1. 14.3 Change of Variables, Polar Coordinates
The equation for this surface is
ρ= sinφ *cos(2θ) (in spherical coordinates)

2. The region R consists of all points between concentric circles of radii 1 and 3 this is called a Polar sector

3. R R

4. A small rectangle in on the left has an area of dydx A small piece of area of the portion on the right could be found by using length times width. The width is rdө the length is dr Hence dydx is equivalent to rdrdө

5. In three dimensions, polar (cylindrical coordinates) look like this.

6. Change of Variables to Polar Form
Recall:
dy dx = r dr dө

7. Use the order dr dө
Use the order dө dr

8. Example 2 Let R be the annular region lying between the two circles
Evaluate the integral

9. Example 2 Solution

10. Problem 18 Evaluate the integral by converting it to polar coordinates
Note: do this problem in 3 steps
Draw a picture of the domain to restate the limits of integration
Change the differentials (to match the limits of integration)
Use Algebra and substitution to change the integrand

11. 18 solution

12. Problem 22
Combine the sum of the two iterated integrals into a single iterated integral by converting to polar coordinates. Evaluate the resulting integral.

13. 22 solution

14. Problem 24
Use polar coordinates to set up and evaluate the double integral

15. Problem 24

17. Figure 14.25

18. Figure 14.26

19. Figure 14.27