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# Accumulate This!

Lesson 14 of 17

## Objective: SWBAT compute integrals with variable upper limits of integration.

## Big Idea: Why do integrals with variable upper limits of integration produce functions? Students answer this question with multiple representations.

*67 minutes*

#### Warm-Up + Homework Review

*40 min*

As usual class begins with a set of Warmup problems. As I circulate the room, I plan to watch for students who approach Warm-up #2 graphically and come up with only one value for *x*, typically the solution greater than 3. Students who approach this problem algebraically will more readily obtain both solutions since the algebraic approach involves solving a quadratic equation. I will point out the existence of two solutions to students who only find one solution, perhaps just by piquing their curiosity by saying, “I’ve noticed some other students found two solutions. How is that possible?” Then, I will let students wrestle with this possibility.

When going over Accumulate_This!_Warmup as a class, I will be sure to connect the algebraic and graphical approaches to the problems. I want my students to understand how switching the limits of integration switches the magnitude of positive and negative areas under the function. This concept tends to be an area of weakness, so I will also look for evidence in last night’s homework problems.

I will also be on the watch for students who solve algebraically by factoring expressions and misapplying the Zero Product Property (e.g., setting each factor equal to 14 after failing to move all terms to one side before factoring). Rather than reteaching the **Zero Product Property**, I will ask students to resubstitute solutions into the original equation. This will make the problem evident. I expect my students to be able to debug their original process.

Warm-up #3 reminds students that critical points occur where the first derivative equals zero or is undefined. I want my students to investigate these critical points graphically to improve student understanding and retention.

Students usually struggle with some of the tasks on last night’s AP question set for homework. So, I allow alot of time to review these questions, including the errors behind the most popular distractors. Last night’s homework solutions appear in the In The Classroom file.

*expand content*

- UNIT 1: Back to School
- UNIT 2: Limits and Derivatives
- UNIT 3: Formalizing Derivatives and Techniques for Differentiation
- UNIT 4: Applications of Differentiation, Part 1
- UNIT 5: Applications of Differentiation, Part 2
- UNIT 6: The Integral
- UNIT 7: Applications of Integration
- UNIT 8: Differential Equations
- UNIT 9: Full Course Review via Motion
- UNIT 10: The Final Stretch - Preparing for the AP Exam

- LESSON 1: Limits and l'Hospital
- LESSON 2: Know Your Limits
- LESSON 3: Local Linearization, 1st and 2nd Derivative Tests, and Computing Derivatives
- LESSON 4: Derivatives Algebraically and Graphically
- LESSON 5: The Calculus of Motion
- LESSON 6: Motion - Velocity on Intervals
- LESSON 7: Motion - Distance vs Displacement
- LESSON 8: Motion - With Multiple Derivatives
- LESSON 9: Motion and Optimization
- LESSON 10: Calculus and My Car's Dashboard
- LESSON 11: Rockin' Related Rates
- LESSON 12: Meet My Friend Riemann
- LESSON 13: Cookies and Pi
- LESSON 14: Accumulate This!
- LESSON 15: Wait, the Interval Width Varies?
- LESSON 16: Integrating Areas to Get Volumes
- LESSON 17: More Areas and Volumes