The purpose of this lab is to use Maple to introduce you to the notion of improper integral and to give you practice with this concept by using it to prove convergence or divergence of integrals involving unbounded integrands or unbounded intervals or both.

To assist you, there is a worksheet associated with this lab that you can copy into your home directory with the following command, which must be run in a terminal window, not in Maple.

cp /math/calclab/MA1023/Improper_int_start.mws My_Documents

Our basic theorem for is that the integral exists if is continuous on the closed interval . We have actually gone beyond this theorem a few times, and integrated functions that were bounded and had a finite number of jump discontinuities on . However, we don't have any theory to help us deal with integrals involving one or more of the following.

- Functions , for example rational functions, that have vertical asymptotes in (or are not bounded on ).
- Integrals where the interval is unbounded, for example intervals like , , or .

We have already seen at least one example of the problems you can run
into if the function is unbounded. Recall the clearly absurd result

that is obtained by blindly applying the FTOC. The second type of problem, where the interval of integration is unbounded, occurs often in applications of calculus, such as the Laplace and Fourier transforms used to solve differential equations. It also occurs in testing certain kinds of infinite series for convergence or divergence, as we will learn later.

We start with the following definition.

- The interval of integration is unbounded.
- The function has an infinite discontinuity at some point in . That is, .

To see how to handle the problem of an unbounded integrand, we start with the following special cases.

Cases where has an infinite discontinuity only at an interior
point are handled by writing

and using the definitions to see if the integrals on the right-hand side exist. If

> ex1 := int(1/x,x=a..2);

> limit(ex1,a=0,right);The example above used the

> int(1/x,x=0..2);

These are handled in a similar fashion by using limits. The definition we need the most is given below.

The other two cases are handled similarly. You are asked to provide suitable definitions for them in one of the exercises.

> ex2:=int(1/x^2,x=2..a); > limit(ex2,a=infinity);This command shows that Maple takes the limit definition into account in the

> int(1/x^2,x=2..infinity);

- The gamma function is an example of an improper integral often used to approximate non-integer factorials and is defined below:

Evaluate by calculating the improper integral for and for each integer value of , check your answer by calculating . - Both of the following improper integrals given below do not exist. Show, by calculating a limit, why they do not exist.
- A
- , interval .
- B
- , interval .

- Recall from Calculus II that the volume of a solid of revolution formed by rotating about the -axis over the interval is

and its surface area is

- A
- Find the volume of the solid obtained by revolving the curve about the -axis, between and . Repeat this using .
- B
- Plot both functions on the same graph.
- C
- Find the surface area of each solid of revolution.
- D
- Is it possible to have a finite volume but an infinite surface area?

2009-01-20