Introduction to Integral Calculus Introduction （积分学概论介绍）.pdfVIP

Introduction to Integral Calculus Introduction It is interesting to note that the beginnings of integral calculus actually predate differential calculus, although the latter is presented first in most text books. However in regards to formal, mature mathematical processes the differential calculus developed first. The first steps towards integral calculus actually began in ancient Greece. In the third century B.C., Aristotle became interested in areas defined by certain curves. He used rectangles to approximate these regions, and then used smaller and smaller rectangles, so that the approximation became better and better. You might note that this is not unlike some of the early methods trying to approximate the slope of a line that eventually led to differential calculus. He called this procedure the method of exhaustion. The famous mathematician Riemann would later generalize this procedure, using the concepts of limits that were developed for differential calculus. The process, often referred to as a “Riemann Sum”, is similar to Aristotles rectangles, but the rectangles need not have a uniform thickness. Also, Riemanns method generalizes to higher dimensions, e.g. computing the volume bounded by a surface. There is an interesting Java applet on the web that illustrates how Riemann Sums work. /instruct/bwagner/applets/RiemannSums.html Since Riemann Sums are not used today to calculate the area of an shape defined by a curve, the specific mathematics of them are not crucial to our discussion of integral calculus. There are, however, several interesting websites that do discuss the mathematics of the Riemann Sums: /encyclopedia/RiemannSum.html 1 /wiki/Riemann_sum Differential calculus was primarily concerned with the slope of a line tangent to a curve at a given point. This was helpful in a varie