lect13_7

# lect13_7 - Triple Integrals in Spherical Coordinates(13.7 1...

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
-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.
This is the end of the preview. Sign up to access the rest of the document.

### Page1 / 3

lect13_7 - Triple Integrals in Spherical Coordinates(13.7 1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online