- Note that is continuous everywhere, so it is bounded on any finite interval.
- , so is unbounded on .
- , so even though is not continuous at , it is bounded on . Hence is proper.
- has a vertical asymptote at , so it is unbounded on .
We can use limits to integrate functions on unbounded domains or functions with unbounded range.
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 .
This says that the outputs of are trapped between the horizontal lines and . Thus the output values cannot become arbitrarily large in either the positive or negative sense 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.
- The interval of integration is infinite.
- The function is unbounded on the interval of integration.
Consider the expression What does this expression mean? Let’s consider a particular example and see if we can make sense of it.
We interpret definite integrals as giving us the net area underneath the graph of the function over the given interval. If we used this interpretation here, this notation should mean that we want to find the area under from to . This idea needs to be more precisely defined. Let’s consider the definite integral
where is some fixed number.
This integral gives us the area underneath from to . If we wanted to make sense of the integral to , we could think of letting get larger and larger and seeing how this affects the area.
That is, we interpret the integral from 1 to as a limit of a definite integral:
We can interpret this result to mean that the area under from to is infinite.
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.
We calculate the definite integral
This gives the area under on the interval .
Now we take the limit of this area as .
In this case, when we take the limit, we get a finite number, namely . This suggests that we can consider the area under from to to be equal to despite the fact that the region is unbounded.
Thus in the two examples we have studied so far we have
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.
- Let be a continuous function on .
- Let be a continuous function on .
- Let be a continuous function on . Let be any real number.
An improper integral is said to converge if its corresponding limit exists and is equal to a real number. Otherwise, the improper integral is said to diverge.
Now we must look at each integral separately.
Similarly we obtain
Now we take the limits of each integral
Hence we have
In this example, both integrals converge and therefore the entire integral converges.
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.
How then do we work with integrals of unbounded functions? Can we define this integral in terms of integrals we know how to compute?
Suppose lies strictly between and . Then consider the definite integral
The function is now bounded on this smaller interval.
Now we can examine what will happen if we let begin to move towards from the right. What will happen to the area under the curve? Just as we did in the case of unbounded integrals, we can make sense of this integral by defining it as the limit of a definite integral
We can then evaluate our original improper integral:
Isn’t it quite surprising that the area bounded by a curve which has a vertical asymptote can be finite? Consider the graph of on the interval :
Essentially, the function approaches the vertical asymptote at “fast enough” so that only a finite amount of area is accumulated.
Now we look at another example:
Let us evaluate these limits one at a time.
We can stop here. Since one limit was infinite, the integral is divergent, and hence does not exist (DNE).
earlier and saw that this integral converges. We remarked at the time this was due to how rapidly the function approached as .
However the integral
diverges. The issue is that the integral is approaching the vertical asymptote too slowly and thus the area builds up too quickly. This is due to the fact that the manner in which approaches the horizontal asymptote is different from the manner in which it approaches the vertical asymptote .
what would have happened if had we not noticed the vertical asymptote in the integrand at ?
We probably would have blindly computed:
But the integrand is always positive, so this answer of is complete nonsense! The reason this error occurs is because this expression is not a definite integral. It only has meaning if we interpret it as a limit of definite integrals
Be on the lookout for vertical asymptotes!
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.
Since lies in the middle of the interval and the upper limit of integration is infinity, we need to break the integral into three pieces
Since the integrand is a rational function, where the denominator is reducible, we could apply partial fractions decomposition to get
Now we find the antiderivative to get
Now we evaluate one integral at a time. Let’s start with
Thus this integral diverges, so the entire integral must diverge.