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.
1. A line
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.
2. A parabola
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.
3. Reciprocal of a power function
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.
4. Reciprocal of a power function with different exponent
Select the fourth example, which uses a different exponent. Does this converge or diverge? What would you guess about bigger exponents?
5. Reciprocal of a power function with an exponent of 0.5
Select the fifth example, which uses an exponent of 0.5 (i.e., a square root). Does this converge or diverge?
6. Reciprocal of a power function with an exponent of 1
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/pan 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.
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Other 'Integration Techniques' topics
Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.