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Definite integrals

Indefinite integrals



The integral is one of two main concepts embodied in the branch of mathematics known as calculus. Interpreted graphically, an integral of a function corresponds to the area under the graph of that function. In concept, the area under the curve made by plotting the function is approximated by a series of rectangles. As the number of these rectangles approaches infinity, the approximation approaches a limiting value that is called the value of the integral. The integral provides a means of determining the areas of those irregular figures whose areas cannot be calculated in any other way (such as by multiple applications of formulas for simple geometric shapes). When an integral represents an area, it is called a definite integral, because it has a definite numerical value; integrals may also take the form of functions, however, without single numerical values, in which they are called indefinite integrals.

The integral is the inverse of the other main concept of calculus, the derivative, and thus provides a way of identifying functional relationships when only a rate of change is known. When an integral represents a function whose derivative is known, it is called an indefinite integral and is a function, not a number. Fermat, the great French mathematician, was probably the first to calculate areas by using the method of integration.

Definite integrals

A definite integral represents the area under a curve, but as such, it is much more useful than merely a means of calculating irregular areas. To illustrate the importance of this concept to the sciences consider the following example. The work done on a piston, during the power stroke of an internal combustion engine, is equal to the product of the force acting on the piston times the displacement of the piston (the distance the piston travels after ignition). Engineers can easily measure the force on a piston by measuring the pressure in the cylinder (the force is the pressure times the cross sectional area of the piston). At the same time, they measure the displacement of the piston. The work done decreases as the displacement increases, until the piston reaches the bottom of its stroke. Because area is the product of width times height, the area under the curve is equal to the product of force times displacement, or the work done on the piston between the top of the stroke and the bottom.

The area under this curve can be approximated by drawing a number of rectangles, each of them h units wide. The height of each rectangle is equal to the value of the function at the leading edge of each rectangle. Suppose we are interested in finding the work done between two values of the displacement, a and b. Then the area is approximated by Area = hf (a) + hf (a+ h) + hf (a+2h) +... + hf (a+(n-1)h) + hf (b-h). In this approximation n corresponds to the number of rectangles. If n is allowed to become very large, then h becomes very small. Applying the theory of limits to this problem shows that in most ordinary cases this results in the sum approaching a limiting value. When this is the case the limiting value is called the value of the integral from a to b and is written:

Where the integral sign (an elongated s) is intended to indicate that it is a sum of areas between x = a and x = b. The notation f (x)dx is intended to convey the fact that these areas have a height given by f (x), and an infinitely small width, denoted by dx.

Indefinite integrals

An indefinite integral is the inverse of a derivative. According to the fundamental theorem of calculus, if the integral of a function f (x) equals F(x) + K, then the derivative of F(x) equals f (x). This is true for any numerical value of the constant K, and so the integral is called indefinite.

The inverse relationship between derivative and integral has two very important consequences. First, in many practical applications, the functional relationship between two quantities is unknown, and not easily measured. However, the rate at which one of these quantities changes with respect to the other is known, or easily measured (for instance the previous example of the work done on a piston). Knowing the rate at which one quantity changes with respect to another means that the derivative of the one with


Fundamental theorem of calculus The fundamental theorem of calculus states that the derivative and integral are related to each other in inverse fashion. That is, the derivative of the integral of a function returns the original function, and vice versa.

Limit A limit is a value that a sequence or function tends toward. When the sum of an infinite number of terms has a limit, it means that it has a finite value.

Rate A rate is a comparison of the change in one quantity with the simultaneous change in another, where the comparison is made in the form of a ratio.

respect to the other is known (since that is just the definition of derivative). Thus, the underlying functional relationship between two quantities can be found by taking the integral of the derivative. The second important consequence arises in evaluating definite integrals. Many times it is exceedingly difficult, if not impossible, to find the value of the integral. However, a relatively easy method, by comparison, is to find a function whose derivative is the function to be integrated, which is then the integral.


There are many applications in business, economics and the sciences, including all aspects of engineering, where the integral is of great practical importance. Finding the areas of irregular shapes, the volumes of solids of revolution, and the lengths of irregular shaped curves are important applications. In addition, integrals find application in the calculation of energy consumption, power usage, refrigeration requirements and innumerable other applications.



Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.

Morgan, Augustus De. The Differential and Integral Calculus. Whitefish, MT: Kessinger Publishing, 2004.

Verberg, Dale, et al. Calculus. 9th ed. Upper Saddle River, NJ: Prentice Hall, 2006.

J. R. Maddocks

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