1. Section 4.2 Calculus AP/Dual, Revised ©2015 viet.dang@humble.k12.tx.us
Riemann’s Sum
Section 4.2
Calculus AP/Dual, Revised ©2015
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4.2 - Riemann's Sum

2. Summation Notation
is the representation of the summation from 𝒊 = 𝟏 to 𝒊 = 𝒏
𝒊 is the index of summation
𝒂𝒊 is the 𝒊𝒕𝒉 term of the sum
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3. How to Approximate the Area Under the Curve?
Left Handed Rule It is an overestimation
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4.2 - Riemann's Sum

4. How to Approximate the Area Under the Curve?
Right Handed Sum It is an underestimation
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5. How to Approximate the Area Under the Curve?
Midpoint Rule
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4.2 - Riemann's Sum

6. Riemann’s Sum
In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph
If 𝒇 is closed on the interval from 𝒂, 𝒃 and the limit exists:
Then 𝒇 in integrable on 𝒂, 𝒃 and the limit is denoted by:
Where 𝒇 𝒙𝒊 is the height of the function at the value, 𝒙𝒊
Partitions: ∆𝒙= 𝒃−𝒂 𝒏
𝒙𝒊=𝒂+𝒊 ∆𝒙
𝐥𝐢𝐦 𝒏→∞ 𝒊=𝟎 𝒏 𝒇 𝒎 𝒊 ∆ 𝒙 𝒊 𝒐𝒓 𝐥𝐢𝐦 𝒏→∞ 𝒊=𝟎 𝒏 𝒇 𝑴 𝒊 ∆ 𝒙 𝒊
lim 𝑛→∞ 𝑖=0 𝑛 𝑓 𝑥 𝑖 ∆ 𝑥 𝑖 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
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4.2 - Riemann's Sum

7. Breaking Up Riemann’s Sum
Height of the Rectangle
𝐥𝐢𝐦 𝒏→∞ 𝒊=𝟏 𝒏 𝒇 𝒙 𝒊 ∆ 𝒙 𝒊 = 𝒂 𝒃 𝒇 𝒙 𝒅𝒙
Width of the Rectangle =
∆𝒙= 𝒃−𝒂 𝒏
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8. Breakdown of Riemann’s Sum
Left hand approximation is the left of the sub-interval is used to determine the height of the rectangle.
Right hand approximation is the right of the sub-interval is used to determine the height of the rectangle.
Midpoint approximation is the midpoint of each sub-interval us used to determine the height of each rectangle.
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4.2 - Riemann's Sum

9. Steps
Identify the missing parts of 𝒂, 𝒃 and the amount of subintervals, 𝒏
The Riemann Sum Formula are as follows:
Establish the amount of rectangles width (partitions) used to approximating an integral, ∆𝒙= 𝒃−𝒂 𝒏
Identify which summation is asked and apply the equation and repeat the process
𝐥𝐢𝐦 𝒏→∞ 𝒊=𝟎 𝒏 𝒇 𝒙 𝒊 ∆ 𝒙 𝒊 = 𝒂 𝒃 𝒇 𝒙 𝒅𝒙
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4.2 - Riemann's Sum

10. File.
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11. Example 1
Calculate the left & right hand sum and midpoint sum for 𝒚=𝒙𝟐+𝟐 with 𝟐 equal intervals from 𝟎, 𝟖 .
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4.2 - Riemann's Sum

12. Example 1
Calculate the left & right hand sum and midpoint sum for 𝒚=𝒙𝟐+𝟐 with 𝟐 equal intervals from 𝟎, 𝟖 .
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4.2 - Riemann's Sum

13. Example 1
Calculate the left & right hand sum and midpoint sum for 𝒚=𝒙𝟐+𝟐 with 𝟐 equal intervals from 𝟎, 𝟖 .
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4.2 - Riemann's Sum

14. Example 1
Calculate the left & right hand sum and midpoint sum for 𝒚=𝒙𝟐+𝟐 with 𝟐 equal intervals from 𝟎, 𝟖 .
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4.2 - Riemann's Sum

15. Example 1
Calculate the left & right hand sum and midpoint sum for 𝒚=𝒙𝟐+𝟐 with 𝟐 equal intervals from 𝟎, 𝟖 .
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4.2 - Riemann's Sum

16. Example 2
Calculate the left, right, and midpoint sum for 𝟎 𝟐 𝒙 𝟐 +𝟏 𝒅𝒙 with 𝟒 equal intervals from 𝟎, 𝟐 .
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4.2 - Riemann's Sum

17. Example 2
Calculate the left, right, and midpoint sum for 𝟎 𝟐 𝒙 𝟐 +𝟏 𝒅𝒙 with 𝟒 equal intervals from 𝟎, 𝟐 .
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4.2 - Riemann's Sum

18. Example 2
Calculate the left, right, and midpoint sum for 𝟎 𝟐 𝒙 𝟐 +𝟏 𝒅𝒙 with 𝟒 equal intervals from 𝟎, 𝟐 .
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4.2 - Riemann's Sum

19. Example 2
Calculate the left, right, and midpoint sum for 𝟎 𝟐 𝒙 𝟐 +𝟏 𝒅𝒙 with 𝟒 equal intervals from 𝟎, 𝟐 .
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4.2 - Riemann's Sum

20. Example 2
Calculate the left, right, and midpoint sum for 𝟎 𝟐 𝒙 𝟐 +𝟏 𝒅𝒙 with 𝟒 equal intervals from 𝟎, 𝟐 .
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21. Example 3
Evaluate the Riemann sum for 𝒇 𝒙 =𝒙𝟑−𝟔𝒙, taking the sample points to be right endpoints and 𝒂=𝟎, 𝒃=𝟑, and 𝒏=𝟔.
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22. Example 3
Evaluate the Riemann sum for 𝒇 𝒙 =𝒙𝟑−𝟔𝒙, taking the sample points to be right endpoints and 𝒂=𝟎, 𝒃=𝟑, and 𝒏=𝟔.
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23. Your Turn
Calculate the left, right, and midpoint sum for 𝒇 𝒙 = 𝒙 with 𝟒 equal intervals from [𝟎, 𝟏].
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24. Example 4
Calculate the left hand sum for 𝒚=𝒙−𝟐 𝐜𝐨𝐬⁡𝟐𝒙 with 𝟒 equal intervals from 𝟎, 𝟐 .
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25. Example 5
Use the graph below to calculate the left, right, and midpoint sum with 𝟑 equal intervals from [𝟎, 𝟔].
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26. Example 5
Use the graph below to calculate the left, right, and midpoint sum with 𝟑 equal intervals from [𝟎, 𝟔].
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27. Example 6
The function 𝒇 is continuous on the closed interval [𝟐, 𝟏𝟒] and has the values as shown in the table below. Using the subintervals [𝟐, 𝟓], [𝟓, 𝟏𝟎] and [𝟏𝟎, 𝟏𝟒], what is the approximation of 𝟐 𝟏𝟒 𝒇 𝒙 𝒅𝒙 found using the right Riemann Sum? 𝒙 𝟐 𝟓 𝟏𝟎 𝟏𝟒
𝒇(𝒙) 𝟏𝟐 𝟐𝟖 𝟑𝟒 𝟑𝟎
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28. Your Turn
The function 𝒇 is continuous on the closed interval 𝟎, 𝟏𝟎 and has the values as shown in the table below. What is the approximation of 𝟎 𝟏𝟎 𝒇 𝒙 𝒅𝒙 found using the left Riemann Sum? 𝒙 𝟎 𝟒 𝟔 𝟕 𝟏𝟎
𝒇(𝒙) 𝟓 𝟑 𝟐
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29. AP Multiple Choice Practice Question
Selected values of the continuous function, 𝒇 𝒙 are given in the table. Using 4 subintervals of Left Riemann’s Sum, determine the approximation for 𝟎 𝟏𝟎𝟎 𝒇 𝒙 𝒅𝒙 (A) 𝟏𝟕,𝟏𝟒𝟎 (B) 𝟏𝟕,𝟕𝟗𝟓 (C) 𝟏𝟖,𝟒𝟐𝟓 (D) 𝟏𝟖,𝟒𝟓𝟎 𝒙 𝟎 𝟒𝟎 𝟕𝟎 𝟗𝟎
𝟏𝟎𝟎
𝒇 𝒙
𝟏𝟓𝟎
𝟏𝟖𝟎
𝟏𝟗𝟓
𝟏𝟖𝟒
𝟏𝟕𝟐
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30. Assignment
Worksheet
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4.2 - Riemann's Sum