lect13_2

lect13_2 - Area, Volume and Center of Mass - (13.2) 1....

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Area, Volume and Center of Mass - (13.2) 1. Areas and Volumes Recall: For a given region R ! ! x , y " | a ! x ! b , g 1 ! x " ! y ! g 2 ! x " , "" R f ! x , y " dA ! " a b " g 1 ! x " g 2 ! x " f ! x , y " dydx . For a given region R ! ! x , y " | h 1 ! y " ! x ! h 2 ! y " , c ! y ! d , R f ! x , y " dA ! " c d " h 1 ! y " h 2 ! y " f ! x , y " dxdy . When f ! x , y " ! 1 for ! x , y " in R , R f ! x , y " dA ! the area of the region R . When f ! x , y " # 0 for ! x , y " in R , R f ! x , y " dA ! the volume of the solid region under the surface z ! f ! x , y " over the region R . Example Find the volume of the tetrahedron bounded by x " 2 y " 3 z ! 6 and 3 coordinate planes. The solid region: 0 ! z ! 1 3 ! 6 $ x $ 2 y " ,0 ! y ! 1 2 ! 6 $ x " ! x ! 6. 0 0.5 1.5 2 0.5 1 1.5 2 2.5 3 y 2 4 6 x the solid region 0 0.5 1 1.5 2 2.5 123456 x the region in the xy $ plane V ! " 0 6 " 0 ! 6 $ x " /2 1 3 ! 6 $ x $ 2 y " dydx ! 1 3 " 0 6 !! 6 $ x " y $ y 2 " | 0 ! 6 $ x " /2 dx ! 1 3 " 0 6 1 2 ! 6 $ x " 2 $ 1 4 ! 6 $ x " 2 dx ! 1 12 " 0 6 ! 6 $ x " 2 dx ! $ 1 36 ! 6 $ x " 3 | 0 6 ! $ 1 36 ! 0 $ 6 3 " ! 6 Example Find the volume of the solid bounded by z ! 4 $ x 2 , x " y ! 2, 3 coordinate plane and in the first octant. The solid region: 0 ! z ! 4 $ x 2 ! y ! 2 $ x ! x ! 2. 1
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0 1 2 3 4 0.5 1 1.5 2 z 0.5 1 1.5 2 x the solid region 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x the region in the xy $
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lect13_2 - Area, Volume and Center of Mass - (13.2) 1....

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