MA101: Single-Variable Calculus I

8.3: Volume of Solids of Revolution

• When we are presented with a solid that was produced by rotating a curve around an axis, there are two sensible ways to take that solid apart: slice it thinly perpendicularly to the axis, into disks (or washers, if the solid had a hole in the middle), or peel layers from around the outside like the paper wrapper of a crayon. The latter method is known as the shell method and produces thin cylinders. In both cases, we find the area of the thin segments and add them up to find the volume; as usual, when we have infinitely many pieces, this addition is really integration.
  

• 8.3.1: Disks and Washers

  • University of Wisconsin: H. Jerome Keisler's "Elementary Calculus, Chapter 6: Applications of the Integral, Section 6.2: Volumes of Solids of Revolution" URL

    Read Section 6.2 (pages 308 through 318).
    

  • Massachusetts Institute of Technology: David Jerison's "Lecture 22: Volumes by Disks and Shells" Page

    Watch this video. Dr. Jerison elaborates on some tangential material for a few minutes in the middle, but returns to the essential material very quickly.
    

  • Temple University: Gerardo Mendoza and Dan Reich's "Calculus on the Web" URL

    Click on the "Index" for Book II. Scroll down to "2. Applications of Integration," and click button 119 (Solid of Revolution - Washers). Work on problems 1-12. If at any time a problem set seems too easy for you, feel free to move on.
    

• 8.3.2: Cylindrical Shells

  • Temple University: Gerardo Mendoza and Dan Reich's "Calculus on the Web" URL

    Click on the "Index" for Book II. Scroll down to "2. Applications of Integration," and click button 120 (Solid of Revolution - Shells). Work on problems 5-17. If at any time a problem set seems too easy for you, feel free to move on.