Sec. 6.2a - Volumes of Solids of Revolution (using disks)

Overview

This section introduces a beautiful use of definite integrals: Finding the volume of certain 3D shapes. Specifically, we will focus on 3D shapes that have circular cross sections. These objects are called solids of revolution, since they have rotational symmetry. You may find it hard to visualize what is happening with these shapes, so we’ll spend a fair amount of time in class practicing this. Visualization is a skill that you can learn – start now, with the resources below!

Basic Learning Objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

• Experience various visualizations of volumes of revolution.

• Given the volume of a representative slice of a 3D object, write an integral that represents the exact volume of the 3D object.

Advanced Learning Objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

• Given a 2D region bounded by simple functions, sketch: A representative rectangular slice, a representative revolved “ring”, “disk”, or “washer”, and the final 3D shape obtained by revolving it around an axis.

• Given a 2D region bounded by simple functions, set up and evaluate an integral that gives the volume of a solid of revolution around any vertical or horizontal axis.

• Choose the most appropriate direction to slice a region, so as to create a simple integral representing its volume.

• Recognize situations in which many “thin” objects are being added together, and use an integral to evaluate these sums exactly.

To prepare for class

• Read the Introduction of Section 6.2 in Active Calculus.

• Watch the following video (by GVSUmath) - note that there is a lot of information in this video, so you may want to view it more than once:

• Read subsection 6.2.1 in Active Calculus: The Volume of a Solid of Revolution

• Do the Preview Activity for this section (on WeBWorK if required by your teacher).

• Watch the following video (by GVSUmath) which shows how we can find the volume of any solid object if we can express the area of its cross-sections by a formula:

• Watch the following videos (by GVSUmath) which show how this becomes easier when the object has rotational symmetry:

• Watch the following video (by Amy Liu) up until 4:50 - it gives many different real-world examples where computing the volume of solids of revolution come into play. Skip the remainder of the video until after this class (it discusses the method of cylindrical shells, which will be the topic for the following class.)

• Do: Here are some sites with nice visualizations and interactive applets for solids of revolution. Play around with each of these to help you solidify your understanding:

  • Surface of Revolution on Desmos 3D (1 contour curve) (by Gabriel Indurskis) This shows the surface obtained when rotating a given contour curve around either the \(x\)-axis or \(y\)-axis.

  • Surface of Revolution on Desmos 3D (2 contour curves) (by Gabriel Indurskis) This shows the two surfaces (“inner” and “outer”) obtained when rotating two given contour curves around the \(x\)-axis.

  • Shodor Solids of Revolution (very flexible online visualization: click on “revolve”, then drag the image to rotate the view)

  • Wolfram Alpha Volume Calculator (Wolfram Alpha, does the calculation but also shows a picture)

  • Wireframe interactive demo, using handdrawn curve (by selim_tezel, make sure to read the instructions)

  • Paul’s Online Math Notes: Volume with Rings (by Paul Dawkins)

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