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2. AREA, VOLUME, AND CENTER OF MASS 125 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.