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.

\\storage\academics\math\calclab

when you hit enter, you can then choose MA1023 and then choose the worksheet

Improper_int_start_B14.mw

Remember to immediately save it in your own home directory. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting the exercises.

- 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, .

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);

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:

- A)
- Verify that for .
- B)
- Verify that
- C)
- Verify Euler's reflection formula:

You may need to use the command.

- Determine which of the regions described below have finite area.
- A)
- The region below , above the axis, over the interval .
- B)
- The region below , above the axis, over the interval .
- C)
- The region below , above the axis, over the interval .

- Evaluate the following integral in Maple by splitting the range of integration into two ranges and completing two improper integrals. Be sure to use limits for each integral.

- Recall from Calculus II how to compute the volume and surface area of a solid of revolution. The formula for volume and surface area of rotated about the -axis over the interval
is given by:

- (A)
- Find the volume of the solid obtained by revolving the curve about the x-axis, between and . Then find the volume of the solid by revolving the curve about the x-axis, between and .
- (B)
- Find the surface area of each solid of revolution.
- (C)
- Is it possible to have a finite volume but an infinite surface area?

2014-11-06