lect13_6

lect13_6 - Triple Integrals in Cylindrical Coordinates -...

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Triple Integrals in Cylindrical Coordinates - (13.6) Cylindrical Coordinates Consider ! x , y " in polar coordinates: x ! r cos ! y ! r sin ! z ! z , r ! x 2 " y 2 tan ! ! y x Triple Integrals in Cylindrical Coordinates Let Q be a solid in space and f ! x , y , z " be defined on Q .I f Q : g 1 ! x , y " ! z ! g 2 ! x , y " for ! x , y " in R , then """ Q f ! x , y , z " dV ! "" R " g 1 ! x , y " g 2 ! x , y " f ! x , y , z " dz dA ! R " g 1 r cos ! , r sin ! g 2 r cos ! , r sin ! f ! r cos ! , r sin ! , z " dz dA dA ! rdrd ! or dA ! rd ! dr Example Set up the triple integral Q f ! x , y , z " dV where Q : above z ! x 2 " y 2 and below z ! 8 # x 2 # y 2 . 0 0.5 1 1.5 2 -2 -1 1 2 v -2 -1 1 2 u Intersection of z ! x 2 " y 2 and z ! 8 # x 2 # y 2 : x 2 " y 2 ! 8 # x 2 # y 2 ,2 ! x 2 " y 2 " ! 8, x 2 " y 2 ! 4 Q : x 2 " y 2 ! z ! 8 # x 2 # y 2 , R : x 2 " y 2 ! 4 # Q : r ! z ! 8 # r 2 , R :0 ! ! ! 2 " ,0 ! r ! 2 Q f ! x , y , z " dV ! " 0 2 " 0 2 " " r 8 # r 2 f r cos ! , r sin ! , z dz rd ! dr Example Set up the triple integral Q f ! x , y , z " dV where Q : above z ! x 2 " y 2 and below z ! 9. 1
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0 5 -2 2 u -2 2 v Intersection of z ! x 2 " y 2 and z ! 9: x 2 " y 2 ! 9 Q : x 2 " y 2 ! z ! 9, R : x 2 " y 2 ! 9 # Q : r 2 ! z ! 9, R :0 ! !
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lect13_6 - Triple Integrals in Cylindrical Coordinates -...

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