Accumulation AP Calculus AB Day 13 Instructional Focus: Evaluate definite integral numerically.

Foerster: Exploration 1.4 Modified Introduce how to evaluate definite integrals on a calculator. Foerster: Text page 22 #s 1, 2, 4

Rocket Problem: Ella Vader (Darth’s daughter) is driving in her rocket ship. At time t = 0 min, she fires her rocket engine. The ship speeds up for a while, then slows down as Alderaan’s gravity takes its effect. The graph of her velocity, 𝑣 𝑡 , in miles per minute, is shown below. Have a student read the context and have another student summarize the context. Have another student describe how the graph is consistent with the context described in words.

Estimate the distance in Problem 1 graphically by counting squares. What mathematical concept would you use to estimate the distance Ella goes between t = 0 and t = 8. Estimate the distance in Problem 1 graphically by counting squares. Chunking: #s 1-2 Have students in their groups complete Problems 1 and 2. Discussion: Have a student explain why a definite integral can be used to find the distance Ella travels in the 8 minutes. Be sure they explain how the unites are miles. There are about 60.8 squares or 1520 miles.

Ella figures that her velocity is given by 𝑣 𝑡 = 𝑡 3 −21 𝑡 2 +100𝑡+110 Plot this graph on your grapher. Does the graph confirm or refute what Ella figures? Tell how you arrive at your conclusion. Chunking #3 Be sure the students put the function v(t) in Y1.

Divide the region under the grap from t = 0 and t = 8, which represents the distance, into four vertical strips of equal width. Draw four trapezoids whose area approximate the areas of these strips and whose parallel sides extend from the x-axis to the graph. By finding the areas of these trapezoids, estimate the distance Ella goes. Does your answer agree with the answer to Problem 2? Chunking #4 As an introduction to the problems 4 and 5, remind students that we previously use trapezoids to estimate a definite integral when all we were given was a table of values. Now we will use a function rule 𝑣 𝑡 = 𝑡 3 −21 𝑡 2 +100𝑡+110 to generate the values we need. Have the students draw the four trapezoids on the graph given. Make sure they have done them correctly. Now the students will calculate the areas of the four trapezoids. 344+472+408+248=1472 miles.

Do you think the trapezoidal sum from Problem 4 overestimates or underestimates the actual distance Ella travels? Explain how you arrived at your conclusion. Chunking #5 This is a very important concept. Allow students to discuss this with each other and come to a decision. Then have students share their thoughts. Big idea is that the trapezoidal sum underestimates because the curve is concave down. The trapezoidal sum overestimates when the curve is concave up.

A trapezoidal sum is a Riemann sum, 𝑅 𝑛 = 𝑘=1 𝑛 𝑓 𝑐 𝑘 ∆ 𝑥 𝑘 , where 𝑓 𝑐 𝑘 is the average height. The exact value of the definite integral 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 is the limit of the estimates by trapezoids as more and more trapezoids are used or as the width of the trapezoids approaches zero.

Double the number of trapezoids to 8 and estimate the distance Ella travels. Does your answer agree with the answer to Problem 2? Do you think the trapezoidal sum still underestimates the actual distance Ella travels? Chunking #6 Have the students complete this problem using the previous directions to efficiently calculate this trapezoidal sum. If you want you can show a summation way to enter this without evaluating each trapezoid separately. Answer for eight trapezoids is 1508. Still should be an underestimate. You can have your students try to find a limit by increasing the number of trapezoids. Be patient because these calculations take a while. 50 trapezoids gives 1519.6928 100 trapezoids gives 1519.9232 150 trapezoids gives 1519.965867

When a function rule is given, like 𝑣 𝑡 = 𝑡 3 −21 𝑡 2 +100𝑡+110 graphing calculators have a built in operation to evaluate (estimate very accurately) a definite integral.

Use your graphing calculator to evaluate 0 8 𝑣 𝑡 𝑑𝑡 Use your graphing calculator to evaluate 0 8 𝑣 𝑡 𝑑𝑡 . Does your answer agree with the answer to Problem 2? Do you think the trapezoidal sums still underestimate the actual distance Ella travels? Chunking #7 Make sure the students can evaluate a definite integral on their calculator. 1520 All of the trapezoidal sums we did are less than 1520 but eventually got really close.

What is the fastest Ella went. At what time was that What is the fastest Ella went? At what time was that? Show how you arrived at your answers. Approximately what was Ella’s rate of change of velocity when t = 5? Was she speeding up or slowing down at this time? How do you know? Based on the equation in Problem 3, there are positive values of t at which Ella is stopped. What is the first such time? How did you find your answer? Chunking #s 8-10 These problems revisit rate of change problems. Have students complete these problems. 8. According to the graph, the fastest Ella went was 𝑣 3.0418… =248.0209… miles per minute. Students can use MAXIMUM feature at this point in the course. 9. 𝑣 ′ 5 =−35 miles per minute per minute; She is slowing down because the rate of change is negative. Note: Calculator may say -34.99999999 10. 𝑣 𝑡 =0⇒𝑡=11

Rate of Change and Accumulation Quiz