MA 242.003

1. MA 242.003 • Day 46 – March 19, 2013 • Section 9.7: Spherical Coordinates • Section 12.8: Triple Integrals in Spherical Coordinates

2. Section 12.8 Triple Integrals in Spherical Coordinates Goal: Use spherical coordinates to compute a triple integral that has spherical symmetry.

3. Section 12.8 Triple Integrals in Cylindrical Coordinates Goal: Use spherical coordinates to compute a triple integral that has spherical symmetry. Spheres

4. Section 12.8 Triple Integrals in Cylindrical Coordinates Goal: Use spherical coordinates to compute a triple integral that has spherical symmetry. Spheres Cones

5. To study spherical coordinates to use with triple integration we must: 1. Define spherical Coordinates (section 9.7)

6. To study spherical coordinates to use with triple integration we must: 1. Define spherical Coordinates (section 9.7) 2. Set up the transformation equations

7. To study spherical coordinates to use with triple integration we must: 1. Define spherical Coordinates (section 9.7) 2. Set up the transformation equations 3. Study the spherical coordinate Coordinate Surfaces

8. To study cylindrical coordinates to use with double integration we must: 1. Define Cylindrical Coordinates (section 9.7) 2. Set up the transformation equations 3. Study the cylindrical coordinate Coordinate Surfaces 4. Define the volume element in spherical coordinates:

9. To study cylindrical coordinates to use with double integration we must: 1. Define Cylindrical Coordinates (section 9.7) 2. Set up the transformation equations 3. Study the cylindrical coordinate Coordinate Surfaces 4. Define the volume element in spherical coordinates: in cylindrical coordinates

10. To study cylindrical coordinates to use with double integration we must: 1. Define Cylindrical Coordinates (section 9.7) 2. Set up the transformation equations 3. Study the cylindrical coordinate Coordinate Surfaces 4. Define the volume element in spherical coordinates: in cylindrical coordinates in Cartesian coordinates

11. 1. Define Spherical Coordinates

12. 2. Set up the Transformation Equations To transform integrands to spherical coordinates To transform equations of boundary surfaces

13. 2. Set up the Transformation Equations To transform integrands to spherical coordinates To transform equations of boundary surfaces

14. 2. Set up the Transformation Equations To transform integrands to spherical coordinates To transform equations of boundary surfaces

15. 2. Set up the Transformation Equations To transform integrands to spherical coordinates To transform equations of boundary surfaces

17. 3. Study the Spherical coordinate Coordinate Surfaces Definition: A coordinate surface (in any coordinate system) is a surface traced out by setting one coordinate constant, and then letting the other coordinates range over there possible values.

18. 3. Spherical coordinate Coordinate Surfaces The = constant coordinate surfaces The = constant coordinate surfaces

19. 3. Spherical coordinate Coordinate Surfaces The = constant coordinate surfaces

20. 3. Spherical coordinate Coordinate Surfaces Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.

21. 3. Spherical coordinate Coordinate Surfaces Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces. A rectangular box in Cartesian coordinates

22. 3. Spherical coordinate Coordinate Surfaces Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces. A cylindrical box in cylindrical coordinates

23. 3. Spherical coordinate Coordinate Surfaces Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces. A spherical box in spherical coordinates

27. 4. Define the volume element in spherical coordinates:

28. Section 12.8 Triple Integrals in Cylindrical Coordinates Goal: Use spherical coordinates to compute a triple integral that has spherical symmetry. Spheres Cones

29. Fubini’s Theorem in spherical coordinates Plausibility argument: Let f(x,y,z) be continuous on the spherical box (spherical wedge) described by

30. Fubini’s Theorem in spherical coordinates Plausibility argument: Let f(x,y,z) be continuous on the spherical box (spherical wedge) described by Partitioning using spherical boxes and using the spherical volume element for each sub box we find

31. The following approximation of a triple Riemann sum

32. The following approximation of a triple Riemann sum But this is an actual triple Riemann sum for the function

33. The following approximation of a triple Riemann sum But this is an actual triple Riemann sum for the function

35. (Continuation of example)

37. (Continuation of example)

38. (Continuation of example)

40. (Continuation of example)

42. (Continuation of example)