THE FUNDAMental THEOREM OF CALCULUs AN INTRODUCTION TO THE FUNDAMENTAL THEOREM In the previous section, we went through all four problems in the discovery project, and you’ll agree that each problem yielded unprecedented but hugely significant results. In this section, we’ll discuss each problem and the implications of each result obtained. These results form the basis of one of the most important theorems in calculus: the Fundamental Theorem of Calculus. As we all know, calculus is categorized into two branches: Differential Calculus and Integral Calculus. Integral calculus arose from the area problem, while differential calculus arose from an apparently unrelated problem, the tangent problem. The fundamental theorem of calculus not only establishes a precise connection between these two branches of calculus, it also goes on to show that these two branches are actually inverse processes; that is, integration is the inverse of differentiation, and vice versa. The discovery of this relationship was made by Isaac Barrow (1630 – 1677), who also happened to be Sir Isaac Newton’s teacher in Cambridge. Years later, it was Leibniz and Newton who actually exploited this relationship and developed calculus into the systematic mathematical method it has become today. More importantly, this theorem enabled these mathematicians to compute integrals easily without having to compute limits of sums. In the late 16th century, there were intense disputes between Newton’s followers and those of Leibniz as to who invented calculus first. Newton arrived at his version of calculus first, but for fears of disparagement and controversy, he did not publish it immediately. It was in 1684 that Leibniz’s version of calculus was first published. The truth of the matter is that both men invented calculus autonomously. You can read an article about this controversy at wikipedia.org. You can also read more about Sir Isaac Newton and Gottfried Wilhelm Leibniz at wikipedia.org. The Fundamental Theorem is divided into two parts. So, for the rest of this section, we’ll discuss the results of the problems we dealt with in the discovery project, from which we’ll derive the two parts of the fundamental theorem. First, let’s analyze problem 1. We found the area under the line y = 2t + 1 using geometry and from the geometrical method used, an expression was derived for the area (in terms of the limit), which was:
A(x) = t2 + t – 2 So far, we have simply performed the process of integration. Now here’s where the whole process takes an interesting twist: We differentiate the area function above and we actually got back the original equation y = 2t + 1. The problem simply teaches us that integration and differentiation are inverse processes: If we obtain the integral of a function on a given interval and then differentiate the result, we end up with the original equation. Thus, it’s basically an issue of one process reversing the other, and vice versa. A similar situation occurs in problem 2: We are given an integral from which we derived an explicit expression for the area:
A(x)
=
4/3
+
x
+
After that, we differentiate A(x) and obtain
A’(x)
=
1 + x2
Now, let’s go back to the integral in question:
A(x)
=
∫
x -1
(1 + t2) dt
(x3/3)