8.4 Volumes of Solids Formed by Revolving Regions Around an Axis

In this lesson, you will learn to find the volume of a solid formed by revolving a region around an axis that produces a cross-section that is either a disc or a washer.

Lesson Objectives

Upon completion of the lesson 8.4, you will be able to:

• • Use the disc method to calculate the volume of a solid formed by revolving a region about an axis that produces a cross-section that is a disc.

  • Use the washer method to calculate the volume of a solid formed by revolving a region between curves about an axis that produces a cross-section that is a washer.

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

• 1. YouTube videos: Solid of Revolution

  2. Part 1:

• 1. Part 2:

• 1. Part 4:

• 1. Part 6:

• 1. YouTube videos: Volumes of Revolution: Disk/Washers

  2. Example 1:

• 1. Example 3:

• 1. Example 4:

• 1. YouTube video: Volumes of Revolution: Disk/Washer about vertical lines

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.

• 1. Volumes of Solids of Revolution - Disk & Washers [Eastern Iowa Community Colleges]

  2. (rotation about the x-axis, rotation about the y-axis, examples with solutions)

  3. Volumes of Solids of Revolution/Methods of Rings [Paul's Online Math Notes]

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

• 1. Calculus/Volume of Solids of Revolution/solutions [Wikibooks]

  2. Note: Do problems #1 - 4 only. To get the solution, click on the triangular figure right before the problem number.

  3. Volumes of Solids of Revolution/Method of Rings [Paul's Online Math Notes]

  4. Note: Skip problem 5.

Additional Resources

Below are additional resources that help reinforce the content for this module.

• 1. YouTube video: Solid of Revolution - Part 3

• 1. Volume Calculations 2: Solids of Revolution [University of Houston]

  2. Volumes of Solids [Bellevue College]

  3. Note: Read Volumes of Revolved Regions and Wrap Up from pp. 385 - 388. Do the Practice 5 - 7. The answers to the practice problems are on p. 393.

  4. Area & Volume Tutorial [Nipissing University]

  5. Note: Scroll down the section titled Volumes. Read the sections titled Volumes, Variation of Volume Problems, the Disk Method, the Washer Method and view examples 4 & 5.