MA 242.003

1. MA 242.003 • Day 45 – March 18, 2013 • Section 9.7: Cylindrical Coordinates • Section 12.8: Triple Integrals in Cylindrical Coordinates

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

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

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

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

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

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

8. To study cylindrical coordinates to use with triple 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 cylindrical coordinates:

9. 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 cylindrical coordinates: recall the polar coordinate area element:

10. 1. Define Cylindrical Coordinates

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

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

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

18. 3. Study the Cylindrical coordinate Coordinate Surfaces Definition: A coordinate surface (in any coordinate system) is a surface traced out byone coordinate constant, and then letting the other coordinates range over theirpossible values. Example: The x = 1 coordinate surface is a plane

19. 3. Study the cylindrical coordinate Coordinate Surfaces Definition: A coordinate surface (in any coordinate system) is a surface traced out byone coordinate constant, and then letting the other coordinates range over theirpossible values. Example: The x = 1 coordinate surface is a plane Definition: A box like region is a region enclosed by three pairs of congruent coordinate surfaces.

20. 3. Cylindrical coordinate Coordinate Surfaces The r = constant coordinate surfaces The = constant coordinate surfaces The z = constant coordinate surfaces

21. 3. Cylindrical coordinate Coordinate Surfaces The = constant coordinate surfaces

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

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

24. 3. Cylindrical 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 A cylindrical box in cylindrical coordinates

28. 4. Define the volume element in cylindrical coordinates:

29. Section 12.8 Triple Integrals in Cylindrical Coordinates Goal: Use cylindrical coordinates to compute a triple integral that has cylindrical symmetry. Cylinders Cones

33. z

34. z

36. (Continuation of example)

38. (Continuation of example)

40. (Continuation of example)

42. (Continuation of example)