What happens if one of the limits of integration for a definite integral is infinity? Does the integral have a value? Or, what if the value of the integrand goes to infinity at one of the limits? We can treat these cases using limits. For example, if we wanted to find the value of a definite integral with an infinite limit, we can instead replace the infinite limit with a variable, and then take the limit as this variable goes to infinity: . We can then just find the antiderivative for the integral, evaluate it at the two integration limits, and then see if the limit of this result exists. Similarly, suppose a function goes to infinity at x = b (e.g., has a vertical asymptote there). We can use this same technique to evaluate this integral:(this assumes that a < b, so the limit approaches b from the left). We can handle problematic lower limits just as easily. If both limits are a problem, or if the problem occurs between the two limits, we can split the integral up into two separate integrals, each with only one problematic limit.
Try the following:
- The applet initially shows a line. We want to know whether has a value. Symbolically, we would do the following: . This last limit does not exist because it is unbounded. We can see this from the applet, which shows a table of values for the integral for different values of b. As b gets bigger, so does the value. You can also see this from the graph, where it is clear that as b gets bigger (try moving the b slider), the area keeps increasing to infinity.
- Select the second example, showing a parabola. Like the previous example, as b increases, we add more and more area, so is also unbounded. You can see that the integral of any power function, from 1 to infinity, is unbounded if the exponent is greater than 1. Reminder: numbers like 1.2345E6 are in scientific notation and is the same as 1.2345 x 106.
- Select the third example. Here we have taken the reciprocal of the power function. Now notice that as b gets bigger, the area seems to be heading towards 1. In fact, if we find the antiderivative and evaluate the limit, we get a value of 1 for this integral. This integral is said to converge, while the examples we looked at above, where the limit did not exist, are said to diverge.
- Select the fourth example, which uses a different exponent. Does this converge or diverge? What would you guess about bigger exponents?
- Select the fifth example, which uses an exponent of 0.5 (i.e., a square root). Does this converge or diverge?
- Select the sixth example which uses an exponent of 1. Does this converge or diverge? This gives rise to the p-test, which says for integrals like , the integral converges if p > 1 and diverges if p ≤ 1.
- Select the seventh example, where we want to know the value of . In this case, the problem is that at x = 2, the integrand goes to infinity. We can treat this case using a limit, "sneaking up" on 2 from the left. The table of values shows what happens, and as you can see, the values seem to converge on 2 (which is the value of this integral).
- Select the eighth example, using an exponent of 2. Does this converge or diverge?
- Select the ninth example. You can enter your own function, set a as you would like, zoom/apn the graph, and edit the b values in the table (just double click on a table cell to edit it; press Enter when done). Note that if you pick very large values for b (e.g., bigger than 1000), the applet may take some time to recompute the table values.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.