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hwsol_19

# hwsol_19 - Homework Assignment 19(13.2 Solutions Page 1037...

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Homework Assignment 19 - (13.2) - Solutions Page 1037: 12*, 14*, 20*, Sketch first the solid for each problem. 28*, 30*, 34*, 35*, 42*, 51* 12. x ! 2 y ! 3 z " 6, x " 0, y " 0, z " 0 The solid: ! 1 3 ! 6 ! x ! 2 y " " z " 0, R :0 " y " 1 2 ! 6 ! x " ,0 " x " 6 -2 0 1 2 1 2 3 y 1 2 3 4 5 6 z V " ! # 0 6 # 0 ! 6 ! x " /2 # 0 ! ! 6 ! x ! 2 y " /3 dzdydx " 6 14. z " x 2 ! y 2 , z " 0, x " 0, x " 1, y " 0, y " 1 The solid: 0 " z " x 2 ! y 2 , R " x " 1, 0 " y " 1 V " # 0 1 # 0 1 ! x 2 ! y 2 " dxdy " 2 3 0 1 -1 -0.5 0.5 1 v 0.5 1 u 20. z " x 2 ! y 2 ! 1, z " 0, y " x 2 , y " 2 x ! 3 1

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0 -5 5 v 5 u Intersections of y " x 2 and y " 2 x ! 3: x 2 " 2 x ! 3, x 2 ! 2 x ! 3 " ! x ! 3 "! x ! 1 " " 0, x " 3, x " ! 1 V " # ! 1 3 # x 2 2 x ! 3 ! x 2 ! y 2 ! 1 " dydx " 20896 105 " 199.01 28. y " x 4 , y " x 2 , ! ! x , y " " 4 Intersections of y " x 4 and y " x 2 : x 2 ! x 2 ! 1 " " 0, x " 0, and x "# 1. The region: R :0 " x " 1, x 4 " y " x 2 m " # 0 1 # x 4 x 2 4 dydx " 8 15 , M x " # 0 1 # x 4 x 2 4 ydydx " 8 45 , M y " # 0 1 # x 4 x 2 4 xdydx " 1 3 x \$ " M y m " 1 3 15 8 " 5 8 , y \$ " M x m " 8 45 15 8 " 1 3 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1 x 30. x " y 2 , x " 4, ! ! x , y " " y ! 3 When x " 4, y 2. The region: ! 2 " y " 2, y 2 " x " 4 m " # ! 2 2 # y 2 4 ! y ! 3 " dxdy " 32, M x " # ! 2 2 # y 2 4 y ! y ! 3 " dxdy " 128 15 , M y " # ! 2 2 # y 2 4 x ! y ! 3 " dxdy " 384 5 x \$ " M y m " 384 5 ! 32 " " 12 5 , y \$ " M x m " 128 15 ! 32 " " 4 15 2
-2 -1 0 1 2 1234 34. The lamina is symmetric about the x -axis, that means ! a " y " a and h 1 ! y " " x " h 2 ! y " where both h 1 ! y " and h 2 ! y " are even in y . The center of mass is located on the x ! axis if y \$ " 0. y \$ " 0if M x " 0, that is ## R y ! ! x , y " dA " # ! a a # h 1 ! y " h 2 ! y " y ! ! x , y " dxdy " 0. If ! ! x , y " is even in y then # h 1 ! y " h 2 ! y " ! ! x , y " dx is even in y and y # h 1 ! y " h 2 ! y " ! ! x , y " dx is odd in y and therefore # ! a a # h 1 ! y " h 2 ! y " y ! ! x , y " dxdy " # ! a a y # h 1 ! y " h 2 ! y " ! ! x , y " dxdy " 0 35. Similarly, if the lamina is symmetric about the y ! axis, then the condition for x \$ " ! ! x , y " is symmetric w.r.t. y ! axis, that is ! ! ! x , y " " ! ! x , y " .

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hwsol_19 - Homework Assignment 19(13.2 Solutions Page 1037...

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