Accumulation AP Calculus AB Days 11-12 Instructional Focus: Estimate the definite integral using a Riemann Sum using left, right or midpoint sample points.

Exploration Riemann Sums and Definite Integrals Discuss why a Riemann sum underestimates or overestimates. The purpose of this activity is to move from calculating values of definite integrals by geometry and counting squares or from data tables with trapezoids and Riemann Sums. In this activity a rule for the function is given the formal definition of definite integrals is given and applied.

The symbol 1 4 𝑥 2 𝑑𝑥 stands for the definite integral of 𝑥 2 from x = 1 to x = 4. By counting squares, find an approximation of 1 4 𝑥 2 𝑑𝑥 graphed. Chunking: Problem 1 Answers will vary and this part uses the previous strategy of counting squares. Answer should be approximately 21 (the exact answer).

Find the values of 𝑓 𝑥 = 𝑥 2 for the values of x = c. On the figure, draw rectangles between vertical grid lines with altitudes equal to 𝑓 𝑐 , for the values of c in problem 2, starting at x = 1 and ending at x = 4. Do these rectangles over or under estimate 1 4 𝑥 2 𝑑𝑥? The sum of the areas of the rectangles mentioned in problem 3 is called a left-endpoint Riemann sum. Evaluate this sum. (In this particular case this is also the lower Riemann sum.) c f(c) 1 2 3 Chunking: Problems 2-4 Relate this table for construction of a Riemann sum to the earlier work done using data and Riemann sums. This table will produce a left Riemann sum. These rectangles will under estimate because the function is increasing. Be sure to emphasize this concept in the discussion. 1 4 𝑥 2 𝑑𝑥≈ 1 1 + 4 1 + 9 1 =14

Find the values of 𝑓 𝑥 = 𝑥 2 for the values of x = c. On the figure above, draw rectangles between vertical grid lines with altitudes equal to 𝑓 𝑐 , for the values of c in problem 5, starting at x = 1 and ending at x = 4. Do these rectangles over or under estimate 1 4 𝑥 2 𝑑𝑥 ? The sum of the areas of the rectangles mentioned in problem 6 is called a right-endpoint Riemann sum. Evaluate this sum. (In this particular case this is also the upper Riemann sum.) c f(c) 2 3 4 Chunking: Problems 5-7 Relate this table for construction of a Riemann sum to the earlier work done using data and Riemann sums. This table will produce a right Riemann sum. These rectangles will over estimate because the function is increasing. Be sure to emphasize this concept in the discussion. 1 4 𝑥 2 𝑑𝑥 ≈ 4 1 + 9 1 + 16 1 =29

Find the values of 𝑓 𝑥 = 𝑥 2 for the values of x = c. On the figure, draw rectangles between vertical grid lines with altitudes equal to 𝑓 𝑐 , for the values of c in problem 8, starting at x = 1 and ending at x = 4. The sum of the areas of these rectangles is called a midpoint Riemann sum. Evaluate this sum. c f(c) 1.5 2.5 3.5 Chunking: Problems 8-9 Relate this table for construction of a Riemann sum to the earlier work done using data and Riemann sums. This table will produce a midpoint Riemann sum. 1 4 𝑥 2 𝑑𝑥 ≈ 2.25 1 + 6.25 1 + 12.25 1 =20.75

Definition: Riemann Sum A Riemann sum, 𝑅 𝑛 , for a function f on the interval 𝑎, 𝑏 is a sum of the form 𝑅 𝑛 = 𝑘=1 𝑛 𝑓 𝑐 𝑘 ∆ 𝑥 𝑘 where the interval 𝑎, 𝑏 is divided (partitioned) into n subintervals of widths ∆ 𝑥 𝑘 , and the numbers 𝑐 𝑘 are sample points, one in each interval.

In problem 8, the value of n is 3, so 𝑅 3 = 𝑘=1 3 𝑐 𝑘 2 ∆ 𝑥 𝑘 = 𝑐 1 2 ∆ 𝑥 1 + 𝑐 2 2 ∆ 𝑥 2 + 𝑐 3 2 ∆ 𝑥 3 What are the values of 𝑐 𝑘 and ∆ 𝑥 𝑘 ? 𝑐 1 = ∆ 𝑥 1 = 𝑐 2 = ∆ 𝑥 2 = 𝑐 3 = ∆ 𝑥 3 = What is the value of 𝑅 3 ? Chunking: Problem 10 Share the definition of a Riemann sum with the students. The purpose of these questions is to make an abstract definition more concrete. Make sure the students see to connection of the work done in problems 2-9. In the formal definition ∆ 𝑥 𝑘 do not have to be equal. But for simplification of set up and calculations, generally we make them the same. Technology note: Storing the function in Y1 using the ANS key can help with this cumulative sum.

The exact value of 1 4 𝑥 2 𝑑𝑥 is between what two values? Using equal values of ∆ 𝑥 𝑘 , evaluate the left-endpoint Riemann sum, 𝑅 6 , for 1 4 𝑥 2 𝑑𝑥 . Explain your answer. Using equal values of ∆ 𝑥 𝑘 , evaluate the right-endpoint Riemann sum, 𝑅 6 , for 1 4 𝑥 2 𝑑𝑥 . Explain your answer. The exact value of 1 4 𝑥 2 𝑑𝑥 is between what two values? Chunking: Problems 11-13 11. 17.375 12. 24.875 13. The exact value is between 17.375 and 24.875