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The Definite Integral
A definite integral is the area of the region in xyplane bound by graph of f, xaxis, and vertical lines x = a and x = b.
Learning Objective

Compute the definite integral of a function over a set interval
Key Points

Integration is an important concept in mathematics andtogether 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

definite integral
the integral of a function between an upper and lower limit

antiderivative
an indefinite integral
Full Text
Integration is an important concept in mathematics andtogether with its inverse, differentiation is one of the two main operations in calculus.
Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral
The integrals discussed in this atom are termed definite integrals.
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 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
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 y = f(x) between x = 0 and x = 1 with
We ask, "What is the area under the function f, in the interval from 0 to 1?
" and call this (yet unknown) area the integral of f.
The notation for this integral will be
As a first approximation, look at the unit square given by the sides x = 0 to x = 1, y = f(0) = 0, and y = f(1) = 1.
Its area is exactly 1.
As it is, the true value of the integral must be somewhat less.
Decreasing the width of the approximation rectangles should yield a better result, so we will cross the interval in five steps, using the approximation points 0, 1/5, 2/5, and so on, up to 1.
Fit a box for each step using the right end height of each curve piece, thus obtaining
Notice that we are taking a finite sum of many function values of f, multiplied with the differences of two subsequent approximation points. We can easily see that the approximation is still too large. Using more steps produces a closer approximation, but will never be exact: replacing the 5 subintervals by twelve as depicted, we will get an approximate value for the area of 0.6203, which is too small. The key idea is the transition from adding a finite number of differences of approximation points multiplied by their respective function values to using an infinite number of fine, or infinitesimal, steps.
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, f(x) = x^{1/2}, the theorem says to look at the antiderivative, F(x) = (2/3)x3/2, and simply take F(1) − F(0), where 0 and 1 are the boundaries of the interval [0,1].
So the exact value of the area under the curve is computed formally as
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Key Term Reference
 approximation
 Appears in this related concepts: Numerical Integration, Roundoff Error, and Uniform Electric Field
 area
 Appears in this related concepts: Approximate Integration, Double Integrals Over Rectangles, and Area Between Curves
 curve
 Appears in this related concepts: Arc Length and Surface Area, Calculus with Parametric Curves, and Arc Length and Speed
 differentiation
 Appears in this related concepts: The Natural Exponential Function: Differentiation and Integration, Caulobacter Differentiation, and Considering the Environment
 function
 Appears in this related concepts: Limit of a Function, Average Value of a Function, and Modal Mixture
 graph
 Appears in this related concepts: Graphing on Computers and Calculators, Graphing Equations, and Equations and Their Solutions
 infinitesimal
 Appears in this related concepts: Differentials, Area and Distances, and The Fundamental Theorem of Calculus
 integral
 Appears in this related concepts: The Correction Factor, Integration By Parts, and Integration Using Tables and Computers
 inverse
 Appears in this related concepts: Inverse Functions, Hyperbolic Functions, and The Law of Universal Gravitation
 root
 Appears in this related concepts: Newton's Method, Radical Functions, and Radical Equations
 variable
 Appears in this related concepts: Related Rates, Calculating the NPV, and Fundamentals of Statistics
 volume
 Appears in this related concepts: Physical and Chemical Properties of Matter, Volume and Density, and Shape and Volume
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Cite This Source
Source: Boundless. “The Definite Integral.” Boundless Calculus. Boundless, 02 Jul. 2014. Retrieved 25 Mar. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/derivativesandintegrals2/integrals11/thedefiniteintegral652945/