Homework 13 Solutions

Homework 13 Solutions - Homework 13 APPM 2350, 2010 1....

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Homework 13 APPM 2350, 2010 1. Using Green’s theorem (1), we can choose α ( x, y )=0and β ( x, y )= x so that Area = ±± R dA = ² C xdy Using the given parametrization of the deltoid, we have ² C xdy = ± t =2 π t =0 (2 cos t +cos(2 t ))(2 cos t - 2cos(2 t )) dt = ± 2 π 0 (4 cos 2 t - 4cos t cos(2 t )+4cos t cos(2 t ) - 2cos 2 (2 t )) dt = ± 2 π 0 (2(1 + cos(2 t )) - (1 + cos(4 t )) dt = ± 2 π 0 (1 + 2 cos(2 t ) - cos(4 t )) dt =2 π 2. We assume that the density of the object ρ is constant. The x and y centers of mass are given by x = ³ R xρdA ³ R ρdA y = ³ R yρdA ³ R ρdA = ³ R xdA ³ R dA = ³ R ydA ³ R dA = ³ R xdA A = ³ R ydA A Recall Green’s theorem: For continuously di±erentiable α ( x, y )and β ( x, y )wehave ±± R ´ ∂β ∂x - ∂α ∂y µ dA = ² C αdx + βdy (1) Consider computing x .W ec h o o s e α ( x, y )and β ( x, y )suchtha tthed i±e renceo fpa r t
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Homework 13 Solutions - Homework 13 APPM 2350, 2010 1....

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