1. Polar Area Day 3 Section 10.5 Calculus BC AP/Dual, Revised ©2016
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10.5: Polar Area: Day 3 with Common Interior

2. Different Types of Areas
Common Interior
Add both parts
Limits
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10.5: Polar Area: Day 3 with Common Interior

3. 10.5: Polar Area: Day 3 with Common Interior
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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10.5: Polar Area: Day 3 with Common Interior

4. 10.5: Polar Area: Day 3 with Common Interior
Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
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10.5: Polar Area: Day 3 with Common Interior

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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10.5: Polar Area: Day 3 with Common Interior

6. 10.5: Polar Area: Day 3 with Common Interior
Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
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10.5: Polar Area: Day 3 with Common Interior

7. 10.5: Polar Area: Day 3 with Common Interior
Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
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10.5: Polar Area: Day 3 with Common Interior

8. 10.5: Polar Area: Day 3 with Common Interior
Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
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10.5: Polar Area: Day 3 with Common Interior

9. 10.5: Polar Area: Day 3 with Common Interior
Example 1
Find the common interior area of 𝒓=𝟐 𝟏+ 𝐜𝐨𝐬 𝜽 and 𝒓=𝟐 𝟏− 𝐜𝐨𝐬 𝜽 .
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10.5: Polar Area: Day 3 with Common Interior

10. 10.5: Polar Area: Day 3 with Common Interior
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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10.5: Polar Area: Day 3 with Common Interior

11. 10.5: Polar Area: Day 3 with Common Interior
Example 2
Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽.
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10.5: Polar Area: Day 3 with Common Interior

12. 10.5: Polar Area: Day 3 with Common Interior
Example 2
Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽.
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10.5: Polar Area: Day 3 with Common Interior

13. 10.5: Polar Area: Day 3 with Common Interior
Example 2
Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽.
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10.5: Polar Area: Day 3 with Common Interior

14. 10.5: Polar Area: Day 3 with Common Interior
Example 2
Find the common interior of 𝒓=𝟑 𝐜𝐨𝐬 𝜽 and 𝒓=𝟏+𝐜𝐨𝐬 𝜽.
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10.5: Polar Area: Day 3 with Common Interior

15. 10.5: Polar Area: Day 3 with Common Interior
Your Turn
Find the common interior of 𝒓=𝟒 𝐬𝐢𝐧 𝜽 and 𝒓=𝟐.
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10.5: Polar Area: Day 3 with Common Interior

16. 10.5: Polar Area: Day 2 Loops and Inside/Outside
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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10.5: Polar Area: Day 2 Loops and Inside/Outside

17. 10.5: Polar Area: Day 2 Loops and Inside/Outside
Example 3
Setup the equation for the area between 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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10.5: Polar Area: Day 2 Loops and Inside/Outside

18. 10.5: Polar Area: Day 2 Loops and Inside/Outside
Example 3
Setup the equation for the area between 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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10.5: Polar Area: Day 2 Loops and Inside/Outside

19. 10.5: Polar Area: Day 2 Loops and Inside/Outside
Example 3
Setup the equation for the area between the loop of 𝒓=𝟏+𝟐 𝐬𝐢𝐧 𝜽
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10.5: Polar Area: Day 2 Loops and Inside/Outside

20. 10.5: Polar Area: Day 3 with Common Interior
Example 5
Find the area between the two spirals of 𝒓=𝜽 and 𝒓=𝟐𝜽 for 𝟎≤𝜽≤𝟐𝝅.
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10.5: Polar Area: Day 3 with Common Interior

21. 10.5: Polar Area: Day 3 with Common Interior
Example 6
The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (calc allowed)
Let 𝑹 be the region that is inside of the graph of 𝒓=𝟐 and also inside the graph of 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 , as shaded in the figure. Find the area.
A particle is moving with nonzero velocity along the polar curve given by 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 has position of 𝒙 𝒕 ,𝒚 𝒕 at time 𝒕, with 𝜽=𝟎 with 𝒕=𝟎. This particle moves along the curve so that 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 . Find the value of 𝒅𝒓 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle.
For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle.
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10.5: Polar Area: Day 3 with Common Interior

22. 10.5: Polar Area: Day 3 with Common Interior
Example 6a
The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 .
Let 𝑹 be the region that is inside of the graph of 𝒓=𝟐 and also inside the graph of 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 , as shaded in the figure. Find the area.
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10.5: Polar Area: Day 3 with Common Interior

23. 10.5: Polar Area: Day 3 with Common Interior
Example 6a
The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 .
Let 𝑹 be the region that is inside of the graph of 𝒓=𝟐 and also inside the graph of 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 , as shaded in the figure. Find the area.
4/14/2019 1:09 PM
10.5: Polar Area: Day 3 with Common Interior

24. 10.5: Polar Area: Day 3 with Common Interior
Example 6b
The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (b) A particle is moving with nonzero velocity along the polar curve given by 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 has position of 𝒙 𝒕 ,𝒚 𝒕 at time 𝒕, with 𝜽=𝟎 with 𝒕=𝟎. This particle moves along the curve so that 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 . Find the value of 𝒅𝒓 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle.
4/14/2019 1:09 PM
10.5: Polar Area: Day 3 with Common Interior

25. 10.5: Polar Area: Day 3 with Common Interior
Example 6b
The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (b) A particle is moving with nonzero velocity along the polar curve given by 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 has position of 𝒙 𝒕 ,𝒚 𝒕 at time 𝒕, with 𝜽=𝟎 with 𝒕=𝟎. This particle moves along the curve so that 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 . Find the value of 𝒅𝒓 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle.
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10.5: Polar Area: Day 3 with Common Interior

26. 10.5: Polar Area: Day 3 with Common Interior
Example 6c
The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (c) For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle.
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10.5: Polar Area: Day 3 with Common Interior

27. 10.5: Polar Area: Day 3 with Common Interior
Example 6c
The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (c) For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle.
4/14/2019 1:09 PM
10.5: Polar Area: Day 3 with Common Interior

28. 10.5: Polar Area: Day 3 with Common Interior
Example 6c
The graphs of the polar curves of 𝒓=𝟐 and 𝒓=𝟑+𝟐 𝐜𝐨𝐬 𝜽 are shown. The curves intersect when 𝜽= 𝟐𝝅 𝟑 and 𝜽= 𝟒𝝅 𝟑 . (c) For the particle described in part (b), 𝒅𝒚 𝒅𝒕 = 𝒅𝒓 𝒅𝜽 , find the value of 𝒅𝒚 𝒅𝒕 at 𝜽= 𝝅 𝟑 and interpret answer in terms of motion of the particle.
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10.5: Polar Area: Day 3 with Common Interior

29. 10.5: Polar Area: Day 3 with Common Interior
Example 7
A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅.
Using the calculator:
Find the area bounded by the curve and 𝒙-axis
Find the angle 𝜽 that corresponds to the point on the curve with 𝒙-coordinate 𝒙=−𝟐.
For 𝝅 𝟑 <𝜽< 𝟐𝝅 𝟑 , 𝒅𝒓 𝒅𝜽 is negative. What does your answer tell you about 𝒓? What does it tell you about the curve?
At what angle 𝜽 is the interval 𝟎≤𝜽≤ 𝝅 𝟐 the curve farthest away from the origin. Justify.
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10.5: Polar Area: Day 3 with Common Interior

30. 10.5: Polar Area: Day 3 with Common Interior
Example 7a
A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅.
Using the calculator:
Find the area bounded by the curve and x-axis
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10.5: Polar Area: Day 3 with Common Interior

31. 10.5: Polar Area: Day 3 with Common Interior
Example 7b
A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. Using the calculator: b) Find the angle 𝜽 that corresponds to the point on the curve with 𝒙-coordinate 𝒙=−𝟐.
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10.5: Polar Area: Day 3 with Common Interior

32. 10.5: Polar Area: Day 3 with Common Interior
Example 7c
A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. For 𝝅 𝟑 <𝜽< 𝟐𝝅 𝟑 , 𝒅𝒓 𝒅𝜽 is negative. What does your answer tell you about 𝒓? What does it tell you about the curve?
𝒅𝒓 𝒅𝜽 is how fast the radius is changing with respects to rotation. Since 𝒅𝒓 𝒅𝜽 is negative it is away from the pole/origin.
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10.5: Polar Area: Day 3 with Common Interior

33. 10.5: Polar Area: Day 3 with Common Interior
Example 7d
A curve is drawn in the 𝒙𝒚-plane and is described by the equation in polar coordinates 𝒓=𝜽+𝐬𝐢𝐧 𝟐𝜽 for 𝟎≤𝜽≤𝝅. At what angle 𝜽 is the interval 𝟎≤𝜽≤ 𝝅 𝟐 the curve farthest away from the origin. Justify.
4/14/2019 1:09 PM
10.5: Polar Area: Day 3 with Common Interior