We can use limits to integrate functions on unbounded domains or functions with unbounded range.

Recall that we introduced the definite integral

as a limit of Riemann sums. This limit need not always exist, as it depended on the properties of the function on the given interval . When the function is continuous on , this definite integral represented the net between the graph of and the -axis on .

Actually computing the limit of Riemann sums is an arduous task, but in the case when is continuous on the finite interval , the Fundamental Theorem of Calculus comes to the rescue and assures us that

where is an antiderivative of .

Note that in order to apply the Fundamental Theorem of Calculus, there are two requirements.

  • The interval over which we integrated, , must be a finite length interval, and
  • The function must be bounded on .

It turns out that there are many instances where these limitations are a problem. For instance, there are applications in statistics, physics, and engineering where we need to integrate over unbounded regions or we need to integrate unbounded functions. In this section, we will generalize the notion of the integral in such a way to overcome each of the restrictions above.

Which of the following integrals are improper according to the previous definition?

Unbounded intervals

Consider the expression What does this expression mean? Let’s consider a particular example and see if we can make sense of it.

It might seem like we should always get an answer of infinity or negative infinity if we integrate over an unbounded region, but the next example shows that is not always the case.

Thus in the two examples we have studied so far we have

and

Why do we get such different answer in these two cases? The difference lies in the speed with which each function approaches as . Although both functions approach as , the function becomes smaller much more rapidly than . We can see in the graphs above that shrinks down to zero much more quickly. The key idea is that the area that we are accumulating as is shrinking rapidly enough that the total area stays bounded. This is a theme we will see again when we study convergence of series.

We now give a precise definition for integrals over unbounded regions.

Unbounded functions

We have just considered definite integrals where the interval of integration was unbounded. We now consider another type of improper integral, where the interval is finite but the function is unbounded on the interval.

Let’s begin with some examples.

Now we look at another example:

We can now generalize the previous two examples to give a definition for such improper integrals of unbounded functions.

Let’s look at a couple more examples.