1. MAT 1236 Calculus 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

9. 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/m23 74/readings/sphcoord/ http://www.math.umn.edu/~nykamp/m23 74/readings/sphcoord/ Variable = constants

12. Example 2 (a)

13. Example 2 (b)

14. Example 2 (c)

15. 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