1. Limits of Riemann’s Sum
Section 4.2B
Calculus AP/Dual, Revised ©2017
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§4.2B: Limits of Riemann's Sum

2. §4.2B: Limits of Riemann's Sum
Summation Notation
𝐥𝐢𝐦 𝒏→∞ 𝒌=𝟎 𝒏 𝒇 𝒙 𝒌 ∆ 𝒙 𝒌 = 𝒂 𝒃 𝒇 𝒙 𝒅𝒙
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§4.2B: Limits of Riemann's Sum

3. Breaking Up Riemann’s Sum
Height of the Rectangle
𝐥𝐢𝐦 𝒏→∞ 𝒌=𝟏 𝒏 𝒇 𝒙 𝒌 ∆ 𝒙 𝒌 = 𝒂 𝒃 𝒇 𝒙 𝒅𝒙
Width of the Rectangle =
∆𝒙= 𝒃−𝒂 𝒏
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§4.2B: Limits of Riemann's Sum

4. §4.2B: Limits of Riemann's Sum
Review
The Limit Equation for Riemann’s Sum It is more traditional to use 𝒌=𝟎 for left or midpoint sums, and 𝒌=𝟏 for right sums.
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§4.2B: Limits of Riemann's Sum

5. Sigma Notation Breakdowns
𝒌=𝟏 𝒏 𝟏 =𝒏
𝒌=𝟏 𝒏 𝒌 = 𝒏(𝒏+𝟏) 𝟐
𝒌=𝟏 𝒏 𝒌 𝟐 = 𝒏(𝒏+𝟏)(𝟐𝒏+𝟏) 𝟔
𝒌=𝟏 𝒏 𝒌 𝟑 = 𝒏 𝟐 𝒏+𝟏 𝟐 𝟒
𝒌=𝟎 𝒏 𝒓 𝒌 = 𝟏− 𝒓 𝒏+𝟏 𝟏−𝒓 , 𝒓≠𝟏
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§4.2B: Limits of Riemann's Sum

6. §4.2B: Limits of Riemann's Sum
Example 1
Write the limit which is equal to 𝟎 𝟓 𝒙 𝒅𝒙 and then solve for the integral.
11/22/2018 9:52 PM
§4.2B: Limits of Riemann's Sum

7. §4.2B: Limits of Riemann's Sum
Example 1
Write the limit which is equal to 𝟎 𝟓 𝒙 𝒅𝒙 and then solve for the integral.
11/22/2018 9:52 PM
§4.2B: Limits of Riemann's Sum

8. §4.2B: Limits of Riemann's Sum
Example 1
Write the limit which is equal to 𝟎 𝟓 𝒙 𝒅𝒙 and then solve for the integral.
11/22/2018 9:52 PM
§4.2B: Limits of Riemann's Sum

9. §4.2B: Limits of Riemann's Sum
Example 1
Write the limit which is equal to 𝟎 𝟓 𝒙 𝒅𝒙 and then solve for the integral.
11/22/2018 9:52 PM
§4.2B: Limits of Riemann's Sum

10. §4.2B: Limits of Riemann's Sum
Example 1
Write the limit which is equal to 𝟎 𝟓 𝒙 𝒅𝒙 and then solve for the integral.
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§4.2B: Limits of Riemann's Sum

11. §4.2B: Limits of Riemann's Sum
Example 2
Write the limit which is equal to 𝟎 𝟐 𝒙 𝟐 𝒅𝒙 and then solve for the integral.
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§4.2B: Limits of Riemann's Sum

12. §4.2B: Limits of Riemann's Sum
Example 3
Use the graph to write the limit for the area of the shaded region given the graph is 𝒇 𝒙 =𝐥𝐧 𝒙.
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§4.2B: Limits of Riemann's Sum

13. §4.2B: Limits of Riemann's Sum
Example 3
Use the graph to write the limit for the area of the shaded region given the graph is 𝒇 𝒙 =𝐥𝐧 𝒙.
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§4.2B: Limits of Riemann's Sum

14. §4.2B: Limits of Riemann's Sum
Your Turn
Write the limit which is equal to 𝟐 𝟓 𝟑𝒙 𝟐 +𝟏 𝒅𝒙 .
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§4.2B: Limits of Riemann's Sum

15. §4.2B: Limits of Riemann's Sum
Example 4
Write the limit using Integral Notation, 𝐥𝐢𝐦 𝒏→∞ 𝒌=𝟏 𝒏 𝟑 𝒏 𝟑 𝟑𝒌 𝒏 .
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§4.2B: Limits of Riemann's Sum

16. §4.2B: Limits of Riemann's Sum
Example 4
Write the limit using Integral Notation, 𝐥𝐢𝐦 𝒏→∞ 𝒌=𝟏 𝒏 𝟑 𝒏 𝟑 𝟑𝒌 𝒏 .
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§4.2B: Limits of Riemann's Sum

17. §4.2B: Limits of Riemann's Sum
Example 5
Write the limit using Integral Notation, 𝐥𝐢𝐦 𝒏→∞ 𝒌=𝟏 𝒏 𝟐 𝒏 𝟑+ 𝟐𝒌 𝒏 𝟑 + 𝟑+ 𝟐𝒌 𝒏 .
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§4.2B: Limits of Riemann's Sum

18. §4.2B: Limits of Riemann's Sum
Example 5
Write the limit using Integral Notation, 𝐥𝐢𝐦 𝒏→∞ 𝒌=𝟏 𝒏 𝟐 𝒏 𝟑+ 𝟐𝒌 𝒏 𝟑 + 𝟑+ 𝟐𝒌 𝒏 .
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§4.2B: Limits of Riemann's Sum

19. §4.2B: Limits of Riemann's Sum
Example 6
Write the limit using Integral Notation, 𝐥𝐢𝐦 𝒏→∞ 𝒌=𝟏 𝒏 𝟏+ 𝟏+ 𝟕𝒌 𝒏 𝟓 𝟕 𝒏 .
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§4.2B: Limits of Riemann's Sum

20. §4.2B: Limits of Riemann's Sum
Your Turn
Write the limit using Integral Notation, 𝐥𝐢𝐦 𝒏→∞ 𝒌=𝟏 𝒏 𝟑 𝒏 𝟐+ 𝟑𝒌 𝒏 𝟒 +𝟒 .
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§4.2B: Limits of Riemann's Sum

21. Left and Right Endpoints
Left-Side Endpoint equation: 𝐥𝐢𝐦 𝒏→∞ 𝒌=𝟎 𝒏−𝟏 𝒇 𝑪 𝑲 ∆ 𝒙 𝑲
Right-Side Endpoint equation: 𝐥𝐢𝐦 𝒏→∞ 𝒌=𝟏 𝒏 𝒇 𝑪 𝑲 ∆ 𝒙 𝑲
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§4.2B: Limits of Riemann's Sum

22. §4.2B: Limits of Riemann's Sum
Example 7
Write a left-endpoint and right-endpoint Riemann Sum 𝑹 𝟒 that approximates 𝟎 𝟏 𝒙 𝒅𝒙. Do not solve.
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§4.2B: Limits of Riemann's Sum

23. §4.2B: Limits of Riemann's Sum
Example 7a
Write a left-endpoint and right-endpoint Riemann Sum 𝑹 𝟒 that approximates 𝟎 𝟏 𝒙 𝒅𝒙. Do not solve.
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§4.2B: Limits of Riemann's Sum

24. §4.2B: Limits of Riemann's Sum
Example 7b
Write a left-endpoint and right-endpoint Riemann Sum 𝑹 𝟒 that approximates 𝟎 𝟏 𝒙 𝒅𝒙. Do not solve.
11/22/2018 9:52 PM
§4.2B: Limits of Riemann's Sum

25. §4.2B: Limits of Riemann's Sum
Example 7
Write a left-endpoint and right-endpoint Riemann Sum 𝑹 𝟒 that approximates 𝟎 𝟏 𝒙 𝒅𝒙. Do not solve.
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§4.2B: Limits of Riemann's Sum

26. §4.2B: Limits of Riemann's Sum
Your Turn
Write a left-endpoint and right-endpoint Riemann Sum 𝑹 𝟓𝟎 that approximates 𝟎 𝟏 𝒙 𝒅𝒙. Do not solve.
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§4.2B: Limits of Riemann's Sum

27. AP Multiple Choice Practice Question 1 (non-calculator)
Which of the following approximates the area between 𝒈 𝒙 = 𝟓 𝒙 +𝟐 and the 𝒙-axis on the interval 𝟏,𝟕 using a Right Riemann sum with 9 equal subdivisions? (A) 𝒌=𝟏 𝟗 𝟏𝟓 𝟑+𝟐𝒌 +𝟐 (B) 𝒌=𝟎 𝟖 𝟏𝟓 𝟑+𝟐𝒌 +𝟐 (C) 𝒌=𝟏 𝟗 𝟏𝟓 𝟑+𝟐𝒌 +𝟐 ∙ 𝟐 𝟑 (D) 𝒌=𝟎 𝟖 𝟏𝟓 𝟑+𝟐𝒌 +𝟐 ∙ 𝟐 𝟑
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§4.2B: Limits of Riemann's Sum

28. AP Multiple Choice Practice Question 1 (non-calculator)
Which of the following approximates the area between 𝒈 𝒙 = 𝟓 𝒙 +𝟐 and the 𝒙-axis on the interval 𝟏,𝟕 using a Right Riemann sum with 9 equal subdivisions?
Vocabulary
Process and Connections
Answer
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§4.2B: Limits of Riemann's Sum

29. §4.2B: Limits of Riemann's Sum
Assignment
Worksheet
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§4.2B: Limits of Riemann's Sum