1. 10.3: Polar functions* Learning Goals: Graph using polar form
Convert polar coordinates to rectangular and vice versa
Calculus of polar forms
*BC Calculus Only
©2008 Roy L Gover (

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
r=10
Pole
Polar Axis
Polar Grid
(r,θ )=

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

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

6. Example
Graph:
r=6

7. Example
Graph: θ=

8. Some Classic Polar Graphs
Example
Some Classic Polar Graphs
8 petal rose

9. Some Classic Polar Graphs
Example
Some Classic Polar Graphs
limacon

10. Some Classic Polar Graphs
Example
Some Classic Polar Graphs
Spiral of Archimedes

11. Try This Use your graphing calculator to graph CARDIOID MODE POL Y=

12. Important Idea 8
4 petals
Polar Roses

13. Important Idea 10
8 petals
Polar Roses

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

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

16. Important Idea
What do you observe about the number of petals?

17. Try This
Describe what you would do to find the angle.

18. Try This
Find the angle.

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

20. Try This
Change the polar coordinates
to rectangular coordinates

21. Try This
Change the rectangular coordinates
to polar coordinates

22. Example Change the polar relation to a rectangular
(cartesian) relation.

23. Important Idea
The slope of the tangent line (derivative-rate of change) to the graph of at the point is:
Remember:

24. Example Find the slope of at and use it to
find the equation of the tangent line.
Demana text ex 5 p.552

25. Try This
Use the nDerive feature of your calculator to find the slope of the graph of the polar curve at
Hint: find
-.667

26. Analysis S r
in radians
Area Formula

27. Analysis
Find the area from 0 to
Area1=

28. Analysis
Find the area from 0 to
Area2=

29. Analysis
Find the area from 0 to
Area3=

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

31. Analysis
Find the area from 0 to

32. Example
Find the area enclosed by
Demana Study Guide ex 6. p173

33. Try This
Find the area of the region bounded by
703/10
sq u

34. Set up the integral to find the shaded area.
Try This
Set up the integral to find the shaded area.
Hint: solve for
when r=0
[PI/2,5PI/6]

35. Find the exact area of one petal of
Try This
Hint:
703/4
sq. units

36. Lesson Close
This lesson completes the curriculum for BC Calculus. The topics I have taught have included everything in the College Board syllabus for BC Calculus.