The definition of a definite integral: requires the interval
[a,b] be finite. The Fundamental Theorem of Calculus requires
that f be continuous on [a,b]. In this section, you will be
studying a method of evaluating integrals that fail these
requirements—either because their limits of integration are
infinite, or because a finite number of discontinuities exist on
the interval [a,b]. Integrals that fail either of these
requirements are improper integrals. (If you are not familiar
with L'Hôpital's rule, it is a good idea to review it before
reading this section.) Improper Integrals with Infinite Limits
of Integration Consider the integral Assigning a finite upper
bound in place of infinity gives This improper integral can be
interpreted as the area of the unbounded region between , (the
-axis), and . 1. Suppose exists for all . Then we define - = as
long as this limit exists and is finite. If it does exist we say
the integral is convergent and otherwise we say it is divergent.
2. Similarly if exists for all we