Some of the most useful tools of integration theory will be developed in this chapter. They center on two important operations on functions. One is the formation of the absolute value. The other is the limit of a sequence of functions. The behavior of the generalized Riemann integral with respect to these operations exhibits the strength of this integral definition most vividly.

When*f*is integrable, it is important to be able to tell whether |*f*| is also integrable. A simple criterion is stated in Section 3.2. This criterion has implications for the calculation of the length of curves. These