14.6 triple integrals in cylindrical and spherical coordinates

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14.6 triple integrals in cylindrical and spherical coordinates

  1. 1. 1
  2. 2.  Triple Integrals in Cylindrical Coordinates Useful for circle-symmetrical integration regions and integrand functions Switch to polar coordinates for 2 of the 3 coordinates, leave the third as is x r cos y r sin z z f ( x, y , z ) f (r , , z ) dx dy dz r dr d dz Equivalent to integrate first inz , then in polar coordinates on the projection to thexy plane 2
  3. 3. Example 1 Convert the point r , , z rectangular 5 ,3 6 to 1, 3, 2 to 4, coordinates. Example 2 Convert the point x, y, z cylindrical coordinates. 3
  4. 4. Example 3 Find an equation in cylindrical coordinates for the surface represented z x2 y2 . Example 4 Find an equation in rectangular coordinates for the surface represented by r 3sec . 4
  5. 5. Example 5 Using cylindrical coordinates, evaluate a2 x2 a 0 0 a2 x2 y 2 0 x2 dz d y dx ; a 0 . Example 6 Find the volume of the solid that is bounded above x2 y 2 z 2 9 2 2 x y 4. and below by the sphere inside the cylinder and 5
  6. 6.  Triple Integrals in Spherical Coordinates Switch to spherical coordinates: radius, longitude, latitude x sin cos y sin sin z cos x2 y2 z2 2 6
  7. 7. Switch to rectangular coordinates x2 cos y2 z2 z 1 x2 tan 1 y2 z2 y x 7
  8. 8. Example 7 Convert , , 2, 2 5 , 3 6 to rectangular coordinates and cylindrical coordinates. Example 8 If the rectangular coordinates of point P are 1, 3, 2 find the spherical coordinates of P. 8
  9. 9. Example 9 Find an equation in spherical coordinates for the surface represented by the equation x2 y2 z2. 9
  10. 10. dV 2 sin d d d A typical triple integral in spherical coordinates has the form f x, y, z dV G h2 g2 h1 g1 , , f , , 2 sin d d d 10
  11. 11. Example 10 Use spherical coordinates to find the volume of the x 2 y 2 z 2 4a 2 solid enclosed by the sphere z 0. and the plane 11 Example Find the volume of the solid region Q bounded by the cone z x2 y2 z2 x2 y2 and the sphere 2 z. 11
  12. 12. Example 12 Use spherical coordinates to evaluate 4 y2 2 0 0 8 x2 y 2 x2 y 2 z 2 dz dx d y. 12

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