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 by going to your computer's start menu and choose run. In the run field type:

\\filer\calclab

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

Improper_int_start_B11.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, .

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);

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);

- i
- , for all
- ii
- iii
- where represents the probability that the random variable is greater than or equal to but less than or equal to .

- 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.

- Another example of an improper integral is calculating the probability of a process that can be modeled as a normal distribution with a continuous random variable having mean and standard deviation . For example, the normal distribution is a probability density function widely referred to as the bell-shaped curve. The probability density function for a normal distribution with mean and standard deviation is given by the following equation.

Show that . Intelligence Quotient (IQ) scores are distributed normally with mean 100 and standard deviation 15.- A)
- What is the probability that an IQ score is above 120?
- B)
- What percentage of the population has an IQ score between 85 and 115?
- C)
- Show that the median IQ score is equivalent to the mean in this example. (Hint: Find such that or .)
- D)
- What would be the minimum IQ score for a person to be in the top 1 % of the population?
- E)
- What would be the maximum IQ score for a person to be in the bottom 1 % of the population?

- 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 .

2011-10-31