Elementary Algebra
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Calculus I
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Calculus II
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Calculus III

Course: Calculus II
Topic: Geometry
Subtopic: Volume by Shell Method

Overview

When finding the volume of a solid of revolution, some solids lend themselves nicely to the slicing methods covered in the previous lesson. Others are better suited for a second method for finding volume called the shell method. This method finds the volume of nested cylindrical shells then uses integration to sum them for the total area of the solid.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 6.4.1 Envision a solid of revolution being divided into a series of nested cylindrical shells
• 6.4.2 Set-up and evaluate an integral that gives the volume of a solid of revolution by the shell method when the axis of revolution is vertical
• 6.4.3 Set-up and evaluate an integral that gives the volume of a solid of revolution by the shell method when the axis of revolution is horizontal

Terminology

Terms you should be able to define: cylindrical shell, nested cylinders, shell method

Mini-Lectures and Examples

Supplemental Resources (recommended):

Visualizing the shell method can be hard at first. Exploring these demonstrations may help:

Supplemental Resources (optional)

Lesson: Solids of Revolution by Shells, Math is Fun (introductory information/examples)

PDF: Additional Applications, Dale Hoffman's Contemporary Calculus (starting on page 5, "Volumes of Revolution Using Tubes (Shells)")

rev. 2021-01-09