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Computing Volumes Using Integrals: Sphere, Pyramid, and Solids of Revolution

Discover how to compute volumes using integrals through key examples including a sphere with radius R and a pyramid with a square base of side length b and height h. Explore two methods for calculating the volumes of solids of revolution: the Disk Method and the Shell Method. Each method offers unique advantages depending on the problem at hand. Learn through examples like finding the volume of a cap atop a sphere and analyzing the volume remaining when a hole is drilled through a sphere's center.

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Computing Volumes Using Integrals: Sphere, Pyramid, and Solids of Revolution

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  1. Using Inegrals to Compute Volumes S. Ellermeyer

  2. General Principle

  3. Example 1 Find the volume of a sphere of radius R.

  4. Example 2 Find the volume of a pyramid with square base of side length b and height h.

  5. Volumes of Solids of Revolution • We will consider two methods for computing volumes of solids of revolution. • The Disk (or Slab) Method • The Shell (or Cylinder) Method • Depending on the problem, one of these methods may be easier to use than the other. (In some problems, both methods are about equally good.)

  6. The Disk (or Slab) Method Suppose that a plane region (pictured in red) is revolved about the x axis.

  7. Example 3

  8. Example 4 Find the volume of a “cap” of height h at the top of a sphere of radius R.

  9. The Shell (or Cylinder) Method Suppose that a plane region (pictured in red) is revolved about the y axis.

  10. Example 5 Repeat Example 3 using the shell method:

  11. Example 6 A hole of radius r is drilled through the center of a sphere of radius R (where r<R). Find the volume that remains.

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