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# More Areas and Volumes

Lesson 17 of 17

## Objective: SWBAT explain and solve problems involving integrating cross-sectional areas of solids to compute their volumes.

## Big Idea: Students think algebraically and graphically to analyze common errors in setting up integrals for finding volumes of solids of revolution.

*65 minutes*

#### Warm-Up + Homework Review

*30 min*

Last night’s homework solutions appear in the In The Classroom file.

In today’s class I will begin by asking students to work on the Warm-up problems. Though not noted in the question prompt provided to students, Warm-up #1 should be solved 3 different ways:

- u-substitution with u = sin(x)
- u-substitution with u = cos(x)
- rewriting using the double-angle sin formula

I don’t ask all students to solve this one problem multiple ways right away because:

- I want students to have time for answering the other questions on the warm-up
- The class discussion for this question will involve asking students to reflect on why they chose the approach they used, and which other approaches they may have considered before choosing one particular approach to use.

Often students rush into solving problems with the first approach that comes to mind, and hopefully by this time in the year students have become somewhat more conditioned to (SMP #1) analyze the problem and consider various solution options before beginning to solve the problem. For students who finish early, I might verbally challenge them to integrate using another approach.

Warm-up #3 is designed as a review of the **Fundamental Theorem of Calculus**. I will be on the watch for students who do not see this connection based on the question prompt. As they work I will direct students to focus on structure (MP7) by Rearranging the FTC. Using generic function names allows me to intervene about the application of the FTC without giving away the specific answer to this problem. It is VERY important for students to understand this application of the rearranged FTC as “an initial y-value, plus the change in that y-value over some interval, equals the ending y-value”. To drive this application home with my students (pun intended), I refer students back to the Car Dashboard Video Project. I say:

**If I hadn’t reset my odometer before starting to drive, F(a) would be the initial reading of the odometer, the integral would give the distance I drove, and F(b) would be the final reading on the odometer. Since I did reset the odometer, F(a) = 0, which leaves the integral alone equaling the final reading on the odometer as the displacement from 0, which is what students computed for this project.**

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#### Investigation

*20 min*

This section of the lesson is short since the warm-up and closure sections include fairly substantial work for students to complete and for the whole class to discuss. Based on the class’s progress in yesterday’s lesson (Integrating Area to Get Volume), today’s lesson is a continuation of the areas between curves and volumes of solids review. I will pickup where we left off yesterday in the In The Classroom file and continue as far as possible (revisit that lesson plan for details about reviewing this content).

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- 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