What are integrals?

One of the major problems of calculus is trying to find the area of a continuous function of the interval from point a to point b. By the seventeenth century mathematicians could calculate areas of functions, parabolas, and spirals using a method called "method of exhaustion"(14), created by Archimedes, which was the earliest form of integration. The problem with the method of exhaustion was that it had different procedures and calculation for every different problem - it wasn't universal. Newton and Leibniz created the antiderivative, or the indefinite integral, method for finding areas of polygons. These mathematicians realized that they could find the derivative of the function A(x). An indefinite integral can be defined as follow: "If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x) = F(x) + C,for all x in I where C is a constant" (15).
This is a general list of integral rules (the "§" sign stands for the integral sign):

§ 0 dx = C

§ k dx = kx + C

§ kf(x)dx = k § f(x) dx

§ [f(x) ± g(x)] dx = § f(x) dx ± § g(x) dx

§ xn dx = (xn + 1) / (n+1) + C n cannot = -1

§ cos x dx = sin x + C

§ sin x dx = -cos x + C

§ (sec x)2 dx = tan x + C

§ sec x tan x dx = sec x + C

§ (csc x)2 x dx = -cot x + C

§ csc x cot x dx = -csc x + C

An integral can be used in many ways. One of the ways that an integral can be applied is to find the volume of a solid object that is revolving around a line. The solid is generated by the region. The axis of revolution is the line about which the revolution takes place. The problem of solving for a sold revolving around a line is best illustrated by the following above image. There are many ways to solve for this volume, but one of the most common is the disc method."To find the volume of a solid of revolution with the disc method, us the formula : Volume = V = pi § [R(x)]2 dx" (16), and examine the data below.

14. Archimedes Biography

15. Antiderivatives

16. The Disc Method

Previous Topic

Home

Next Topic