In this section we are going to find the center of mass or centroid of a thin plate with uniform density ρ. The center of mass or centroid of a region is
the point in which the region will be perfectly balanced horizontally if
suspended from that point.
So, let’s suppose that the plate is the region bounded by
the two curves 
and on the interval [a,b]. So, we want to find the center of mass of the
region below.
We’ll first need the mass of this plate. The mass is,
Next we’ll need the moments
of the region. There are two moments,
denoted by Mx and My. The moments measure the tendency of the
region to rotate about the x and y-axis respectively. The moments are given by,
Equations of Moments
The coordinates of the center of mass, ,
are then,
Center of Mass
Coordinates
Note that the density, ρ, of the plate cancels out and so isn’t
really needed.
Let’s work a couple of examples.
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Example 1 Determine
the center of mass for the region bounded by ,
on the interval .
Solution
Here is a sketch of the region with the center of mass
denoted with a dot.
Let’s first get the area of the region.
Now, the moments (without density since it will just drop
out) are,
The coordinates of the center of mass are then,
Again, note that we didn’t put in the density since it
will cancel out.
So, the center of mass for this region is .
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Example 2 Determine
the center of mass for the region bounded by and .
Solution
The two curves intersect at and and here is a sketch of the region with the
center of mass marked with a box.
We’ll first get the area of the region.
Now the moments, again without density, are
The coordinates of the center of mass is then,
The coordinates of the center of mass are then, .
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