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# Volumes of Solids of Revolution

This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

#### 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

To enable students to:
• 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 works best in a small class but can also be carried out in a large lecture setting anytime during or after students have been introduced to the concept of volume of a solid of revolution. In a small class, the instructor can circulate throughout the classroom and help individual pairs of students who are having difficulty constructing and labeling diagrams correctly; in a large lecture setting, this is not possible but, after giving student pairs time to work together on the exercises, the instructor can then present the correct diagrams to the entire class.

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

## Description and Teaching Materials

Activity description:
• 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
https://www.merlot.org/merlot/viewMaterial.htm?id=83836&hitlist=keywords%3DCalculus%2520on%2520the%2520Web&fromUnified=true

## Teaching Notes and Tips

Some students have difficulty constructing and labeling diagrams accurately. At the outset, their chief difficulty lies in determining which volume method to use-disk, washer or shell; afterwards, their chief difficulty arises in determining the appropriate radii when the axis of revolution is not one of the major axes. In these cases, the fact that a radius of revolution needs to be connected to the axis of revolution seems to escape their attention.

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

This activity is graded on the percentage of correctly labeled diagrams on the worksheet; I assign a point value that is relatively small and represents approximately 2% of a semester grade.

## References and Resources

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