Centroids - Centroids Let A be a geometric figure in...

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Centroids Let A be a geometric figure in Euclidean space. The Euclidean space may be R 2 or R 3 . It could in fact be R n . The figure may be any dimension. The coordinates of the centroid or center of mass of the figure are given by the following formulas. x = x ! dA " dA " y = y ! dA " dA " z = z ! dA " dA " Of course, if the figure is in R 2 , only the first two coordinates apply. The dA in the formulas are small strips of the figure that are perpendicular to the direction of the respective x –axis or y –axis or z –axis. The simplest way to understand this is to work through a few examples. 1. Determine the centroid of the portion of the area of a disk of radius R centered at the origin that is in the first quadrant. x = x ! R 2 " x 2 dx 0 R # R 2 " x 2 dx 0 R # = 4 R 3 $ y = y ! R 2 " y 2 dx 0 R # R 2 " y 2 dx 0 R # = 4 R 3 2. Determine the centroid of the area under the curve f ( x ) = x n over the interval [0,1] . x = x ! x n dx 0 1 " x n dx 0 1 " = n + 1 n + 2 y = y ! (1 # y n ) dy 0 1 " # y n ) dy 0 1 " = n + 1 4 n + 2 x n R 2 ! x 2
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This note was uploaded on 07/14/2011 for the course MAC 3473 taught by Professor Block during the Spring '08 term at University of Florida.

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Centroids - Centroids Let A be a geometric figure in...

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