Subsections

# Improper Integrals

## Purpose

The purpose of this lab is to acquaint you with the family of functions known as Gaussian distributions. The two aspects of them we wish to become familiar with are their geometry and various integrals of them.

## A. The Geometry of Gaussian Distributions

Our focus in this part is on the function

f(x)=e-x2

We wish to learn as much as possible about its graph. Please use algebra and Maple to answer the following questions:
1.
Does it have any symmetry?
2.
Does it have any asymptotes?
3.
Does it have any local extrema? If so, where?
4.
Does it have any point(s) of inflection? If so, where?
We now generalize the function a bit by adding a parameter, x0. The function now becomes

f(x)=e-(x-x0)2

5.
So if x0 = 0, it is the same as in part A. Describe the effect of the parameter x0 on graph of f(x).
In this section, we add a second parameter, , and seek its effect. The function now becomes

6.
To see what its effect is, try simultaneously graphing the function for .
7.
Next ,use calculus to see what are the point(s) of inflection of the new function. Your answer here should be consistent with the graphs you just found.
8.
At this point, you should be able to conclude what the effect has on the graph; please describe it.

## B. Definite Integrals of Gaussian Distributions

Consider the initial Gaussian Distribution with x0 = 0 and you looked at earlier

f(x)=e-x2

1.
Your goal here is to apply knowledge of integration to obtain a value for the improper integral

Note that this integral cannot be done using antiderivatives and the Fundamental Theorem of Calculus. The function simply does not have an antiderivative! (try finding one!) Thus the only option you have is a numerical method (your choice). Note: take advantage of symmetry and save some work!

Since you have an improper integral, you must approximate the limit,

To demonstrate this, suppose we instead wanted to find

with the trapezoid rule to 2 places. Using the Maple command
  > evalf(trapezoid(f(x),x=0..a,n));

With and varying both a and n, one gets the following table of values:

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
a 		 n 		 evalf(trapezoid(...) 		 a 		 n 		 evalf(..)

100 		 20 		 2.186 		 600 		 800 		 1.568

100 		 100 		 1.567 		 600 		 1000 		 1.569

100 		 300 		 1.561 		1000 		 1000 		 1.576

100 		 600 		 1.561 		 1000 		 2000		 1.570

200 		 600 		 1.566 		  1000 		 4000 		 1.570

200 		 800 		 1.566 		 4000 		 4000 		 1.576

400 		 800		 1.568 		 4000 		 8000 		 1.571

400 		 800 		 1.568 		 4000 		 10,000 		 1.571

8000 		 10,000 		 1.572

8000 		 15,000 		 1.571

8000 		 20,000 		 1.571

The method here being for a given value of a to take n large enough so that no change is seen to 2 decimal places. Then we increase a and again increase n enough so no change to 2 decimal places is seen. When a is large enough that no change, to 2 places, is seen between two consecutive values of a, then we are done. In the above example, which converges pretty slowly, this happens for a around 4000. We take the value of the integral to be 1.57 to 2 decimal places. Note that we save three places at all times and observe normal roundoff rules (1.568 rounds to 1.57; 1.561 rounds to 1.56 etc). The exact answer is .

2.
Suppose we wished to scale the function f(x) so that the area under the curve from to was one. What value of k would you use to do this if

## C. Effect of the parameter x0

Your goal in this section is to decide if/how the paramter x0 effects the integral of f(x).

1.
What is the integral of from to ?
Instead of doing this numerically, as in A, lets make a change of variable and make use of the result from A. Make the substitution (change of variable) u=x-x0 and determine the integral.
2.
How would you scale it so the integral would become one?

## D. Effect of the Parameter

Here, we consider the function from Part One,

and wish to find its integral from to . Again, instead of doing this numerically, we can make use of earlier results with a change of variable. beginenumerate [1.] Determine the value of the integral

by making the substitution (change of variable)

## E. Putting It all Together

1.
The goal here is simple: pick a constant k so that the function

Refer back to Part Two. From your earlier work,

2.
What vertical line is it symmetric about?
3.
Where are its points of inflection?
4.
What does its graph look like?

## Conclusion

The family of functions you have studied in this lab have many applications in mathematics, science, and engineering because many naturally occuring processes generate data which follows them. Such processes include SAT scores, heights of people, the strength of bolts made on an assembly line, the percentage of people voting for a candidate based upon a sample, and many more. They have a number of different names you may encounter:

and were originally developed by the mathematician/physicist Karl Gauss (1777-1855) for use with astronomical data he collected. The functions are noteworthy as their areas involve improper integrals which cannot be done with antiderivatives.