2. AREA, VOLUME, AND CENTER OF MASS
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Moments and Center of Mass
Consider a lamina (a thin, flat plate) in the shape of a region R 2 R2 whose
density varies throughout the plate.
Our goal is to find the center of mass. But first we need to find the total mass.
Assume we have a mass density function ⇢(x, y).
We partition the region, and if a subregion Ri is small enough, then ⇢(x, y) is
almost constant on Ri, whose mass is then
mi ⇡ ⇢(ui, vi) |{z}
Ai
| {z }
mass/unit area area
where (ui, vi) is an arbitrary point in Ri. Summing
n
X
m⇡
⇢(ui, vi) Ai.
i=1
Then the exact mass is
m = lim
kP k!0
n
X
i=1
⇢(ui, vi) Ai =
ZZ
R
⇢(x, y) dA.