More Areas and Volumes

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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.

Video Narrative - Lesson Overview

Warm-Up + Homework Review

30 minutes

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:

  1. I want students to have time for answering the other questions on the warm-up
  2. 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.



20 minutes

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).

Closure + Homework

15 minutes
Closure + Homework.docx page 1