# Integral - Definite Integrals, Indefinite Integrals - Applications

### function calculus value derivative

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

## User Comments

almost 5 years ago

good

almost 2 years ago

project1:intgrationand application of in eonomics

about 2 years ago

Thanks for your article, it can help me.

Really your post is really very good and I appreciate it.

It’s hard to sort the good from the bad sometimes,

You write very well which is amazing. I really impressed by your post.

Crystal X mengatasi Keputihan

Crystal X Mengobati Keputihan

Cara Mengatasi Keputihan

Cara Mengobati Keputihan

Cara Mengatasi Keputihan Dengan Crystal X

Crystal X mengatasi Keputihan

Crystal X

Crystal X Agar Cepat Hamil

Crystal X

Crystal X Mengobati Kanker Serviks

Crystal X Mengatasi Masalah Haid

Crystal X Merawat Organ Intim

Crystal X mengatasi Keputihan

http://www.crystalxmengatasikeputihan.com/2015/05/cara-mengatasi-keputihan-tanpa-efek.html

Crystal X Mengobati Keputihan

http://www.crystalxmengatasikeputihan.com/2015/04/mengatasi-keputihan-dengan-crystal-x.html

Cara Mengatasi Keputihan Dengan Crystal X

Crystal X Mengatasi Masalah Haid

http://www.crystalxmengatasikeputihan.com

Crystal X Agar Cepat Hamil

http://www.crystalxmengatasikeputihan.com/2015/04/khasiat-crystal-x-merawat-organ-intim.html

Crystal X Mengobati Kanker Serviks

http://www.crystalxmengatasikeputihan.com/2015/04/khasiat-crystal-x-merawat-organ-intim.html

Crystal X Merawat Organ Intim

http://www.crystalxmengatasikeputihan.com/2015/04/khasiat-crystal-x-merawat-organ-intim.html

http://distributor-resmi-nasa.blogspot.com/2013/05/manfaat-dan-bahaya-crystal-x.html

over 6 years ago

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.

Ads by Google

Magento ERP Integration 40+ ERPs/Accounting Pkgs supported Automate - increase profits www.ebridgeconnections.com

The Economist Magazine Get a World view Every Week. 12 Issues for Rs. 500 only! EconomistSubscriptions.com

Science/Maths Class 6-12 Video Lessons, Tests, NCERT Solutions, Activities & More. View! TopperLearning.com/Class6_12

Tech Writing Tool Reviews Free Reviews of Leading Help and Authoring Tools - Flare, Robo, Doc www.tecwriter.com

Listen to an audio version of this article.

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

Read more: Integral - Definite Integrals, Indefinite Integrals - Applications - Calculus, Function, Value, Derivative, Called, and Change http://science.jrank.org/pages/3618/Integral.html#ixzz1EwbhRj1X