Slice it! Rotate it!: Volumes of revolution

The students will first view the animated slicing and rotating of various area about the x-axis or about different lines at internet sites. Then the student will complete a lab experience in which various food items will be examined as to what planar region created them.

A lesson plan for grades 9–12 Mathematics

By Sharon Whitted

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Related topics

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Learning outcomes

Students will:

• visualize the rotation of a function about the x or y-axes or about a horizontal or vertical line.
• take very common food items and imagine these items revolved about an axis.
• mentally visualize the planar region that created the three-dimensional object.

Teacher planning

Time required for lesson

3 hours

Materials/resources

• various shapes of candy (chocolate kisses, M&Ms, etc.)
• bamboo skewers

Technology resources

• computer for each student
• access to the internet

Pre-activities

The students should have already learned how to find area between two curves. They should also understand Riemann sums, how to integrate basic functions, and understand the fundamental theorem of calculus.

Activities

Session One

1. In groups of two the students will access the internet and go to the Applications of Integration site. The site will show the student step by step how to understand how the volume of a three-dimensional shape can be formed by rotating a planar region about an axis. The site goes through many examples and the students should examine each of these. The students will have clearer visual imagery of these problems than merely learning this from a text. The students should discuss the animations with each other and attempt the exercises that the site provides for them to do.
2. Allow students to search for other sites. Have students share sites they find that are good with each other. At the end of this 90-minute period get the whole class together to discuss what they have learned.

Session Two

For the second 90-minute period have students bring in various candies. The students will complete a lab.

1. Imagine the planar region that was revolved about either a horizontal or vertical line to create the various candies on your tray. Draw that planar region on your paper and label the drawing with the candy it represents.
2. Represent a typical slice on your drawing and label it with the calculus symbols as the internet site did. Hint: use ∆x and ƒ(x).
3. Show symbolically the volume of each of the candies by summing up all of the slices to create the volume of the candy.
4. If possible suggest what function could possibly be used to create each candy.

Assessment

The goals of the first 90-minute period will be assessed both by teacher observation and also during the class discussion period each group will discuss how to find the volume of some function revolved about an axis of their choice. They must set up the problem but not actually integrate it.

The second laboratory experience for the students will be assessed through the lab report that each student will turn in at the end of the session.

Supplemental information

I have refined this lesson plan over the last three years. The student feedback has been very good. The students are able to visualize volumes of revolution so much better than in a more traditional approach to teaching this concept.