Summary

Example 4

Some regions R are more easily described by taking y as the independent variable, so that R is the region between

x = f(y) and x = g(y) for c ≤ y ≤ d.

The volumes of the solids of revolution are then computed by integrating with respect to y. Often we have a choice of either x or y as the independent variable.

How can one decide whether to use the Disc or Cylindrical Shell Method? The answer depends on both the axis of rotation and the choice of independent variable. Use the Disc Method when rotating about the axis of the independent variable. Use the Cylindrical Shell Method when rotating about the axis of the dependent variable.

Example 5