Limits of Riemann’s Sum Section 4.2B Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§4.2B: Limits of Riemann's Sum Summation Notation 𝐥𝐢𝐦 𝒏→∞ 𝒌=𝟎 𝒏 𝒇 𝒙 𝒌 ∆ 𝒙 𝒌 = 𝒂 𝒃 𝒇 𝒙 𝒅𝒙 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

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 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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. 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

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 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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

§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

§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

§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

§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

§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. 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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. 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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. 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§4.2B: Limits of Riemann's Sum Your Turn Write the limit which is equal to 𝟐 𝟓 𝟑𝒙 𝟐 +𝟏 𝒅𝒙 . 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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 . 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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 . 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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 . 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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 . 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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 . 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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 . 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

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 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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. 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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. 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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

§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. 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§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. 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

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 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

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 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum

§4.2B: Limits of Riemann's Sum Assignment Worksheet 11/22/2018 9:52 PM §4.2B: Limits of Riemann's Sum