lect13_7

lect13_7 - Triple Integrals in Spherical Coordinates -...

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Triple Integrals in Spherical Coordinates - (13.7) 1. Spherical Coordinates Consider spherical coordinates: x ! ! cos " sin # y ! ! sin " sin # z ! ! cos # where ! ! x 2 " y 2 " z 2 tan " ! y x cos # ! z ! " is counterclockwise from the positive side of the x ! axis, and # is clockwise from the positive side of the z ! axis. Example Give the ! x , y , z " coordinates for the point in spherical coordinates ! ! , " , # " ! 3, $ 4 , 2 $ 3 . x ! 3 cos $ 4 sin 2 $ 3 ! 36 4 , y ! 3 sin $ 4 sin 2 $ 3 ! 4 , z ! 3 cos 2 $ 3 ! ! 3 2 Example Give the ! ! , " , # " coordinates for the point in ! x , y , z " ! ! 1, ! 1,3 " coordinates. ! ! 1 2 " ! ! 1 " 2 " 3 2 ! 11 tan " ! ! 1 1 ! ! 1, since " is in Quadrant IV , " ! 2 $ ! $ 4 ! 7 $ 4 cos # ! 3 11 , # ! cos ! 1 3 11 ! 0.4405107 2. Equations in Spherical Coordinates a. ! ! c # 0 # x 2 " y 2 " z 2 ! c # x 2 " y 2 " z 2 ! c 2 , a sphere with radius c ! " c # 0 # x 2 " y 2 " z 2 " c 2 , a ball with radius c b. " ! c # tan " ! y x ! c # y ! cx , a plane c. # ! c # a cone Example Write the equations in spherical coordinates: a . x 2 " y 2 " z 2 ! 8 b . ! x " 1 " 2 " y 2 " z 2 ! 4 a. x 2 " y 2 " z 2 ! 8 # ! ! 8 b. ! x " 1 " 2 " y 2 " z 2 ! 4 # x 2 " 2 x " 1 " y 2 " z 2 ! 4 # ! 2 " 2 ! cos " sin # ! 3 Example Describe and sketch the solid region Q: 0 " ! " 1, 0 " # " $ 3 , $ 2 " " " $ Q :0 " ! " 1, 0 " # " $ 3 , $ 2 " " " $ , a portion of a unit ball 1
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-1 0.5 1 0.5 v -1 1 u Example Describe the solid Q lying inside x 2 " y 2 " z 2 ! 2 z and z 2 ! x 2 " y 2 in the spherical coordinates. x 2 " y 2 " z 2 ! 2 z # ! 2 ! 2 ! cos # # ! ! 2cos # z 2 ! x 2 " y 2 # ! 2 cos 2 # ! ! 2 sin 2 # # tan 2 # ! 1, # ! $ 4 since z # 0. Q :0 " ! " # ,0 " # " $ 4 " " " 2 $ 0 0.5 -1 -0.5 0.5 1 v -1 -0.5 0.5 1 u 3. Triple Integrals in Spherical Coordinates Let ! ! , " , # " be a point in the space. Consider changes: $ ! , $ # , $ " along ! , " , # , respectively. Then the total change in volume $ V can be computed as follows.
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lect13_7 - Triple Integrals in Spherical Coordinates -...

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