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# Volumes of Cross-Sectional Solids (part 2 of 4)

Lesson 10 of 14

## Objective: SWBAT write variable expressions for areas of basic geometric shapes; SWBAT compute volumes of cross-sectional solids.

## Big Idea: No matter how you slice it, finding volumes of cross-sectional solids is a blast! Today we continue with volumes of cross-sectional solids.

*60 minutes*

#### Setting the Stage

*10 min*

The Bottle Project

Prior to continuing with cross-sectional solids, we first introduce the bottle project. **Distribute the Bottle project Description Sheet and centimeter graph paper, then briefly discuss how both solids of revolution and cross-sectional solids can be used to model many bottles and estimate their volumes.** Also note for students that a primary part of this project is in discussing possible sources of error (DO NOT SHOW TO STUDENTS) in students’ estimates, because for a variety of reasons students will not obtain the exact volumes as listed on the labels. I usually make the project due sometime after two weekends have passed, but not the 3^{rd} weekend; this allows students some flexibility in when they work on the project without making its due date too far away from our study of these topics.

Cross-Sectional Solids Examples Sheet

**Distribute the Cross-Sectional Solids Examples Sheet,** which like the Solids of Revolution sheet gives students several different examples integrating various cross-sectional shapes with respect to *x* and *y*. Briefly discuss each example, observe that the same process was repeated in each example, and note how the pictures were sketched and labeled prior to setting up each integral.

**Resources**: Bottle Project Description sheet, Cross-Sectional Solids Examples sheet

##### Resources (7)

#### Resources

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Here are several excellent applets for visualizing cross-sectional solids.

http://www.stewartcalculus.com/tec/

http://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2xsection2.htm

http://web.monroecc.edu/manila/webfiles/pseeburger/secure/MyLarson/ch7/LC7_2xsection1.htm

http://web.monroecc.edu/manila/webfiles/calcNSF/JavaCode/other/myXSection.htm

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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: Easy as ABC - Areas Between Curves
- LESSON 2: Areas Between Curves - dx or dy?
- LESSON 3: Variable Limits of Integration and the Fundamental Theorem of Calculus
- LESSON 4: Catch-up Day - ABC and FTC
- LESSON 5: Volumes of Solids of Revolution (part 1 of 4)
- LESSON 6: Volumes of Solids of Revolution (part 2 of 4)
- LESSON 7: Volumes of Solids of Revolution (part 3 of 4)
- LESSON 8: Volumes of Solids of Revolution (part 4 of 4)
- LESSON 9: Volumes of Cross-Sectional Solids (part 1 of 4)
- LESSON 10: Volumes of Cross-Sectional Solids (part 2 of 4)
- LESSON 11: Volumes of Cross-Sectional Solids (part 3 of 4)
- LESSON 12: Volumes of Cross-Sectional Solids (part 4 of 4)
- LESSON 13: Applications and Problem Solving with Volumes of Solids (part 1 of 2)
- LESSON 14: Applications and Problem Solving with Volumes of Solids (part 2 of 2)