1. Give 4 different pairs of polar coordinates that correspond to the Cartesian Coordinates (2,2) Aim: How do we describe Curves in Polar form?

2. Definition A Polar Coordinate System is a method of locating a point ( r, ) in a plane where r is a distance from a point called the pole directed at an angle θ measured counterclockwise from a polar axis.

3. Example Polar Grid Pole Polar Axis r=10 ( r,θ )=

4. r=10 r= ̶ 10 Negative radius means go in the opposite directions Remember to consider the rotation FIRST

5. Example ( r,θ )= Find the polar coordinates:

6. Example ( r,θ )= Find the polar coordinates:

7. Plot Name the point with three other polar coordinates using

8. Important Idea Polar coordinates are related to rectangular coordinates as follows: θ r y x

9. Example Graph: r =6 What about r = ̶ 6?

10. Example Graph: θ=θ= What about

11. Some Classic Polar Graphs Example limacon

12. A Classic Polar Graph Example Spiral of Archimedes

13. Some Classic Polar Graphs Example 8 petal rose

14. Use your graphing calculator to graph MODEPOL GRAPH Y= CARDIOID

15. Polar Roses Important Idea 8 4 petals

16. Polar Roses Important Idea 10 8 petals

17. Polar Roses Try This Name the angle in rads?

18. Important Idea What do you observe about the number of petals? The period?

19. Some Calculus Find the slope of at.

20. We can write polar equations parametrically where

21. What is this telling us about the curve?

22. Area r S in radians

23. Analysis Find the area from 0 to 0 Area 1 =

24. Analysis Find the area from 0 to 0 Area 2 =

25. Analysis Find the area from 0 to 0 Area 3 =

26. Analysis Find the area from 0 to 0 Riemann Sum for total area

27. Analysis Find the area from 0 to 0