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

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

40 minutes

10 minutes

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

7 minutes

3 minutes

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