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Improper Integrals

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.

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See About the calculus applets for operating instructions.

1. A line

The applet initially shows a line. We want to know whether int has a value.
Symbolically, we would do the following: int 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 int 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 ptest the integral converges if p > 1 and diverges if p ≤ 1.

7. 1/(2-x)^0.5

Select the seventh example, where we want to know the value of int 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).

8. 1/(2-x)^2

Select the eighth example, using an exponent of 2. Does this converge or diverge?

Explore

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.

Other 'Integration Techniques' topics

Acknowledgements

Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.