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=πr^{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:

- 6.2.1 Envision the shape of a solid of revolution given the graph of a function and axis of revolution
- 6.2.2 Sketch the slice of a solid of revolution and recognize it as a disk or washer
- 6.2.3 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
- 6.2.4 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
- 6.2.5 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
- 6.2.6 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
- 6.2.7 Envision a solid (one that is not a solid of revolution) being divided into a series of parallel slabs
- 6.2.8 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**

Define: axis of revolution, solid of revolution, slice, disk, washer, method of disks, method of washers, slicing methods, slab, cross-section, cross-section method

**Supplemental Resources (recommended):**

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

- Calculus Applets: Volumes of Revolution
- Larson Calculus: Illustrating the Washing Method
- Wolfram Demo Project: Visualizing Revolutions

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