1. Polar Area Day 2 Section 10.5A Calculus BC AP/Dual, Revised ©2018
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

2. Different Types of Finding Areas
Loops from Limaçon and/or another figure
Inside vs Outside
Area of the inside object MINUS the area of the outside object
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

3. §10.5A: Polar Area with Loops and Inside/Outside
Steps
Determine the intersection by either making the equation to zero (1 equation) or make it equal to each other
Determine the symmetry and apply the equation
Solve and include identities
Identities
𝐬𝐢𝐧 𝟐 𝜽= 𝟏−𝐜𝐨𝐬 𝟐𝜽 𝟐
𝐜𝐨𝐬 𝟐 𝜽= 𝟏+𝐜𝐨𝐬 𝟐𝜽 𝟐
𝐬𝐢𝐧 𝟐𝜽 =𝟐 𝐬𝐢𝐧𝜽 𝐜𝐨𝐬𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

4. §10.5A: Polar Area with Loops and Inside/Outside
Example 1
Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽 𝒓 𝜽 𝟑 𝟎 −𝟑
𝝅/𝟐 𝝅 𝟗
𝟑𝝅/𝟐 𝟐𝝅 𝒓 𝜽 𝟎
𝝅/𝟐 𝝅
𝟑𝝅/𝟐 𝟐𝝅
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

5. §10.5A: Polar Area with Loops and Inside/Outside
Example 1
Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

6. §10.5A: Polar Area with Loops and Inside/Outside
Example 1
Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

7. §10.5A: Polar Area with Loops and Inside/Outside
Example 1
Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

8. §10.5A: Polar Area with Loops and Inside/Outside
Example 1
Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

9. §10.5A: Polar Area with Loops and Inside/Outside
Example 1
Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

10. §10.5A: Polar Area with Loops and Inside/Outside
Example 1
Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

11. §10.5A: Polar Area with Loops and Inside/Outside
Example 1
Find the area inside the loop of 𝒓=𝟑−𝟔 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

12. §10.5A: Polar Area with Loops and Inside/Outside
Example 2
Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽 𝒓 𝜽 𝟏 𝟎 𝟑
𝝅/𝟐 𝝅 −𝟏
𝟑𝝅/𝟐 𝟐𝝅 𝒓 𝜽 𝟎
𝝅/𝟐 𝝅
𝟑𝝅/𝟐 𝟐𝝅
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

13. §10.5A: Polar Area with Loops and Inside/Outside
Example 2
Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

14. §10.5A: Polar Area with Loops and Inside/Outside
Example 2
Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

15. §10.5A: Polar Area with Loops and Inside/Outside
Example 2
Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

16. §10.5A: Polar Area with Loops and Inside/Outside
Example 2
Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

17. §10.5A: Polar Area with Loops and Inside/Outside
Example 2
Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

18. §10.5A: Polar Area with Loops and Inside/Outside
Example 2
Find the area inside the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

19. §10.5A: Polar Area with Loops and Inside/Outside
Your Turn
Find the area inside the loop of 𝒓=𝟐 + 𝟒 𝐜𝐨𝐬 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

20. Inside and Outside Area
Determine the intersection points by making both equations equal to each other
Be aware of the intersection points
Apply symmetry rules
Use the equation: 𝟏 𝟐 𝒂 𝒃 𝑰𝒏𝒔𝒊𝒅𝒆 𝟐 − 𝑶𝒖𝒕𝒔𝒊𝒅𝒆 𝟐
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§10.5A: Polar Area with Loops and Inside/Outside

21. §10.5A: Polar Area with Loops and Inside/Outside
Example 3
Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐.
𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 𝒓 𝜽 𝟑 𝟎 𝟏
𝝅/𝟐 −𝟏 𝝅
𝟑𝝅/𝟐 𝟐𝝅
𝒓=𝟐 𝒓 𝜽 𝟐 𝟎
𝝅/𝟐 𝝅
𝟑𝝅/𝟐 𝟐𝝅
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

22. §10.5A: Polar Area with Loops and Inside/Outside
Example 3
Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐.
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

23. §10.5A: Polar Area with Loops and Inside/Outside
Example 3
Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐.
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

24. §10.5A: Polar Area with Loops and Inside/Outside
Example 3
Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐.
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

25. §10.5A: Polar Area with Loops and Inside/Outside
Example 3
Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐.
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

26. §10.5A: Polar Area with Loops and Inside/Outside
Example 3
Find the area inside 𝒓=𝟏+𝟐 𝐜𝐨𝐬⁡𝜽 but outside the circle 𝒓=𝟐.
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

27. §10.5A: Polar Area with Loops and Inside/Outside
Example 4
Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽
𝒓=𝟑 𝐬𝐢𝐧 𝜽 𝒓 𝜽 𝟎 𝟑
𝝅/𝟐 𝝅 −𝟑
𝟑𝝅/𝟐 𝟐𝝅
𝒓=𝟐−𝐬𝐢𝐧 𝜽 𝒓 𝜽 𝟐 𝟎 𝟏
𝝅/𝟐 𝝅 𝟑
𝟑𝝅/𝟐 𝟐𝝅
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

28. §10.5A: Polar Area with Loops and Inside/Outside
Example 4
Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

29. §10.5A: Polar Area with Loops and Inside/Outside
Example 4
Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

30. §10.5A: Polar Area with Loops and Inside/Outside
Example 4
Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

31. §10.5A: Polar Area with Loops and Inside/Outside
Example 4
Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

32. §10.5A: Polar Area with Loops and Inside/Outside
Example 4
Find the area inside 𝒓=𝟑 𝐬𝐢𝐧 𝜽 and outside 𝒓=𝟐−𝐬𝐢𝐧 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

33. §10.5A: Polar Area with Loops and Inside/Outside
Your Turn
Find the area inside 𝒓=𝟏 and outside 𝒓=𝟏−𝐜𝐨𝐬 𝜽
𝒓=𝟏 𝒓 𝜽 𝟏 𝟎
𝝅/𝟐 𝝅
𝟑𝝅/𝟐 𝟐𝝅
𝒓=𝟏−𝐜𝐨𝐬 𝜽 𝒓 𝜽 𝟎 𝟏
𝝅/𝟐 𝟐 𝝅
𝟑𝝅/𝟐 𝟐𝝅
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

34. §10.5A: Polar Area with Loops and Inside/Outside
Your Turn
Find the area inside 𝒓=𝟏 and outside 𝒓=𝟏−𝐜𝐨𝐬 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

35. §10.5A: Polar Area with Loops and Inside/Outside
Your Turn
Find the area inside 𝒓=𝟏 and outside 𝒓=𝟏−𝐜𝐨𝐬 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

36. §10.5A: Polar Area with Loops and Inside/Outside
Your Turn
Find the area inside 𝒓=𝟏 and outside 𝒓=𝟏−𝐜𝐨𝐬 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

37. §10.5A: Polar Area with Loops and Inside/Outside
Your Turn
Find the area inside 𝒓=𝟏 and outside 𝒓=𝟏−𝐜𝐨𝐬 𝜽
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

38. AP Multiple Choice Practice Question 1 (non-calculator)
The area of the region inside the polar curve 𝒓= 𝟒 𝐬𝐢𝐧 𝜽 but outside the polar curve 𝒓=𝟐 𝟐 is given by (A) 𝟏 𝟐 𝝅/𝟒 𝟑𝝅/𝟒 𝟒 𝐬𝐢𝐧 𝟐 𝜽−𝟐 𝟐 𝟐 𝒅𝜽 (B) 𝟏 𝟐 𝝅/𝟒 𝟑𝝅/𝟒 𝟒 𝐬𝐢𝐧 𝜽−𝟐 𝟐 𝒅𝜽 (C) 𝟏 𝟐 𝝅/𝟒 𝟑𝝅/𝟒 𝟏𝟔 𝐬𝐢𝐧 𝟐 𝜽−𝟖 𝒅𝜽 (D) 𝟏 𝟐 𝝅/𝟒 𝟑𝝅/𝟒 𝟒 𝐬𝐢𝐧 𝟐 𝜽−𝟏 𝒅𝜽
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§10.5A: Polar Area with Loops and Inside/Outside

39. AP Multiple Choice Practice Question 1 (non-calculator)
The area of the region inside the polar curve 𝒓= 𝟒 𝐬𝐢𝐧 𝜽 but outside the polar curve 𝒓=𝟐 𝟐 is given by
Vocabulary
Connections and Process
Answer
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside

40. §10.5A: Polar Area with Loops and Inside/Outside
Assignment
Worksheet
11/24/ :25 AM
§10.5A: Polar Area with Loops and Inside/Outside