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See "The Study of Change" (21 September) and "Close Enough for Scientific Work" (14 September).

28 September 2001

Adding it All Together

by George E. Hrabovsky, President of MAST

News from MAST
Hello everyone. This week we are going to continue with the theme of the last two weeks. Before that, though, I can report that we now have a working autoclave, so our microbiology experiments will be able to continue!

How to Add Things that are Very Small
Like last week, it is best if you follow along with a pen and paper. It might be helpful to review the last two columns. Let us say that we have a function of the form,

We can make a graph of this,

It is important to realize that many such functions are, in fact, derivatives of other functions. So what? If this is true, then we can figure out what the original function is. How do we do this? We first realize that we can divide the t axis of the graph into vertical slices of a fixed length. This length is traditionally called t for the example above. If we center t so that the top center of the slice intersects the line of the graph then we can say that the height of the slice is equal to .=20

We can determine the area of this slice in the normal way, length time width. In this case the width is t and the length is . So we can write,

If we shrink t to the limit where t -> 0 then we know that t becomes dt and becomes the derivative . The area of the slice is then . If we add up all of the slices we will get the total area under the curve. In this case that would give us . In our example this would give us a right triangle with the base being t and the altitude being f(t). The area of a right triangle is simply half the area of a rectangle with similar sides,

This is a peculiar thing. The area under our curve is the same as the function that we took the derivative of last week! This process of finding the area under a curve is called integration and forms another third of calculus. When we write an integral we use an elongated s to denote that we are taking the sum of all values. In our case we would say,

where c is an arbitrary constant (recall that since differentiation is the rate of change then a constant will have no such rate and c will vanish under differentiation).  Finding such formulas is a difficult task, so there are extensive lists of integrals. Programs like Mathematica can do a variety of integrals (and can even be used to correct some of the published tables). It is important to realize that integration and differentiation are exactly inverse operations (much like addition and subtraction).

Books That I Like
Richard Courant, Fritz John, 1989, Introduction to Calculus and Analysis, Volume 1, Springer. This really delves into the theory of basic integration. It is very mathematical.

Morris Kline 1977, Calculus An Intuitive and Physical Approach, John Wiley and Sons (Dover Publications reprint 1998). Spends a lot more time developing the ideas of integration than I can here.

Alexander J. Hahn 1998, Basic Calculus, Springer-Verlag New York. As I said last week, this book keeps the reader motivated by presenting the material in a historical context.

Murray R. Speigel, 1968, Schaum's Outline Mathematical Handbook, McGraw-Hill Inc. This book is now in its 34th printing, and is immensely useful. I highly recommend it, if you do not have access to Mathematica (and even if you do, it is a good resource). It is also inexpensive (less than $15).

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Converted by Mathematica      September 27, 2001