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 3rd 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
Here are several excellent applets for visualizing cross-sectional solids.