Volumes

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 6.2 – Volumes
• Section 6.3 – Volumes by Cylindrical Shells

Expected Educational Results

• Objective 11–1: I can set up the integral to compute the volume obtained by rotating a region about an axis.
• Objective 11–2: I can evaluate the integral to compute the volume obtained by rotating a region about an axis using FTC-II.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Volumes

Volume of Revolution – The Disk Method

Activity 01

In the CalcPlot3D applet below,

1. click on checkbox for Show representative rectangle;

2. move the slider for $x$$x$-val;

   1. What is the height of the red representative rectangle? Explain.
   2. What is the width of the red representative rectangle? Explain.

3. move slider to Revolve representative rectangle;

   1. Identify the shape of the object that results when the red representative rectangle is completely rotated. Explain.
   2. How are the radius of the object and the y-values of the function related? Explain.
   3. How “thick” is the object? Explain.
   4. What is the volume of the object? Explain.

4. click on checkbox for Show $n$$n$ washers/shells. Use slider to increase the number of shells, $n$$n$;

   1. What do you notice? Explain.

5. move slider to Revolve the washers/shells;

   1. What do you notice?

6. What is the total volume of the revolved region $0\le x\le 1$$0\leq x\leq 1$? Explain.

7. Try other Top functions and intervals of $x$$x$.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Wednesday, 29 September 2021 - 05:32 (EDT)