1. 14.7 Day 2 Triple Integrals Using Spherical Coordinates and more applications of cylindrical coordinates This is a Klein bottle, It is a 4 dimensional objected depicted here in 3 dimensions This object has only 1 side. More information about the Klein bottle can be found at http://www-maths.mcs.standrews.ac.uk/images/klein.html

2. Conversions between Spherical and other Coordinate systems

3. Converting the differential (finding the Jacobian) dxdydz=ρ sinφ dρdφdθ 2 Why? To find volume of the box at the left, use V=lwh V = dρ * ρdφ * rdθ (the r is from cylindrical coordinates) From chapter 11 r = ρsin φ Hence dxdydz=ρ sinφ dρdφdθ 2

4. Example 4

5. Example 4 Solution

6. Example 4 explanation

7. Problem 22

8. Problem 22 Solution

9. (the really sad part of this example is that the example provided by the teacher is also incorrect)

10. Problem 14 (spherical coordinates only) Convert the integral from rectangular to spherical coordinates

11. Problem 14 (spherical coordinates only)

12. Problem 14 Solution (cylindrical) (from yesterday)