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Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
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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 Slicing Methods

Overview

The previous lesson covered finding area of a bounded region in 2D. In this lesson we introduce using integrals to find the volume of certain 3D solids. 3D solids can take on a usual form such as a cylinder or cone. Known formulas such as V=(pi r^2 h)/3 can be used to find their volume. But some 3D solids are unusual in their form, yet do have rotational symmetry about a central axis. The volume of such solids of revolution can be found using integration via what is called the slicing methods (method of disks and method of washers). The volume of solids that are not formed by revolving about an axis can be found by the cross section method where the volume of thin slabs are summed to find the total volume. To assist in setting up the integrals, draw lots of sketches!

Objectives

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

• 6.3.1 Envision the shape of a solid of revolution given the graph of a function and axis of revolution
• 6.3.2 Sketch the slice of a solid of revolution and recognize it as a disk or washer
• 6.3.3 Set-up and evaluate an integral that gives the volume of a solid of revolution by the disk method when the axis of revolution is horizontal
• 6.3.4 Set-up and evaluate an integral that gives the volume of a solid of revolution by the washer method when the axis of revolution is horizontal
• 6.3.5 Set-up and evaluate an integral that gives the volume of a solid of revolution by the disk method when the axis of revolution is vertical
• 6.3.6 Set-up and evaluate an integral that gives the volume of a solid of revolution by the washer method when the axis of revolution is vertical
• 6.3.7 Envision a solid (one that is not a solid of revolution) being divided into a series of parallel slabs
• 6.3.8 Set-up and evaluate an integral that gives the volume of a solid (one that is not a solid of revolution) by the cross-section method with slabs taken perpendicular to the base

Terminology

Terms you should be able to define: axis of revolution, solid of revolution, slice, disk, washer, method of disks, method of washers, slicing methods, slab, cross-section, cross-section method

Mini-Lectures and Examples

Supplemental Resources (recommended)

Visualizing the disk/washer methods can be hard at first. Exploring these demonstrations may help:

Supplemental Resources (optional)

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

Lesson: More Volumes, Paul's Online Notes for Calc I covers finding volume of non-revolution solids by cross-section method

Examples: Volume of a Solid with a Known Cross Section, Math24's Calculus includes several examples of finding volume of non-revolution solids by cross-section method, but watch that you only study the functions in x and y, not the parametric (in terms of t) nor polar (in terms of theta).

rev. 2021-01-16