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Weighted Average The centroid or center of mass of a planar region is the point at which that region balances perfectly, like a plate on the end of a stick. The coordinates of the centroid are given by weighted averages. � x dA The x coordinate of the centroid is x̄ = � , where dA is an inﬁntessimal dA portion of area; the weighting function in this average� is just x. y dA Similarly, the y coordinate of the centroid is ȳ = � . dA Find the centroid (x̄, ȳ) of the parabolic region bounded by x = −1, x = 3, y = (x − 1)2 and y = 4. Solution y 4 −1 3 x We might guess from the symmetry of the region that x̄ = 1. We’ll do the computation for practice. Since we’re integrating with respect to x, our area dA will be a vertical rectangle with base at y = (x − 1)2 , top at y = 4 and width dx. �3 x̄ = �−31 � 3 −1 2 x(4 − (x − 1) ) dx x(4 − (x − 1)2 ) dx −1 (4 − (x − 1)2 ) dx � 3 = −1 � = 3x − x3 + 2x2 dx 3 2 1 4 2 3 x − x + x 2 4 3 1 �3 −1 � � � � 3 1 2 3 1 2 · 9 − · 81 + · 27 − − − 2 4 3 2 4 3 2 10 3 = = � 3 −1 (4 − (x − 1)2 ) dx � 3 = −1 3 − x2 + 2x dx �3 1 3 2 = 3x − x + x 3 −1 � � � � 1 1 = 3 · 3 − · 27 + 9 − −3 + + 1 3 3 2 = 10 3 � 10 23 = 1. 10 23 To ﬁnd ȳ we will be integrating with respect to y. Our area dA will have width equal to the width of the region and height dy. The left side of the region √ √ has the equation x = 1 − y and the right side has equation x = 1 + y, so the √ √ √ width is (1 + y) − (1 − y) = 2 y. We conclude that x̄ = √ y(2 y) dy �4 √ 2 y dy 0 �4 ȳ = 4 � √ 0 y(2 y) dy 4 � = y 3/2 dy 2 0 0 �4 2 5/2 �� 2· y � 5 0 4 5 (2 − 0) 5 27 5 = = = � 4 √ 2 y dy 4 � = 0 y 1/2 dy 2 0 = = = 2 �4 2 3/2 �� 2· y � 3 0 4 3 · (2 − 0) 3 25 3 27 5 25 3 12 = 2.4. 5 The center of mass of the parabolic region is at (1, 2.4); this seems plausible given that the top of the parabola (y = 4) will weigh more than the bottom (y = 0). Hence, ȳ = = 3 MIT OpenCourseWare http://ocw.mit.edu 18.01SC Single Variable Calculus�� Fall 2010 �� For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.