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Integral - Definite Integrals, Indefinite Integrals - Applications

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The integral is one of two main concepts embodied in the branch of mathematics known as calculus, and it corresponds to the area under the graph of a function. The area under a curve is approximated by a series of rectangles. As the number of these rectangles approaches infinity, the approximation approaches a limiting value, called the value of the integral. In this sense, the integral gives meaning to the concept of area, since it 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 simple geometric formulas). When an integral represents an area, it is called a definite integral, because it has a definite numerical value.

The integral is also 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.


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.


Resources

Books

Abbot, P., and M.E. Wardle. Teach Yourself Calculus. Lincolnwood, IL: NTC Publishing, 1992.

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

Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. New York: CRC Press, 1998.


J.R. Maddocks

KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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.

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The integral is one of two main concepts embodied in the branch of mathematics known as calculus, and it corresponds to the area under the graph of a function. The area under a curve is approximated by a series of rectangles. As the number of these rectangles approaches infinity, the approximation approaches a limiting value, called the value of the integral. In this sense, the integral gives meaning to the concept of area, since it 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 simple geometric formulas). When an integral represents an area, it is called a definite integral, because it has a definite numerical value.

The integral is also 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.


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.


Resources
Books
Abbot, P., and M.E. Wardle. Teach Yourself Calculus. Lincolnwood, IL: NTC Publishing, 1992.

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

Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. New York: CRC Press, 1998.


J.R. Maddocks

KEY TERMS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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.

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Additional Topics

Integral - 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 displacemen…

Integral - 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 conseq



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