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17 October 2003

Notions of Integration

by George E. Hrabovsky, President, MAST

Where We Have Been

Last time we explored how to determine if a critical point is the maximum or the minimum by taking the second derivative. I made a stupid mistake that is quite common; I reversed the directions for the values given.  If the value of the second derivative at a critical point is greater than 0, then the critical point is a minimum; and if the second derivative of the critical point is less than 0 then the point is a maximum. I apologize for this error, there is no excuse for it.

 I would like to thank Elm Grove for pointing this out to me.

A New Problem

While in your lab you are making a set of measurements at equal time intervals . You are understandably excited about this and you work hard to graph the data and then rush off to show the data to your troublesome associate. She looks over your data and smiles, "Have you determined the function for this data curve?"

"I will try to fit the curve as soon as I get back to my desk," you remark.

 "I don't mean fitting the curve, I mean integrating it."

 "Huh?"

She smiles, "Let us say that you have a function like velocity, we know that this is distance traveled over time traveled.  We plot the velocity versus time. Now if we think about it long enough it will dawn on us that if we add up all of the values of velocity within some time interval we will be making an approximation of the area under our curve within the interval. Using velocity, denoted by the symbol as the distance traveled , then the area will be . Thus the area will tell us how far the object moves at any given time."

"So, if I add up all of my data it give me the area?"

"There is a little more to it than that. Let me explain," she says, moving to her blackboard.

"Let's say that we have a curve like this," and she draws on the blackboard,

and continues, "Now, lets say that this is a velocity curve and we want to know the function for the position at any time along the curve. We begin by setting our initial time."

"Then we choose some later time, say ."

"The problem is then to find the area of this figure beneath the curve and bounded by 0 and . How do we do that?"

 You shake your head and she nods, "This is problem that vexed the best mathematicians in the world for thousands of years. It was eventually realized that we can fill the space between 0 and with rectangles whose length extends to the function line and whose width is a specific interval of time, ."

"The area of each rectangle is the length, the value of our function at that location, times the width of the rectangle, ."

"We then add all of the areas to get the area under the curve."

You look at the diagram and make an objection, "As you get closer to it looks like you get more error."

Your friend smiles, "That is correct. So you have to choose to make as small as possible. The best way to do this is to find it in the limit of 0. We start with the summation,"

 Confused, you ask, "What is that?"

She answers, "The is the Greek letter Sigma, and it stands for summation. What follows on the right is what is to be summed, the small expression beneath the sigma tells us where to start the summation, and the symbol on top tells us how many terms to have in the sum. For example,"

 You nod, "I see."
 She continues, "If we take the limit of the sum as we get,"

"this is the definition of the definite integral of the function from 0 to . We write it like this,"

The Math Challenge from Last Time

Here are some classic theorems about critical points and the second derivative test:

Theorem 1:

If has an extreme value at a point , then either is not differentiable at the point or .

Theorem 2:

If is a critical point of and if exists and is nonzero, then has a minimum at the critical point if and a maximum if .

The Math Challenge

Can you prove the two theorems?

Math Resources for Integrals

Online:

http://www.ma.utexas.edu/users/kawasaki/mathPages.dir/

This is a neat site.  For a more challenging page look at this one:

http://www.mathpages.com/home/icalculu.htm

Both sites are very cool.  Another very nice site is:

http://www.ugrad.math.ubc.ca/coursedoc/math101/

Here is an online table of integrals

http://www.math2.org/math/integrals.htm

Here is a site powered by WebMathematica:

http://www.calc101.com/

Of course there is the Wolfram research Integrator:

http://integrals.wolfram.com/

Books:

Richard A. Silverman, 1969, Modern Calculus and Analytic Geometry, MacMillan Company, New York (Dover Publications has reprinted this book with corrections in 2002). This has a rigorous, but readable chapter on integrals.

Created by Mathematica  (October 17, 2003)