A student will be able to:
- Compute by hand the integrals of a wide variety of functions by using the technique of Improper Integration.
- Combine this technique with other integration techniques to integrate.
- Distinguish between proper and improper integrals.
The concept of improper integrals is an extension to the concept of definite integrals. The reason for the term improper is because those integrals either
- include integration over infinite limits or
- the integrand may become infinite within the limits of integration.
We will take each case separately. Recall that in the definition of definite integral we assume that the interval of integration is finite and the function is continuous on this interval.
Integration Over Infinite Limits
If the integrand is continuous over the interval then the improper integral in this case is defined as
If the integration of the improper integral exists, then we say that it converges. But if the limit of integration fails to exist, then the improper integral is said to diverge. The integral above has an important geometric interpretation that you need to keep in mind. Recall that, geometrically, the definite integral represents the area under the curve. Similarly, the integral is a definite integral that represents the area under the curve over the interval as the figure below shows. However, as approaches , this area will expand to the area under the curve of and over the entire interval Therefore, the improper integral can be thought of as the area under the function over the interval
We notice immediately that the integral is an improper integral because the upper limit of integration approaches infinity. First, replace the infinite upper limit by the finite limit and take the limit of to approach infinity:
Thus the integral diverges.
Thus the integration converges to
What we need to do first is to split the integral into two intervals and So the integral becomes
Next, evaluate each improper integral separately. Evaluating the first integral on the right,
Evaluating the second integral on the right,
Adding the two results,
Remark: In the previous example, we split the integral at However, we could have split the integral at any value of without affecting the convergence or divergence of the integral. The choice is completely arbitrary. This is a famous theorem that we will not prove here. That is,
Integrands with Infinite Discontinuities
This is another type of integral that arises when the integrand has a vertical asymptote (an infinite discontinuity) at the limit of integration or at some point in the interval of integration. Recall from Chapter 5 in the Lesson on Definite Integrals that in order for the function to be integrable, it must be bounded on the interval Otherwise, the function is not integrable and thus does not exist. For example, the integral
develops an infinite discontinuity at because the integrand approaches infinity at this point. However, it is continuous on the two intervals and Looking at the integral more carefully, we may split the interval and integrate between those two intervals to see if the integral converges.
We next evaluate each improper integral. Integrating the first integral on the right hand side,
The integral diverges because is undefined, and thus there is no reason to evaluate the second integral. We conclude that the original integral diverges and has no finite value.
So the integral converges to .
In Chapter 5 you learned to find the volume of a solid by revolving a curve. Let the curve be and revolving about the axis. What is the volume of revolution?
From the figure above, the area of the region to be revolved is given by . Thus the volume of the solid is
As you can see, we need to integrate by parts twice:
At this stage, we take the limit as approaches infinity. Notice that the when you substitute infinity into the function, the denominator of the expression being an exponential function, will approach infinity at a much faster rate than will the numerator. Thus this expression will approach zero at infinity. Hence
So the volume of the solid is
This can be a tough integral! To simplify, rewrite the integrand as
Substitute into the integral:
Using substitution, let
Returning to our integral with infinite limits, we split it into two regions. Choose as the split point the convenient
Taking each integral separately,
Thus the integral converges to
For a video presentation of Improper Integrals (22.0), see Improper Integrals, www.justmathtutoring.com (6:23).
For a video presentation of Improper Integrals with Infinity in the Upper and Lower Limits (22.0), see Improper Integrals, www.justmathtutoring.com (7:55).
- Determine whether the following integrals are improper. If so, explain why.
Evaluate the integral or state that it diverges.
- The region between the axis and the curve for is revolved about the axis.
- Find the volume of revolution,
- Find the surface area of the volume generated,