# Volumes of Solids of Revolution

#### Summary

This write-pair-share activity presents Calculus II students with a worksheet containing several exercises that require them to find the volumes of solids of revolution using disk, washer and shell methods and to sketch three-dimensional representations of the resulting solids. Afterwards, a Web-based tool is used to produce graphs of the solids and an interactive applet provides additional practice and feedback.

This activity inevitably brings to light student misunderstandings concerning the various radii involved and enables them to discover the cause of their misunderstandings and resultant errors. The write-pair-share mode is helpful in that it allows students to get immediate feedback from their partners on their attempts at drawing and labeling diagrams.

## Learning Goals

- develop their understanding of the methods of finding volumes of solids of revolution
- develop the skills of drawing and labeling diagrams and conceptual models
- bring to light and ultimately correct common misconceptions concerning the radii involved in volume calculations
- exercise the skill of constructing three-dimensional sketches of conceptual models

## Context for Use

This activity takes approximately 40 minutes to complete, not including the final phase involving additional practice.

## Description and Teaching Materials

- Students are given the write-pair-share activity worksheet (Rich Text File 25kB Jul25 06) and allowed time to work together in pairs. For several different axes of revolution, students are directed to draw large diagrams of the function being revolved, clearly label the important features in each diagram, calculate the respective volumes and construct three-dimensional sketches of the resulting solids of revolution.
- Afterwards, the instructor shares the correct answers with the students in order to correct any misunderstandings concerning the radii or method involved in each exercise.
- In addition, the instructor produces rotatable 3-D graphs of the solids of revolution by using the graphing applet listed below and the related Graphing Guide (Rich Text File 33kB Jul25 06). This allows students to clearly and easily see the results of choosing different axes of revolution.
- Mathematics Visualization Toolkit (MVT)-an excellent set of graphing tools

http://amath.colorado.edu/java/

Access path: Applications/Solids of Revolution App - MERLOT description of this resource

http://www.merlot.org/artifact/ArtifactDetail.po?oid=1400000000000026304 - Lastly, students are directed to practice additional exercises involving volumes of solids of revolution through an interactive Java applet. Approximately one-third to one-half of the exercises in each set involve axes of revolution that are different than the major axes. This practice can be assigned for homework or can take place in a computer lab or in a wireless classroom with laptops.
- Calculus on the Web (COW)-a collection of interactive exercises

http://cow.math.temple.edu/~cow/

Access path: Calculus Book II/Applications of Integration/Volume/Solids of Revolution-washers and shells - MERLOT description of this resource

http://www.merlot.org/artifact/ArtifactDetail.po?oid=1010000000000239400

## Teaching Notes and Tips

In addition, many students claim to have little or no skill in constructing three-dimensional sketches. While their efforts are generally somewhat ineffective, the MVT graphing tool is simple and easy to use and provides immediate feedback.

No matter how many examples may be presented by an instructor, student understanding develops only through practice and feedback. The worksheet and subsequent interactive exercises are designed to provide both of these ingredients for success.

## Assessment

## References and Resources

- Volume of Solid of Revolution

A tutorial on finding volumes using examples that illustrate the disk and shell methods as well as horizontal and vertical axes of revolution; clear, multi-colored diagrams of important features; some animations; includes a brief review of the disk, washer and shell methods

MERLOT description of the Volume of Solid of Revolution site.

- Volumes-Visual Calculus

Tutorials and animations (Flash and Java) on finding volumes of solids of revolution; helpful for students who are visual learners

MERLOT description of the Volumes-Visual Calculus site.

Direct link to the Volumes-Visual Calculus site. - Gallery of Volume Visualizations

An extensive gallery of colorful visualizations for disk, washer, shell and cross-section methods of finding volume; useful for viewing the incremental approach in constructing three-dimensional solids and an excellent source for classroom demos

MERLOT description of Gallery of Volume Visualizations site.