Integral Calculus

Type of physical science: Mathematical methods

Field of study: Calculus

A field of study that is the companion, complement, and inverse of differential calculus, integral calculus is indispensable in solving a variety of physical problems, from determining the paths of motion of heavenly bodies to calculating the flow of current through an electrical network.

Overview

To understand integral calculus, one must start with the problem of finding the area bounded by the graph of a continuous function y = f(x), the x-axis, the line x = a, and the line x = b.

The basic idea of integral calculus is to cut this area vertically into thin slices and approximate the area of each slice with the area of a rectangle. The height of a typical rectangle will be the value of the function f(x) and the width of this rectangle will be denoted by Δx. The desired area will be approximated by summing the areas f(x) x Δx as x goes from a to b. By letting the widths of these rectangles approach zero, the approximating sums will approach the desired area called the "definite integral of f(x) from a to b" and designated by ∫^b_a f(x)dx. This notation, formulated by Gottfried Wilhelm Leibniz, contains all the pertinent information about the area one is to find. The function f(x) is called the "integrand," the numbers a and b are the "limits of integration," and the variable x is the "variable of integration." Thus, if one is to find the area below the function f(x) = 3x², between x = 1 and x = 4, one would write ∫⁴₁ 3x²dx to represent this area.

The most useful result of integral calculus is the fundamental theorem of calculus. This theorem provides a connecting link between the evaluation of a definite integral and the concept of the derivative In symbols; the fundamental theorem of calculus says that if G(x) is any antiderivative of f(x), then ∫^b_a f(x)dx = G(b) - G(a). This means that one can evaluate the definite integral of f(x) from a to b if one can find an antiderivative, G(x), for the function f(x). For example, once it is determined that the derivative of G(x) = x³ is f(x) = 3x², making G(x) an antiderivative for f(x), one can calculate the area bounded by f(x) between x = 1 and x = 4 as follows: ∫⁴₁ f(x)dx = ∫⁴₁ 3x²dx = G - G = 4³ - 1³ = 63 square units.

In terms of the integral notation, the most general antiderivative of a function f(x) is called the "indefinite integral of f(x)" and denoted by ∫f(x)dx. As an example, when f(x) = 3x², then ∫3x²dx = x³ + C. The indefinite integral of a general function f(x) is a function G(x), containing an arbitrary constant C, whose derivative is f(x). Thus, finding an indefinite integral is the reverse or inverse of finding a derivative.

Integral calculus may be used to derive formulas for the area of a circle, the volume of a cone, the volume of a torus, and many other areas and volumes. If calculating areas and volumes were the only uses for integrals, they might be of interest only to mathematicians.

Calculus textbooks are filled with properties of integrals and theorems about integrals, but the importance of the integral to the physical sciences cannot be overemphasized. If a physical problem is modeled in such a way that a quantity of interest can be approximated by the summation of values f(x) x Δx, as x ranges from a to b, then this quantity may be modeled as the integral ∫^b_a f(x)dx. The intuitive ease with which the symbols of integral calculus can be manipulated may permit one to deal with complicated concepts easily in a wide variety of disciplines.

Applications

The integral may be used to model problems involving work. In the terminology of physics, the work needed to move an object a specified distance when opposed by a constant force is given by the product of the opposing force and the distance moved (work equals force times distance). This formula is adequate for situations where the force is constant, but what if the force changes from one point to another along the path where the work is to be done?

Assuming that the force is given by a continuous function, f(x), one may approximate the work done in moving the small step from x to x + Δx by f(x) x Δx (force times distance). By adding all these small steps of work as x goes from a to b, an approximation can be obtained to the work done in moving from a to b that is also an approximation to the definite integral of f(x) from a to b. Letting the lengths of these small intervals go to zero, it can be concluded that ∫^b_a f(x)dx is the work done in moving from a to b. Armed with this idea, scientists may calculate the work required to perform a wide variety of tasks from putting a satellite into orbit to pumping water from one tank to another.

As an example, suppose someone must push the end of a spring, thus compressing the spring, along a straight line from point a to point b. How much work will be expended to do this task? Hooke's law states that the force required to hold a spring x units shorter than its natural length is proportional to x. That is, if the coordinate system is set up so that the spring is relaxed when x = 0, then the variable resisting force will be f(x) = kx, where k is a positive constant called the spring constant.

Thus, the work required to move the end of the spring from x = a to x = b will be given by the integral ∫^b_a kx dx. This integral may be evaluated as k(b² - a²)/2. This example also illustrates the need for the "dx" in the notation for the integral, which indicates that the variable of integration is x and not k.

Integrals have been generalized in many different directions. The original concept involved a positive function, whose integral evaluates to a positive number (area). In many physical applications, the quantity modeled as an integral may be negative, so the concept was generalized to include functions that may be negative. Yet, what if the spring in the example above were stretched when beginning to move from a to b? The spring would be pulling the person from a to b, or that person would be doing "negative work."

Generalizing the definition of "integral" to include functions that may be negative conforms to this physical model.

The rate of decay of radium is proportional to the amount present at a given time. That is, if Q denotes the amount of radium present at time t, the rate of change of Q will be the derivative of Q with respect to t or Q, and Q = kQ for some constant k. This equation involving Q and its derivative is called a differential equation. The solution to this differential equation will be a function Q = f(t) that gives the amount of radium present at time t. This problem is readily solved using integral calculus, as are similar problems involving radioactive decay. Many other physical situations are described or modeled in terms of differential equations. Since the integral is the reverse or inverse of the derivative, it plays an important role in the solution of many of these differential equations. One technique of solving a large category of differential equations was developed by Pierre-Simon Laplace in 1779. It involves a transformation that is defined in terms of an integral. With the Laplace transform, a differential equation is converted into a simpler problem that may be solved by algebraic methods. The algebraic answer is then transformed into a solution for the differential equation.

The theory of integral calculus proves that if f(x) is any continuous function, then it has an antiderivative or indefinite integral G(x). There are large categories of functions, f(x), whose integrals may be evaluted directly using multiplication, division, addition, subtraction, and the like. In any calculus textbook, entire chapters are devoted to techniques of integration or methods of evaluating integrals in terms of the calculator functions. Many textbooks include a table of integrals, which contains many of these results so that one may easily find the indefinite integral of f(x) if it conforms to one of the general forms listed in the table. Nevertheless, there are many functions, f(x), that are expressed in terms of the functions on a scientific calculator, but whose antiderivative, G(x), cannot be expressed in terms of these same calculator functions.

One important function whose antiderivative cannot be expressed in terms of calculator functions is the bell-shaped "normal probability distribution" shown in the figure. The scores of intelligence tests satisfy a normal probability distribution, which means that the probability that an individual score will fall between a and b is given by the integral ∫^b_a f(x)dx, where f(x) is shaped like the function in the figure. When a scientist collects data to determine the average tread life of tires, the average height of ten-year-old orange trees, or the average number of kilometers per gallon a car gets, one expects these data to be normally distributed. For each of these examples, there is a different bell-shaped function f(x) whose definite integral from a to b gives the probability that a data value will fall between a and b. Statistics books contain tables that permit one to calculate these integrals to four decimal places.

Mathematicians have devised ingenious methods for approximating the value of an integral ∫^b_a f(x)dx in case this value cannot be found exactly in terms of the calculator functions. One plan involves dividing the interval from a to b into a large number of small intervals (as in the definition of the integral), approximating f(x) on each small interval by a function whose integral can be calculated on that interval, and adding all these results to get an approximation to the integral from a to b. The simplest of these schemes, called the trapezoidal rule, approximates f(x) by a straight line segment on each small interval. As noted in the figure, one can see that the definition of the integral approximates the area on each small interval by the area of a rectangle whose height is given by the value of the function at the left end point of the interval. A better approximation may be obtained to the area by using trapezoids whose sides and bottoms are the same as the rectangles, but whose tops are slanted to make trapezoids. For many functions, the trapezoidal rule will give acceptable accuracy with a relatively small number of intervals. Some hand-held calculators have a trapezoidal rule program built in so that the calculator can be given a function f(x), the limits of integration a and b, and the number of small intervals to use. The program does the rest to give a trapezoidal rule approximation to the integral ∫^b_a f(x)dx.

One novel method of approximating the definite integral of f(x) from a to b is called the Monte Carlo method. For example, suppose that someone is tossing stones through a rectangular, open window that has the x-axis at the bottom, x = a as its left side, x = b as its right side, and a graph of the function f(x) contained inside the window. A count is given of how many stones go below the graph of f(x) and the total number of tosses that go through the window. The ratio of stones below the graph to the total number of tosses will approximate the ratio of the integral to the area of the rectangular window. Thus, ∫^b_a f(x)dx is approximately equal to (stones below the graph) x (area of window)/(total tosses). As with most of the numerical methods, the stones are not actually tossed; this process is simulated with a computer.

Context

Integral calculus has its origins in the problem of finding the area of a two- dimensional region bounded by a curve. The Greek mathematician Eudoxus of Cnidus (c. 390-335 B.C.) developed the "method of exhaustion" for proving theorems about areas and volumes. Archimedes of Syracuse (287-212B.C.) made ingenious use of Eudoxus' method of exhaustion to find the areas of regions bounded by parabolic curves and circular curves. The idea was to inscribe larger and larger polygons of known area inside the curved region so that the area would eventually be "exhausted." The desired area would be approached by the areas of the inscribed polygons. In order to calculate the area bounded by more complicated curves, new mathematical machinery was needed. The modern decimal number system made calculations much easier than the cumbersome number system used by the ancient Greek mathematicians.

The coordinate geometry of Rene Descartes (1596-1650) was invaluable to the ultimate development of the calculus. With the Cartesian coordinate system, one can associate an algebraic equation or formula with a geometric curve. Mathematicians could now discuss a curve in terms of a simple equation in place of a complicated geometric definition.

Meanwhile, scientists were finding more reasons to calculate areas bounded by curves.

Johannes Kepler (1571-1630) discovered that the planets moved in elliptical (not circular, as previously believed) paths around the sun. His second law stated that the line joining the sun and a planet sweeps out equal areas over equal time intervals. This was merely one more motivation for the development of integral calculus.

In the middle of the seventeenth century, two mathematical giants entered the picture: Sir Isaac Newton (1642-1727) in England and Gottfried Wilhelm Leibniz (1646-1716) in Germany. They are each credited with the discovery of integral calculus, and it seems that each one developed the tools of calculus somewhat independently. Newton was content to discover mathematical truths without publishing them.

On the other hand, Leibniz did publish his results and deserves some credit for being the first to share his work with the world. The integral sign, , first used by Leibniz, was invented because it resembled a long letter "S" as in "summation." Eventually, Newton's work was published, but Leibniz's integral sign is still in use today.

Integral calculus has become such a valuable tool in the study of most branches of science and engineering that it is virtually inseparable from them. To understand these areas of endeavor fully, integral calculus must be understood.

Principal terms

ANTIDERIVATIVE OF F(x): a function G(x) whose derivative is f(x)

CONTINUOUS FUNCTION: a function f with the property that as the number u gets close to x, the functional value f(u) gets close to f(x)

DERIVATIVE OF G'(x): a function, denoted by G(x), that gives the slope of the tangent line to the graph of the function G at the point (x, G(x))

FUNCTION: a rule or formula, sometimes denoted by f or f(x), that gives a number y = f(x) for each number x in a specified interval; f(x) is the value of the function at x

GRAPH OF A FUNCTION: the set of points (x,f(x))

relative to a rectangular coordinate system

RECTANGULAR COORDINATE SYSTEM: a system where each geometric point is associated with a pair of numbers (x,y) called its coordinates; the x and y coordinates of a point are the directed distances from a fixed pair of intersecting lines called the y-axis and the x-axis

TRAPEZOID: a quadrilateral having two parallel sides

Bibliography

Downing, Douglas. CALCULUS THE EASY WAY. Woodbury, N.Y.: Barron's Educational Series, 1982. This is a well-written book that gives an introduction to integral calculus. Contains many examples and worked problems, together with many practical applications of integral calculus.

Gordin, William R., and Bernard Sohmer. ADVANCED ALGEBRA AND CALCULUS MADE SIMPLE. Garden City, N.Y.: Made Simple Books, 1959. This book gives a simple, yet comprehensive introduction to integral calculus. Contains many examples, practice exercises, and worked problems.

Hogben, Lancelot. MATHEMATICS FOR THE MILLIONS. New York: W. W. Norton, 1937. This excellent book is written using a historical perspective. Each new concept is motivated with practical applications before it is introduced. The chapter "The Calculus of Newton and Leibnitz" brings calculus alive.

Newman, James R. THE WORLD OF MATHEMATICS. 4 vols. New York: Simon & Schuster, 1956. This four-volume compendium gives a comprehensive historical survey of mathematics from the Rhind Papyrus of Egypt to Albert Einstein's theories. The sections on Newton are of particular interest.

Stewart, James. CALCULUS. Monterey, Calif.: Brooks/Cole, 1987. This calculus textbook gives a thorough treatment of integral calculus, together with many applications. Contains a discussion of the trapezoidal rule and Simpson's rule for numerically approximating definite integrals.

Weast, Robert C. CRC HANDBOOK OF CHEMISTRY AND PHYSICS. 66th ed. Boca Raton, Fla.: CRC Press, 1986. This reference book gives a concise explanation of the mechanical steps involved in evaluating integrals, together with a table of integrals containing 728 formulas. Includes a section on the Laplace transform and its applications.

Graph of a continuous function y = f(x)

Numerical Solutions of Differential Equations

Orthogonal Functions and Expansions