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MA 242.003

MA 242.003 . Day 44 – March 14, 2013 Section 12.7: Triple Integrals. GOAL: To integrate a function f(x,y,z ) over a bounded 3-dimensional solid region in space. . Step 1: Subdivide the box into subboxes . Generalization to bounded regions (solids) E in 3-space:.

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MA 242.003

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  1. MA 242.003 • Day 44 – March 14, 2013 • Section 12.7: Triple Integrals

  2. GOAL: To integrate a function f(x,y,z) over a bounded 3-dimensional solid region in space.

  3. Step 1: Subdivide the box into subboxes.

  4. Generalization to bounded regions (solids) E in 3-space:

  5. Generalization to bounded regions (solids) E in 3-space: 1. To integrate f(x,y,z) over E we enclose E in a box B 2. Then define F(x,y,z) to agree with f(x,y,z) on E, but is 0 for points of B outside E. 3. Then Fubini’s theorem applies, and we define

  6. Definition: A solid region E is said to be of type 1 if it lies between the graphs of two continuous functions of x and y, that is

  7. Using techniques similar to what was needed for double integrals one can show that

  8. When the formula Specializes to

  9. When the formula Specializes to

  10. (continuation of problem 11)

  11. Definition: A solid region E is said to be of type 2 if it lies between the graphs of two continuous functions of y and z, that is

  12. Definition: A solid region E is said to be of type 2 if it lies between the graphs of two continuous functions of y and z, that is

  13. (continuation of problem 17)

  14. Definition: A solid region E is said to be of type 3 if it lies between the graphs of two continuous functions of x and z, that is

  15. Definition: A solid region E is said to be of type 3 if it lies between the graphs of two continuous functions of x and z, that is

  16. (continuation of problem 18)

  17. An Application of Triple Integration The volume of the solid occupying the 3-dimensional region E is

  18. An Application of Triple Integration The volume of the solid occupying the 3-dimensional region E is

  19. An Application of Triple Integration The volume of the 3-dimensional region E is The area of the region D is

  20. (continuation of problem 20)

  21. #33

  22. (continuation of problem 33)

  23. (continuation of problem 33)

  24. (see maple worksheet)

  25. (continuation of problem 38)

  26. (continuation of problem 43)

  27. (continuation of problem )

  28. (continuation of problem )

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