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. 2. Set up the transformation equations To study cylindrical coordinates to use with triple integration we must:

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

8. 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: To study cylindrical coordinates to use with triple integration we must:

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 a.To transform integrands to cylindrical coordinates b.To transform equations of boundary surfaces

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

13. 2. Set up the Transformation Equations a.To transform integrands to cylindrical coordinates b.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 by one coordinate constant, and then letting the other coordinates range over their possible values. Example: The x = 1 coordinate surface is a plane

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

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)