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The Definite Integral
A definite integral is the area of the region in the
Learning Objective

Compute the definite integral of a function over a set interval
Key Points
 Integration is an important concept in mathematics and—together with its inverse, differentiation—is one of the two main operations in calculus.
 Integration is connected with differentiation through the fundamental theorem of calculus: if
$f$ is a continuous realvalued function defined on a closed interval$[a, b]$ , then, once an antiderivative$F$ of$f$ is known, the definite integral of$f$ over that interval is given by$\int_{a}^{b}f(x)dx = F(b)  F(a)$ .  Definite integrals appear in many practical situations, and their actual calculation is important in the type of precision engineering (of any discipline) that requires exact and rigorous values.
Terms

integration
the operation of finding the region in the
$xy$ plane bound by the function 
antiderivative
an indefinite integral

definite integral
the integral of a function between an upper and lower limit
Full Text
Integration is an important concept in mathematics and—together with its inverse, differentiation—is one of the two main operations in calculus. Given a function
Definite Integral
A definite integral of a function can be represented as the signed area of the region bounded by its graph.
The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if
Definite integrals appear in many practical situations. If a swimming pool is rectangular with a flat bottom, then from its length, width, and depth we can easily determine the volume of water it can contain (to fill it), the area of its surface (to cover it), and the length of its edge (to rope it). But if it is oval with a rounded bottom, all of these quantities call for integrals. Practical approximations may suffice for such trivial examples, but precision engineering (of any discipline) requires exact and rigorous values for these elements.
For example, consider the curve
We ask, "What is the area under the function
As a first approximation, look at the unit square given by the sides
Notice that we are taking a finite sum of many function values of
As for the actual calculation of integrals, the fundamental theorem of calculus, due to Newton and Leibniz, is the fundamental link between the operations of differentiating and integrating. Applied to the square root curve,
and simply take
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Key Term Reference
 approximation
 Appears in these related concepts: Numerical Integration, Roundoff Error, and The Discrete Fourier Transform
 area
 Appears in these related concepts: Double Integrals Over Rectangles, Area Between Curves, and Introduction to Circles
 curve
 Appears in these related concepts: Integration By Parts, Arc Length and Surface Area, and Arc Length and Speed
 differentiation
 Appears in these related concepts: WBC Formation, Considering the Environment, and Socioemotional Development in Adolescence
 function
 Appears in these related concepts: Solving Differential Equations, Visualizing Domain and Range, and The Vertical Line Test
 graph
 Appears in these related concepts: Graphing Equations, Graphical Representations of Functions, and Graphs of Equations as Graphs of Solutions
 infinitesimal
 Appears in these related concepts: Differentials, Area and Distances, and The Fundamental Theorem of Calculus
 integral
 Appears in these related concepts: The Correction Factor, Expected Value, and Indefinite Integrals and the Net Change Theorem
 inverse
 Appears in these related concepts: Inverse Functions, Hyperbolic Functions, and The Law of Universal Gravitation
 root
 Appears in these related concepts: Radical Equations, Rational Exponents, and Introduction to Radicals
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
 volume
 Appears in these related concepts: Volumes, Volume and Density, and Shape and Volume
Sources
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Cite This Source
Source: Boundless. “The Definite Integral.” Boundless Calculus. Boundless, 26 May. 2016. Retrieved 31 Jul. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/derivativesandintegrals2/integrals11/thedefiniteintegral652945/