By the seventeenth century, mathematicians and other thinkers were faced with *old problems* in *new clothings*. The problems of lengths, areas, volumes and motion were old problems that ancient Egyptians, Babylonians and Greeks encountered and to which they devised significant practical and theoretical solution methods. But early in the seventeenth century, the need to devise solution methods for the motion of projectiles; the motion of the moon and planet to aid navigation; the motion of light to improve the design of the newly discovered telescope and microscope became paramount to mathematicians and scientists. Furthermore, *heliocentric astronomy* as originated by Copernicus and Kepler had replaced the Aristotelian laws of motion. So, new methods of analysis were required. From this background, came the new science of motion that was created by Galileo and Newton, and in the process, the mathematics of *functions* and *calculus*. In essence, *calculus* which consists of differential calculus and *integral calculus* was *spirited* by the desire to solve *P _{j}* problems.

**The Infinitesimal And Continuity**

The most familiar meaning assigned to the *integral sign* (*∫*) is that of *summation*. But *summation* of what? Why is the *sigma sign* (*Σ*) not sufficient for all types of *summation*? In order to answer these questions one must understand the notion of the *infinitesimal* and how mathematicians have used it to establish continuity. To the mathematician, an *infinitesimal* is an extremely small space of the space of interest; and *continuity* means *no-jump*. This *no-jump* definition is explained in the continuum. Most matter and natural phenomena exist as continuums. For example, the smooth surface of a table is a continuum, distance traveled between stops (without jumps) is a continuum. In fact, *existence* in general, is a continuum. In order to establish a continuum, the mathematician considers several infinitesimals in the space of interest; establishes a relationship that is dependent on the infinitesimals; sums the relationships and then sends the infinitesimal towards zero (never quite arriving at zero). Consequently, the discrete summation represented by the *sigma sign* (*Σ*) is replaced by a continuous summation represented by the *integral sign* (*∫*) as the infinitesimal approaches zero. The mathematical concepts of infinitesimals and continuity have served humanity greatly: it is at the heart of all advanced human knowledge. They are often used to establish solution methods for containership problems such as the length of a curve, areas under curves, solids of revolutions; and motion and change problems involving rate of change.

**Length Of A Curve**

Once the concept of a straight line as defined by Euclid was understood, it was easy to observe that many paths of motion are not straight lines. These paths that are not straight lines are called *curves* (Examples of curves are the circumference of a circle and the path of a projectile). An important aspect of motion is to determine the length of the path traveled. When the path is a straight line or a circular curve, theorems of Euclidean geometry can accurately determine its length, and until calculus was discovered, these theorems were also sufficient for approximating the lengths of non-circular curves. After integral calculus was discovered, it was used to calculate lengths of curves. Consider the curve in figure 3.1:

In figure 3.1, the infinitesimal secant *ds* is considered. Its length is determined by the Pythagoras theorem and then summed over the given interval to determine the length of the curve.

**Area under A curve And Riemann Sums**

The concept of infinitesimal is also used in determining the area under a curve. Consider the area under the curve in figure 3.2. In figure 3.2, the area of the infinitesimal rectangle ABCD is determined and summed over the given interval. This sum is called the *Riemann sum*. The *Riemann sum* approaches a limiting value, *L*, represented by the definite integral *∫ _{a} ^{b} f(x)dx*, as

**Fundamental Theorem Of Calculus**

Equations (1) and (2) of figure 3.3 are the two forms of the *Fundamental Theorem of Calculus*. The differrence between equation (1) in figure 3.3 and the following expression for antiderivatives:
*∫f(x)dx = F(x) + C* (where f(x) is a derivative of F(x) and C is a constant) is the presence of the lower and upper limits *a* and *b* (these limits make this integral a definite integral as opposed to the indefinite integral used for the antiderivatives) and the absence of C in equation (1) in figure 3.3. Now lets informally show why equation (2) of figure 3.3 is true:

Let *F(x)* be continuous in *a ≤ x ≤ b* and *F'(x) exist in a < x < b*.

Then by the *Law of the mean* (also known as the mean value theorem):
*F(b) - F(a) = F'(ξ)(b-a) a < x < b*.

When the interval from a to b is divided into sub-intervals as in figure 3.2, the Riemann sum becomes:
*∑ _{i=1}^{n} F'(x_{i})Δx*.

As

Thus, in the limit:

In essence, all expressions of integral calculus are expressions of Pj Problems.