Chapter 6.2 - Volume: The Disk Method

Objectives:

• • Find the volume of a solid of revolution using the disk method.

  • Find the volume of a solid of revolution using the washer method.

  • Find the volume of a solid with known cross sections.

Volumes of Revolution, Disk & Washer pt1 (for Day #64 lesson)

Revolving regions about a given horizontal line produces a volume with infinitesimally-thin circular (or disk-shaped) cross sections. The definite integral adds up the infinitely-many volumes of each cross section to give the total volume.

Volumes of Revolution, Disk & Washer pt2 (for Day #65 lesson)

The fist exercise involves both of the two things that make typically make volumes of revolution tricky: (a) a hole through the middle of the volume, requiring the "washer method," and (b) a centerline that is not simply the x-axis. The second exercise involves a shape centered along the y-axis instead of the x-axis.

Volumes With Known Cross Section (for Day #65 lesson)

In two dimensions, the definite integral essentially adds up an infinite number of infinitesimal rectangular areas. Now we expand that concept to three dimensions, where the definite integral will add up an infinite number of infinitesimal slices. As long as each cross-section maintains a predictable shape and size, we can compute the total volume.

Volume Activity - Vases

In this activity you will use Calculus to calculate the volume of a vase.

If we're using Geogebra for this activity, get Mr. W to tell you how to access the related Geogebra file.

If we're using Desmos for this activity, click on the link below the image you want to use. Image resolution is low in Desmos versions, but it's good enough for this activity.

Volume Activity - Booty Pillow

Solution

Here is my paper solution (link will be provided after activity is completed), and here is a Geogebra solution (link will be provided after activity is completed) that models the Booty Pillow. You may find it interesting to "disect" the Geogebra file to see how I made it and pick up a few techniques along the way.

Info on your Q4 Exam

This exam will be split over two days. To help me grade the exams quickly, you'll be given the Free Response sections first.

First Day:

Section 2A:

2 Free Response, WITH CALCULATOR (30 minutes)

Section 2B:

4 Free Response, NO CALCULATOR (60 minutes)

Second Day:

Section 1A:

28 Multiple Choice, NO CALCULATOR (55 minutes)

Section 1B:

17 Multiple Choice, WITH CALCULATOR (50 minutes)