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Copyright 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
NumericalDifferentiation and
IntegrationIntroduction
Newton-Cotes Integration FormulasIntegration of EquationsNumerical Differentiation
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Copyright 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Introduction
Calculus is the mathematics of change. Because engineersmust continuously deal with systems and processes thatchange, calculus is an essential tool of engineering.
Standing in the heart of calculus are the mathematicalconcepts of differentiation and integration :
b
a
ii x
ii
dx x f I
x x f x x f
dxdy
x
x f x x f
x
y
)(
)()(lim
)()(
0
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NumericalDifferentiation
and Integration
Newton-CotesIntegrationFormulas
Integration ofEquations
NumericalDifferentiation
High-accuracyformulas
Richardsonextrapolation
Unequal-spaced data
GaussQuadrature
Trapezoidal ruleSimpsons rules
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Graphical definition of a derivative
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Graphical representation of the integral
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Noncomputer Methods forDifferentiation and Integration
The function to be differentiated or integrated willtypically be in one of the following three forms: A simple continuous function such as polynomial, an
exponential, or a trigonometric function. A complicated continuous function that is difficult or
impossible to differentiate or integrate directly.
A tabulated function where values of x and f(x) are given ata number of discrete points, as is often the case withexperimental or field data.
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Newton-Cotes IntegrationFormulas
The Newton-Cotes formulas are the most commonnumerical integration schemes.
They are based on the strategy of replacing acomplicated function or tabulated data with anapproximating function that is easy to integrate:
n is order of polynomial.
nn
nnn
b
an
b
a
xa xa xaa x f
dx x f dx x f I
1110)(
)()(
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Figure 21.1
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Figure 21.2 integral can be approximated using a seriesof polynomials applied piecewise to the function or dataover segments of constant length.
3 straightlinesegmentsare used
8/13/2019 Chapter 6 Enotes
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The Trapezoidal Rule
The Trapezoidal rule is the first of the Newton-Cotes closed integration formulas, correspondingto the case where the polynomial is first order :
The area under this first order polynomial is an
estimate of the integral of f(x) between the limitsof a and b :>>>equation 21.3
b
a
b
adx x f dx x f I )()( 1
Derivation of Trapezoidal Rule >> Box 21.1
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I width x average height
Try out Problem 21.10 for first 2 data
Try out Example 21.1
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The Multiple Application Trapezoidal Rule One way to improve the accuracy of the
trapezoidal rule is to divide the integrationinterval from a to b into n number of segmentsand apply the method to each segment.
The areas of individual segments can then beadded to yield the integral for the entire interval.
General form:>>>equation 21.10
n
n
x
x
x
x
x
x
n
dx x f dx x f dx x f I
xb xan
abh
1
2
1
1
0
)()()(
0
8/13/2019 Chapter 6 Enotes
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General format for multiple-application integrals
Try out Problem 21.10 a
Try out Example 21.2
8/13/2019 Chapter 6 Enotes
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Simpsons Rules
More accurate estimate of an integral is obtained if ahigh-order polynomial is used to connect the points.The formulas that result from taking the integrals
under such polynomials are called Simpsons rules .
Simpsons 1/3 Rule
Results when a second-order Lagrange interpolating polynomial is used.
>>>equation 21.15
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Simpsons 1/3 rule (parabola)
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The Multiple- Application Simpsons 1/3 Rule
Just as the trapezoidal rule, Simpsons rulecan be improved by dividing the integrationinterval into a number of segments of equalwidth.
Yields accurate results and considered superiorto trapezoidal rule for most applications.
Can be employed only if the number ofsegments is even>>>equation 21.18
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8/13/2019 Chapter 6 Enotes
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Simpsons 3/8 Rule Results when a third-order Lagrange
interpolating polynomial is used.>>>equation 21.20>>>n? data points?
Estimate five segments?? Use trapezoidal rule (large truncation error)
Alternative : apply Simpsons 1/3 rule to the
first 2 segments & Simpsons 3/8 rule to thelast three.
Try out Problem 21.10 & Problem 21.11
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Simpsons 3/8 rule (cubic)
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8/13/2019 Chapter 6 Enotes
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Integration of Equations Functions to be integrated
numerically are in two forms: A table of values . We are limited by the
number of points that are given. A function . We can generate as many
values of f(x) as needed to attain
acceptable accuracy.
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Gauss Quadrature Gauss quadrature implements a
strategy of positioning any twopoints on a curve to define a straightline that would balance the positiveand negative errors.
Hence the area evaluated under thisstraight line provides an improvedestimate of the integral.
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Two-point formula General form
>>>equation 22.12
Yield to>>>equation 22.17
Equation 22.23 & 22.24 can be substitutedfor x & dx in the equation to be integrated
Substitution transform the integrationinterval
Try out Problem 22.8
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Figure 22.7
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Higher-point formula General form
>>>equation 22.25
>>>refer Table 22.1
Try out Problem 22.8
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Numerical Differentiation
Notion of numerical differentiationhas been introduced in Chapter 1. Inthis chapter more accurate formulasthat retain more terms will bedeveloped.
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High Accuracy DifferentiationFormulas
High-accuracy divided-differenceformulas can be generated by includingadditional terms from the Taylor series
expansion.>>>Figure 23.1 to 23.3
Try out Problem 23.3
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Richardson Extrapolation There are two ways to improve derivative
estimates when employing finite divideddifferences: Decrease the step size, or Use a higher-order formula that employs more
points.
A third approach, based on Richardsonextrapolation, uses two derivativeestimates to compute a third, moreaccurate approximation.>>>equation 23.8
Try out Problem 23.4
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Derivatives of Unequally Spaced Data
Data from experiments or field studiesare often collected at unequal intervals.One way to handle such data is to fit a
second-order Lagrange interpolatingpolynomial.>>>equation 23.9
Try out Problem 23.26