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MAT 1236 Calculus III

MAT 1236 Calculus III. Section 15.8, 15.9 Triple Integrals in Cylindrical and Spherical Coordinates. http://myhome.spu.edu/lauw. HW. WebAssign 15.8-15.9. Preview.

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MAT 1236 Calculus III

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  1. MAT 1236Calculus III Section 15.8, 15.9 Triple Integrals in Cylindrical and Spherical Coordinates http://myhome.spu.edu/lauw

  2. HW • WebAssign 15.8-15.9

  3. Preview • Double integrals: For regions related to circles, double integrals are better done in polar coordinates (PC) than rectangular coordinates (RC). • Common regions for Triple integrals: • (3D) regions related to cylinders • (3D) regions related to sphere

  4. Recall: 15.7 Example 3 Find the volume of the solid bounded by the paraboloids

  5. Cylindrical Coordinates • Replace the first 2 RC by PC • Direct extension of the PC. • No new formula are needed. We used CC in 15.7 Example 3.

  6. Example 1 Evaluate where E is the solid lies between , above the xy-plane and below the plane

  7. Example 1

  8. Example 1

  9. Example 1 Evaluate where E is the solid lies between , above the xy-plane and below the plane

  10. Spherical Coordinates

  11. New – More Examples • http://www.math.umn.edu/~nykamp/m2374/readings/sphcoord/ • Variable = constants

  12. Example 2 (a)

  13. Example 2 (b)

  14. Example 2 (c)

  15. Spherical Wedge

  16. Spherical Wedge

  17. Example 3 Evaluate where H is the region lies above the xy-plane and below

  18. Remarks • One may attempt to use CC for the integral in example 3.

  19. Bonus Point Friday!? • 2 bonus points for your next exam • Come and listen to an applied math themed presentation (probability, limits, and derivatives) • OMH 139, 5:10 – 5:50 • Fill in a survey • Limited seats, sign up sheet is going around

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