Summary of Volumes of Revolution: Cylindrical Shells

Essential Concepts

  • The method of cylindrical shells is another method for using a definite integral to calculate the volume of a solid of revolution. This method is sometimes preferable to either the method of disks or the method of washers because we integrate with respect to the other variable. In some cases, one integral is substantially more complicated than the other.
  • The geometry of the functions and the difficulty of the integration are the main factors in deciding which integration method to use.

Key Equations

  • Method of Cylindrical Shells
    [latex]V={\displaystyle\int }_{a}^{b}(2\pi xf(x))dx[/latex]


method of cylindrical shells
a method of calculating the volume of a solid of revolution by dividing the solid into nested cylindrical shells; this method is different from the methods of disks or washers in that we integrate with respect to the opposite variable