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**

STUDY: Geometry - Volume by Slicing and Shell Methods

**Supplemental Resources (recommended)**

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

- Wolfram Demo Project: Visualizing Revolutions
- Geogebra: Volume of Solid of Revolution - The Washers Method (slide the vertical bar up/down)
- Geogebra: Solid of Revolution - Washer 1, x axis and Solid of Revolution - Washer 2, y axis

Both excellent animations connecting the 2D graphs with 3D solid. - Shodor Interactivate: Function Revolution revolves function and calculates surface area and volume, but fairly old-style graphics

**Supplemental Resources (optional)**

Video: Volume Calculations I (Cross Sections), Selwyn Hollis's Video Calculus

Video: Volume Calculations II (Disks/Washers), Selwyn Hollis's Video Calculus

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

Lesson: Volume with Rings, Paul's Online Notes for Calc I

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

PDF: Volumes of Solids, Dale Hoffman's Contemporary Calculus

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