. The x coordinate is found by the equation: x_bar = M_y / A. The y coordinate is found by the equation y_bar = M_x / A. The center of mass is (x_bar, y_bar).

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