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1. History of the Integral from the 17th Century
The path to the development of the integral is a branching one, where similar discoveries were made simultaneously by different people. The history of the technique that is currently known as integration began with attempts to find the area underneath curves. The foundations for the discovery of the integral were first laid by Cavalieri, an Italian Mathematician, in around 1635. Cavalieriís work centered around the observation that a curve can be considered to be sketched by a moving point and an area to be sketched by a moving line.
1.2 Cavalieri's Method of Indivisbles
In order to deal with the geometrical notion of a moving point, Cavalieri worked with what he called "indivisibles". That is, if a moving point can be considered to sketch a curve, then Cavalieri viewed the curve as the sum of its points. By this notion, each curve is made up of an infinite number of points, or "indivisibles". Likewise, the "indivisibles" that composed an area were an infinite number of lines. Though Cavalieri was not the first person to consider geometric figures in terms of the infinitesimal (Kepler had done so before him), he was the first to use such a notion in the computation of areas (Hooper 248-250).
In order to introduce Cavalieriís method, consider finding the area of a triangle. For many years, it had been known that the area of a triangle was ½ the area of a rectangle which has the same base and height.
In Figure 1.1, the rectangle has a base of 6 units and a height of 5 units (A = bh, so the total area is 30 units). The total area of the inner rectangular regions can easily be computed by taking the sum of all the inner rectangles. Comparing the two areas:
Using the same methodology, the ratio for a larger rectangle with a greater number of inner subdivisions is computed:
The total area of the inner regions is always one-half the area of the total rectangle. This can be shown formally by using the closed form of the summation for the numerator:
Using the closed form, it can be seen that:
Cavalieri now took a step of great importance to the formation of the integral calculus. He utilized his notion of "indivisibles" to imagine that there were an infinite number of shaded regions. He saw that as the individual shaded regions became small enough to simply be lines, the jagged steps would gradually define a line. As the jagged steps became a line, the shaded region would form a triangle. As the number of shaded regions increases, the ratio remains simply one-half.
Cavalieriís methodology agreed with the long-held result that the area of a triangle was one-half the product of the base and height. He had also shown that his notion of "indivisibles" can be used to successfully describe the area underneath the curve. That is, as the areas of the rectangles turn into lines, their sum does indeed produce the area underneath the curve (in this case, a line). Cavalieri went on to use his method of "indivisibles" to find the area underneath many different curves. However, he was never able to formulate his techniques into a logically consistent foundation that others accepted. Though Cavalieriís techniques clearly worked, it was not until Sir John Wallis of England that the limit was formally introduced in 1656 and the foundation for the integral calculus was solidified (Hooper 249-253).
In order to fully understand Wallisí contributions to the integral calculus, it is first necessary to see how Cavalieriís theoretical techniques can be applied to find the area underneath a curve more complicated than a line. In order to do so, this technique will be applied to find the area underneath the parabola .
Each rectangular region has a base of 1 unit along the x-axis and height of x2 (obtained from the definition of the parabola). The number of rectangular regions will be defined to be m. Cavalieri again attempted to express the area underneath the curve as the ratio of an area that was already known. He considered the area enclosing all of the m rectangles. It can easily be seen from the diagram that the base of this rectangle will be m+1 (there are m rectangles, the first starting at ½ and the last one ending at m + ½). The height of the enclosing rectangle will be m2, from the definition of the parabola. The ratio can now be expressed with the following equation:
Recall that the area of a rectangle is defined by the product of its base and height. It was stated that the bounding rectangle had a base of m+1 and a height of m2, which accounts for the denominator. The numerator is easily explained as well: each of the m rectangles has a base of 1 and a height of its x value squared. Cavalieri now proceeded to calculate the ratio for different values of m. In doing so, he noticed a pattern and was able to establish a closed form for the ratio of the areas:
Cavalieri then utilized his important principle of "indivisibles" to make another important leap in the development of the calculus. He noticed that as he let m grow larger, the term had less influence on the outcome of the result. In modern terms, he
That is, as he lets the number of rectangles grow to infinity, the ratio of the areas will become closer to . Though Cavalieri did not formally introduce the notation for limits, he did utilize the idea in the computation of areas. After using the concept of infinity to describe the ratios of the area, he was able to derive an algebraic expression for the area underneath the parabola. For at any distance x along the x-axis, the height of the parabola would be x2. Therefore, the area of the rectangle enclosing the rectangular subdivisions at a point x was equal to or x3. From his earlier result, the area underneath the parabola is equal to 1/3 the area of the bounding rectangle. In other words:
With this technique, Cavalieri had laid the fundamental building block for integration.
1.3 Wallis' Law for Integration of Polynomials
John Wallisí contribution to the integral calculus was to derive an algebraic law for integration that alleviated the necessity of going through such analysis for each curve. Through examining the relationship between a function and the function that describes its area (henceforth referred to as the area-function), he was able to derive an algebraic law for determining area-functions. Rather than simply present the algebraic relationship (which the reader is doubtless familiar with if (s)he has studied a minimal amount of calculus), we will perform a similar analysis as to what led Wallis to derive his law.
First, consider the graph of the function y = k or:
Clearly, it can be seen from the diagram, that the area underneath the line at any point along the x-axis will be kx or .
Next, consider the graph of the function y = kx :
At any point x along the x-axis, the height will be equal to kx. Since the area forms a triangle, the area underneath the curve can be expressed as ½ the base times the height or . As was already shown above, the area underneath a parabola
y = kx2, can be expressed as . Wallis noticed an algebraic relationship between a function and its associated area-function. That is, the area-function of is . Wallis went on to show that not only does this hold true where n is a natural number (which had been the extent of Cavalieriís work), but that it also worked for negative and fractional exponents. Wallis also showed that the area underneath a polynomial composed of terms with different exponents (e.g. ) can be computed by using his law on each of the terms independently (Hooper 255 - 260).
1.4 Fermat's Approach to Integration
One of the first major uses of infinite series in the development of calculus came from Pierre De Fermatís method of integration. Though previous methods of integration had used the notion of infinite lines describing an area, Fermat was the first to use infinite series in his methodology. The first step in his method involved a unique way of describing the infinite rectangles making up the area under a curve.
Fermat noticed that by dividing the area underneath a curve into successively smaller rectangles as x became closer to zero, an infinite number of such rectangles would describe the area precisely. His methodology was to choose a value 0 < e < 1, such that a rectangle was formed underneath the curve at each power of e times x (see Figure 1.5, NOTE: e was simply Fermatís choice of variable names, not e = 2.71828Ö). Fermat then computed each area individually:
The first equation represents the area of the largest rectangle, the second equation the next rectangle to the left, and so on. The areas are simply found by multiplying the base times the height. The base is known by the power of e, and the height by evaluating at the given x value. The simplifications of each area expression are given in a form that will be useful when attempting to find the infinite sum. Fermatís next step was to compute the infinite sum of these rectangles as the power of e approached infinity.
By determining the sum of each increasing finite series, he was able to develop an expression for the infinite sum.
In order to find a closed form for the expression
...note that the sum is a geometric series of the form:
If 0 < x < 1, the sum is (this can be shown to be true by long dividing (1-x) into 1). Therefore, by substituting back in for x and inserting into the overall equation, the area can be expressed as:
Fermat now wished to express the area entirely in terms of x, and in order to do so substituted , which by simplification and factoring out (1-E):
Fermat now made a step that with the benefit of current knowledge is explainable, but at that time was not properly justified. That is, Fermat said let E = 1 and since and because then e must also equal 1. By substituting 1 for E in the area expression above:
Although this methodology yielded the appropriate result for the area underneath the curve, Fermatís justification of letting E = 1 was not properly formulated. What he actually was doing was taking the limit as E approaches 1 and as E approaches 1 so too will e. As e approaches 1, then e raised to any power will also approach 1, and the infinite sum of the areas underneath the curve has been determined. The notion of a limit was hinted at in Fermatís work, but it was not formally defined until later (Boyer 162 - 169).
Wallis and Fermat's work had laid the groundwork for the modern concept of the integral. However, what Fermat and Wallis had failed to recognize was the relationship between the differential and the integral. That idea would be developed simultaneously by two men: Newton and Leibniz. This would later be known as the Fundamental Theorem of Calculus and, as the name implies, it is a landmark discovery in the history of the Calculus. However, before proceeding on to describe this important theorem, it is first necessary to examine the development of the differential.
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