An Overview of the Course

In the first part of these notes we consider the problem of calculating the areas of various plane figures. The technique we use for finding the area of a figure $A$ will be to construct a sequence $I_n$ of sets contained in $A$, and a sequence $O_n$ of sets containing $A$, such that

1.

The areas of $I_n$ and $O_n$ are easy to calculate.

2.

When $n$ is large then both $I_n$ and $O_n$ are in some sense ``good approximations'' for $A$.

Then by examining the areas of $I_n$ and $O_n$ we will determine the area of $A$. The figure below shows the sorts of sets we might take for $I_n$ and $O_n$ in the case where $A$ is the set of points in the first quadrant inside of the circle $x^2 + y^2 = 1.$

In this example, both of the sets $I_n$ and $O_n$ are composed of a finite number of rectangles of width $${1 \over n}$$, and from the equation of the circle we can calculate the heights of the rectangles, and hence we can find the areas of $I_n$ and $O_n$. From the third figure we see that area$(O_n) -$ area $(I_n) = \displaystyle {1 \over n}$. Hence if $n = 100000$, then either of the numbers area$(I_n)$ or area$(O_n)$ will give the area of the quarter-circle with an error of no more than $10^{-5}$. This calculation will involve taking many square roots, so you probably would not want to carry it out by hand, but with the help of a computer you could easily find the area of the circle to five decimals accuracy. However no amount of computing power would allow you to get thirty decimals of accuracy from this method in a lifetime, and we will need to develop some theory to get better approximations.

In some cases we can find exact areas. For example, we will show that the area of one arch of a sine curve is $2$, and the area bounded by the parabola $y=x^2$ and the line $y=1$ is $${{4\over 3}}$$.

However in other cases the areas are not simply expressible in terms of known numbers. In these cases we define certain numbers in terms of areas, for example we will define

$$\pi = \mbox{the area of a circle of radius 1,}$$

and for all numbers $a > 1$ we will define

$$\begin{eqnarray*} \ln(a) &=& \mbox{the area of the region bounded by the curves}\\ & & y=0,\; xy=1,\; x=1,\; \mbox{ and }x=a. \end{eqnarray*}$$

We will describe methods for calculating these numbers to any degree of accuracy, and then we will consider them to be known numbers, just as you probably now think of $\sqrt{2}$ as being a known number. (Many calculators calculate these numbers almost as easily as they calculate square roots.) The numbers $\ln(a)$ have many interesting properties which we will discuss, and they have many applications to mathematics and science.

Often we consider general classes of figures, in which case we want to find a simple formula giving areas for all of the figures in the class. For example we will express the area of the ellipse bounded by the curve whose equation is

$${x^2\over a^2} + {y^2 \over b^2} = 1$$

by means of a simple formula involving $a$ and $b$.

The mathematical tools that we develop for calculating areas, (i.e. the theory of integration) have many applications that seem to have little to do with area. Consider a moving object that is acted upon by a known force $F(x)$ that depends on the position $x$ of the object. (For example, a rocket propelled upward from the surface of the moon is acted upon by the moon's gravitational attraction, which is given by

$$F(x) = {C\over x^2},$$

where $x$ is the distance from the rocket to the center of the moon, and $C$ is some constant that can be calculated in terms of the mass of the rocket and known information.) Then the amount of work needed to move the object from a position $x=x_0$ to a position $x = x_1$ is equal to the area of the region bounded by the lines $x=x_0$, $x = x_1$, $y=0$ and $y = F(x)$.

In the case of the moon rocket, the work needed to raise the rocket a height $H$ above the surface of the moon is the area bounded by the lines $x=R$, $x = R+H$, $y=0$, and $y = \displaystyle {{C\over x^2}}$, where $R$ is the radius of the moon. After we have developed a little bit of machinery, this will be an easy area to calculate. The amount of work here determines the amount of fuel necessary to raise the rocket.

Some of the ideas used in the theory of integration are thousands of years old. Quite a few of the technical results in the calculations presented in these notes can be found in the writings of Archimedes(287-212 B.C.), although the way the ideas are presented here is not at all like the way they are presented by Archimedes.

In the second part of the notes we study the idea of rate of change. The ideas used in this section began to become common in early seventeenth century, and they have no counterpart in Greek mathematics or physics. The problems considered involve describing motions of moving objects (e.g. cannon balls or planets), or finding tangents to curves. An important example of a rate of change is velocity. The problem of what is meant by the velocity of a moving object at a given instant is a delicate one. At a particular instant of time, the object occupies just one position in space. Hence during that instant the object does not move. If it does not move, it is at rest. If it is at rest, then its velocity must be $0$(?)

The ability to find tangents to curves allows us to find maximum and minimum values of functions. Suppose I want to design a tin can that holds 1000 cc., and requires a minimum amount of tin. It is not hard to find a function $S$ such that for each positive number $h$, the total surface area of a can with height $h$ and volume $1000$ is equal to $S(h)$. The graph of $S$ has the general shape shown in the figure, and the minimum surface area corresponds to the height $h_0$ shown in the figure. This value $h_0$ corresponds to the point on the graph of $S$ where the tangent line is horizontal, i.e. where the slope of the tangent is zero. From the formula for $S(h)$ we will be able to find a formula for the slope of the tangent to the graph of $S$ at an arbitrary height $h$, and to determine when the slope is zero. Thus we will find $h_0$.

The tool for solving rate problems is the derivative, and the process of calculating derivatives is called differentiation. (There are two systems of notation working here. The term differential was introduced by Gottfried Leibniz(1646-1716) to describe a concept that later developed into what Joseph Louis Lagrange(1736-1813) called the derived function. From Lagrange we get our word derivative, but the older name due to Leibniz is still used to describe the general theory - from which differentials in the sense of Leibniz have been banished.) The idea of derivative (or fluxion or differential) appears in the work of Isaac Newton(1642-1727) and of Leibniz, but can be found in various disguises in the work of a number of earlier mathematicians.

As a rule, it is quite easy to calculate the velocity and acceleration of a moving object, if a formula for the position of the object at an arbitrary time is known. However usually no such formula is obvious. Newton's Second Law states that the acceleration of a moving object is proportional to the sum of the forces acting on the object, divided by the mass of the object. Now often we have a good idea of what the forces acting on an object are, so we know the acceleration. The interesting problems involve calculating velocity and position from acceleration. This is a harder problem than the problem going in the opposite direction, but we will find ways of solving this problem in many cases. The natural statements of many physical laws require the notion of derivative for their statements. According to Salomon Bochner

The mathematical concept of derivative is a master concept, one of the most creative concepts in analysis and also in human cognition altogether. Without it there would be no velocity or acceleration or momentum, no density of mass or electric charge or any other density, no gradient of a potential and hence no concept of potential in any part of physics, no wave equation; no mechanics no physics, no technology, nothing[11, page 276].

At the time that ideas associated with differentiation were being developed, it was widely recognized that a logical justification for the subject was completely lacking. However it was generally agreed that the results of the calculations based on differentiation were correct. It took more than a century before a logical basis for derivatives was developed, and the concepts of function and real number and limit and continuity had to be developed before the foundations could be described. The story is probably not complete. The modern ``constructions'' of real numbers based on a general theory of ``sets'' appear to me to be very vague, and more closely related to philosophy than to mathematics. However in these notes we will not worry about the foundations of the real numbers. We will assume that they are there waiting for us to use, but we will need to discuss the concepts of function, limit and continuity in order to get our results.

The fundamental theorem of the calculus says that the theory of integration, and the theory of differentiation are very closely related, and that differentiation techniques can be used for solving integration problems, and vice versa. The fundamental theorem is usually credited to Newton and Leibniz independently, but it can be found in various degrees of generality in a number of earlier writers. It was an idea floating in the air, waiting to be discovered at the close of the seventeenth century.