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Math 251 notes section 13.6 Center of Mass J. Lewis i 1, 2,3 then the center of mass is at If three masses m1 , m 2 , m 3 are located at the points ( x i , y i ) _ _ _ _ ( x , y ) where x the weighted average of the x - values and y is the weighted average of the y - values. _ x 1 3 xi mi , M i 1 1 3 _ y yi mi , M i 1 M m1 m 2 m 3 . In general, for a lamina in the shape of region D with continuous density function ( x, y ), _ _ _ the center of mass is ( x , y ) with x 1 M x ( x , y ) dA , _ y D 1 M y ( x, y ) dA , D M ( x, y ) dA D Examples Find the center of mass for the lamina described by the region D with density function ( x, y ) . 1. D is the rectangle (0,0) and (2,3) as opposite corners, ( x, y ) y . 2. D is the triangle with corners (0,0), (1,1) and (4,0), ( x, y ) x . 3. D is the unit disk x 2 y 2 1, ( x, y ) K ( x 2 y 2 ) , K a constant 4. Find the moment of inertia about the x-axis which is I x y 2 ( x , y ) dA for D D bounded by y sin x, y 0, 0 x and constant density function, .