1. Polar Area Day 3 Section 10.5B Calculus BC AP/Dual, Revised ©2018
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§10.5B: Common Interior & Inner Loop

2. Different Types of Areas
Common Interior
Add both parts
Limits
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§10.5B: Common Interior & Inner Loop

3. §10.5B: Common Interior & Inner Loop
Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
𝒓=𝟐+𝟐 𝐜𝐨𝐬⁡𝜽 𝜽 𝒓 𝟎 𝟒
𝝅/𝟐 𝟐 𝝅
𝟑𝝅/𝟐 𝟐𝝅
𝒓=𝟐−𝟐 𝐜𝐨𝐬 𝜽 𝜽 𝒓 𝟎
𝝅/𝟐 𝟐 𝝅 𝟒
𝟑𝝅/𝟐 𝟐𝝅
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§10.5B: Common Interior & Inner Loop

4. §10.5B: Common Interior & Inner Loop
Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
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§10.5B: Common Interior & Inner Loop

5. Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
Two sides of each circle “turn”
Two sides of each circle “turn”
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§10.5B: Common Interior & Inner Loop

6. §10.5B: Common Interior & Inner Loop
Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
2/24/2019 9:50 AM
§10.5B: Common Interior & Inner Loop

7. §10.5B: Common Interior & Inner Loop
Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
2/24/2019 9:50 AM
§10.5B: Common Interior & Inner Loop

8. §10.5B: Common Interior & Inner Loop
Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
2/24/2019 9:50 AM
§10.5B: Common Interior & Inner Loop

9. §10.5B: Common Interior & Inner Loop
Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
2/24/2019 9:50 AM
§10.5B: Common Interior & Inner Loop

10. §10.5B: Common Interior & Inner Loop
Example 2
Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽.
𝒓=𝟑 𝐜𝐨𝐬 𝜽 𝒓 𝜽 𝟑 𝟎
𝝅/𝟐 –𝟑 𝝅
𝟑𝝅/𝟐 𝟐𝝅
𝒓=𝟏+𝐜𝐨𝐬 𝜽 𝒓 𝜽 𝟐 𝟎 𝟏
𝝅/𝟐 𝝅
𝟑𝝅/𝟐 𝟐𝝅
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§10.5B: Common Interior & Inner Loop

11. §10.5B: Common Interior & Inner Loop
Example 2
Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽.
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§10.5B: Common Interior & Inner Loop

12. §10.5B: Common Interior & Inner Loop
Example 2
Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽.
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§10.5B: Common Interior & Inner Loop

13. §10.5B: Common Interior & Inner Loop
Example 2
Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽.
2/24/2019 9:50 AM
§10.5B: Common Interior & Inner Loop

14. §10.5B: Common Interior & Inner Loop
Example 2
Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽.
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§10.5B: Common Interior & Inner Loop

15. §10.5B: Common Interior & Inner Loop
Example 2
Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽.
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§10.5B: Common Interior & Inner Loop

16. §10.5B: Common Interior & Inner Loop
Your Turn
Find the common interior of 𝒓=𝟒 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐
𝒓=𝟒 𝐜𝐨𝐬 𝜽 𝒓 𝜽 𝟒 𝟎
𝝅/𝟐 −𝟒 𝝅
𝟑𝝅/𝟐 𝟐𝝅
𝒓=𝟐 𝒓 𝜽 𝟐 𝟎
𝝅/𝟐 𝝅
𝟑𝝅/𝟐 𝟐𝝅
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§10.5B: Common Interior & Inner Loop

17. §10.5B: Common Interior & Inner Loop
Your Turn
Find the common interior of 𝒓=𝟒 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐
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§10.5B: Common Interior & Inner Loop

18. §10.5B: Common Interior & Inner Loop
Your Turn
Find the common interior of 𝒓=𝟒 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐
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§10.5B: Common Interior & Inner Loop

19. §10.5B: Common Interior & Inner Loop
Your Turn
Find the common interior of 𝒓=𝟒 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐
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§10.5B: Common Interior & Inner Loop

20. §10.5B: Common Interior & Inner Loop
Your Turn
Find the common interior of 𝒓=𝟒 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐
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§10.5B: Common Interior & Inner Loop

21. §10.5B: Common Interior & Inner Loop
Review
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
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§10.5B: Common Interior & Inner Loop

22. §10.5B: Common Interior & Inner Loop
Polar Sweep file
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§10.5B: Common Interior & Inner Loop

23. §10.5B: Common Interior & Inner Loop
Example 3
Setup the equation for the area between the inner and outer loops 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
2/24/2019 9:50 AM
§10.5B: Common Interior & Inner Loop

24. §10.5B: Common Interior & Inner Loop
Example 3
Setup the equation for the area between the inner and outer loops 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
2/24/2019 9:50 AM
§10.5B: Common Interior & Inner Loop

25. §10.5B: Common Interior & Inner Loop
Example 3
Setup the equation for the area between the inner and outer loops of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽
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§10.5B: Common Interior & Inner Loop

26. §10.5B: Common Interior & Inner Loop
Your Turn
Setup the equation and solve for the area between the loop of 𝒓 𝜽 =𝟑𝜽 𝐬𝐢𝐧 𝜽 from 𝟎,𝟐𝝅 with a calculator
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§10.5B: Common Interior & Inner Loop

27. §10.5B: Common Interior & Inner Loop
Assignment
Worksheet
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§10.5B: Common Interior & Inner Loop