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center of mass of sheet metal with a curved edge

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A thin, uniform piece of sheet metal in an x,y plane has one side along the x axis from (0, 0) to (3, 0), and a vertical side from (3,0) to (3, 5.4). The remaining side is curved between (0,0) and (3, 5.4) on the line y= .6 x^2.

Find xcm, the x coordinate of the c.m. SEE ATTACHMENT #1 for a diagram of the object and parameters. Distances are in meters.

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Step 1.
Each mass element dm, contributes an amount x dm to the numerator of the general equation (2) (on ATTACHMENT #2), for the x coordinate of a center of mass.
The best first step of a solution is a careful diagram showing parameters. Carefully observe ATTACHMENT #2 which shows the element of area dA whose mass is dm at distance x from the origin.

Step 2.
Since the metal is uniform, let the constant area density (mass per unit area) ...

Solution Summary

In a step by step solution, the problem is explained and solved. The center of mass of sheet metal with a curved edge are examined.