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Unformatted text preview: Side note: To do #29 of page 846 you need the following deﬁnition: Defn. Average Value of f (x, y, z ) The average value of f (x, y, z ) over a closed, bounded region R is 13.6 Moments and Centers of Mass
In R2 : Find the center of mass (C.O.M.) of a thin plate of density δ (x) over the region bounded by y = f (x) on top and y = g (x) on bottom from x = a to x = b. C.O.M. = (¯, y ), x¯ where x = ¯ My Mx ,y= ¯ M M
b M = total mass = (area)(density) =
a δ (x) f (x) − g (x) dx Mx and My are moments. Recall moment = (directed distance)(mass)
b Mx = moment about the xaxis =
a y δ (x) f (x) − g (x) dx ˜ b My = moment about the y axis =
a xδ (x) f (x) − g (x) dx ˜ 1 In R3 : There is no need to worry about mixing x’s and y ’s. We can also ﬁnd the center of mass of two kinds of objects since we have a 3rd dimension. Center of Mass of a thin plate of density δ (x, y ) over a region R
Center of Mass: Mass: Moments: 2 Center of Mass of a solid D of density δ (x, y, z )
Center of Mass: Mass: Moments: 3 Example 1  Thin Plate Find the center of mass of a thin plate of density δ (x, y ) = y over the region bounded by y = x2 and y = 2x. 4 Example 2  Solid SETUP ONLY. Find the center of mass of a solid with density δ (x, y, z ) = x + z bounded below by the plane z = 0, above by the paraboloid z = 4 − x2 − y 2 and by the cylinder x2 + y 2 ≤ 1. 5 ...
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This note was uploaded on 03/03/2010 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.
 Spring '03
 MECothren
 Multivariable Calculus

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