Download - CHAPTER 7c. DIFFERENTIATION AND INTEGRATION ' Assakkaf Slide No. 75 Ł A. J. Clark School of Engineering Ł Department of Civil and Environmental Engineering ENCE 203 Œ CHAPTER 7c

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� A. J. Clark School of Engineering �Department of Civil and Environmental Engineering

by

Dr. Ibrahim A. AssakkafSpring 2001

ENCE 203 - Computation Methods in Civil Engineering IIDepartment of Civil and Environmental Engineering

University of Maryland, College Park

CHAPTER 7c. DIFFERENTIATION AND INTEGRATION

© Assakkaf

Slide No. 74

� A. J. Clark School of Engineering � Department of Civil and Environmental Engineering

ENCE 203 � CHAPTER 7c. DIFFERENTIATION AND INTEGRATION

Numerical Integration

� In calculus, integration is used to find the area under the curve.

� In engineering applications, the area under the curve can have physical interpretation and implications.

� For example, it can mean finding the total energy or rate of flow Q through a cross section of a river.

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� Area Under the Curvey = f(x)

xxa xb

( )∫=b

a

x

x

dxxf area

y = f(x)

xxa xb

( )∫=b

a

x

x

dxxf area

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� Examples

( )( ) ( )

cxex

dxxexdxxfydx

xexxfy

x

x

x

+−−=

+−==⇒

+−==

∫∫ ∫cos

4

sin

sin

4

3

3

( ) ( )21

21

1)(1

0

21

0

1

0

=

+=+==

+==

∫∫xxdxxdxxfI

xxfy

3

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� Need for Numerical IntegrationIn general, the function to be integrated will

typically be in one of the following forms:1. A simple continuous linear function such as a

polynomial, an exponential, or trigonometric function, such as

( )( )( ) xxf

exfxxxf

x

sin321

20232

5

+=−=

+−=

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� Need for Numerical Integration2. A complex non-linear continuous function that

is difficult or impossible to integrate directly such as

( )

( ) ( )

( )x

xxxx

exf

exxx

xxf

xxxxf

x

x

cosln3

tan1cos1

sin1

2

4

23

−++

=

−+−=

+++=

4

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� Need for Numerical Integration3. A tabulated continuous function where values

of the independent variable x and f(x) are given at a number of discrete data points as is often the case with experimental or field data such as distance traveled by a car vs. time:

61033015050100D (ft)

1086420t (sec)

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� Need for Numerical Integration� In the first case, the integral of a simple

function may be computed analytically using calculus.

� For the second case, analytical solutions are often impractical and sometimes difficult or impossible to obtain.

� In these situations as well as in the third case, approximate methods must be used.

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� Need for Numerical Integration� Pre computers and computational devices,

a visually oriented approach were used to integrate tabulated data and complicated functions.

� In this approach, the function is plotted on a grid (see figure), and the number of boxes that approximate the area are counted.

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� Need for Numerical Integration� This number is multiplied by the area of

each box to give a rough estimate of the total area under the curve.

� This estimate can be refined at the expense of additional effort by using a finer grid.

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� Need for Numerical Integration

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12x

f (x )

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� Need for Numerical Integration

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12x

f (x )

7

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� Engineering Applications� A surveyor might need to know the area of a

piece of land bounded by a meandering stream and two roads

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� Engineering Applications� A water-resource engineer might need to know

the cross-sectional area of a river to calculate the rate of flow Q.

VAVdAQ == ∫

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� Engineering Applications� A structural engineer might need to determine

the net lateral force due to non-uniform wind blowing against a side of a tall building.

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� Integration Using Interpolating Polynomial

� The general form of an interpolating polynomial is given by

� This polynomial can be integrated analytically as follows:

( ) nn xbxbxbxbbxf +++++= L3

32

210

( )132

11

32

21

0 ++++++=

+−∫ n

xbn

xbxbxbxbdxxfn

nn

nK

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� Integration Using Interpolating Polynomial� The Gregory-Newton method for deriving

an interpolation formula can also be used to evaluate the integral of a function.

� Recall G-N method:

( ) ( ) ( )( ) ( )( )( )( )( ) ( ) ( )( ) ( )nnnn xxxxxxaxxxxxxa

xxxxxxaxxxxaxxaaxf−−−+−−−+

−−−+−−+−+=

+− KK 211121

3214213121

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� Integration Using Interpolating Polynomial

� The terms could be rearrange to form an nth-order polynomial of the type

� This polynomial can be integrated analytically as

( ) 11

21 +− +++= n

nn bxbxbxf L

( ) xbxnbx

nbdxxf n

nn1

211

1 ++ +++

+=∫ L

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� Example 1Derive a second-degree interpolation polynomial to fit the following data points, and then using the fitted polynomial to approximate

Compare your result with that of the exact integral (i.e., x4/4).

2781f(x)321x

( )∫5.2

5.1

dxxf

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� Example 1 (cont�d)� The general form of the interpolation

polynomial is given by

� In our case it is

( ) nn xbxbxbxbbxf +++++= L3

32

210

( ) 2210 xbxbbxf ++=

11

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� Example 1 (cont�d)� We need to find the coefficients b0, b1, and

b2.� We notice that have three unknowns, n = 3

, that require solving 3 simultaneous linear equations using the pair, xi and f(xi), of the given data as follows:

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� Example 1 (cont�d)

( ) ( )( ) ( )

( ) ( )2210

2210

2210

3327

228

111

bbb

bbb

bbb

++=

++=

++=

( ) 2210 xbxbbxf ++=

2781f(x)321x

12

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� Example 1 (cont�d)

� OR

=

2781

931421111

2

1

0

bbb

27938421

210

210

210

=++=++=++

bbbbbbbbb

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� Example 1 (cont�d)� The solution of this set of equations yields

� Therefore, the interpolation polynomial is

� And its anti-derivative (integral) is

−=

6116

2

1

0

bbb

( ) 2210 xbxbbxf ++=

( ) 26116 xxxf +−=

( ) cxxxdxxf ++−=∫ 32

36

2116

13

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� Example 1 (cont�d)The result is

( )

( ) ( ) ( ) ( ) ( ) ( )

5.8 375.3875.11

5.125.12

115.165.225.22

115.26

22

116

3232

5.2

5.1

325.2

5.1

=−=

+−−

+−=

+−=∫ xxxdxxf

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� Example 1 (cont�d)� Evaluation of the exact integral is as

follows:

� In this example, the approximation is identical to the true value.

( ) ( ) ( ) 5.84

5.15.24

445.2

5.1

45.2

5.1

35.2

5.1

=−=== ∫∫xxdxxf

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� The Trapezoidal Rule� The trapezoidal rule approximates the area

of a function defined by a set of discrete points by fitting a trapezoid to each pair of adjacent points that defines the dependent variable and summing the individual areas as shown in the figure.

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� The Trapezoidal Rule

a b

x

f(x)

x4x3x2x1

( )∫=b

a

dxxfI

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� The Trapezoidal Rule� Derivation

� The trapezoidal rule can be derived by fitting a linear interpolating polynomial to each pair of points.

� Using, for example Gregory-Newton formula, an expression for a linear polynomial can be obtained as follows:

( ) ( ) ( )( ) ( )( )( )( )( ) ( ) ( )( ) ( )nnnn xxxxxxaxxxxxxa

xxxxxxaxxxxaxxaaxf−−−+−−−+

−−−+−−+−+=

+− KK 211121

3214213121

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� If we denote the values of the two independent variables as xi and xi+1, then G-N formula gives

( ) ( )( ) ( )( ) 1

21

21

axfxxaaxf

xxaaxf

i

iii

i

=−+=

−+=

f(xi+1)xi+1

f(xi)xi

f(x)x

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( ) ( )( ) ( ) ( ) ( )

( ) ( )ii

ii

iiiiii

i

xxxfxfa

xxaxfxxaaxfxxaaxf

−−=

−+=−+=−+=

+

+

+++

1

12

121211

21

Therefore,

f(xi+1)xi+1

f(xi)xi

f(x)x

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� Hence, the linear polynomial is given by

( ) ( )

( ) ( ) ( ) ( )( )iii

iii

i

xxxx

xfxfxfxf

xxaaxf

−−−+=

−+=

+

+

1

1

21

f(xi+1)xi+1

f(xi)xi

f(x)x

(1)

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� Integrating Eq. 1 between two points, say a and b:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )2

2

2

2

1

1

2

1

1

2

1

1

i

ii

iii

i

ii

iii

b

a

i

ii

iii

b

a

xaxx

xfxfaxf

xbxx

xfxfbxf

xxxx

xfxfxxfdxxf

−−−−−

−−−+=

−−−+=

+

+

+

+

+

+∫

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Or

Letting b = xi+1 and a = xi, results in

( ) ( ) ( ) ( )( ) ( ) ( )

−−−+

−−= +++

+∫ 111

1 2 iiiiiiii

b

a

xfxxfxxfxfabxx

abdxxf

( ) ( ) ( )[ ]iiii

x

x

xfxfxxdxxfi

i

+−= ++∫

+

11

2

1

18

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� The Trapezoidal RuleThe trapezoidal rule can be used to approximate the integral between two points x1 and xn of a function. It is given by

( ) ( ) ( ) ( )∑∫−

=

++

+−≈1

1

11 2

1

n

i

iiii

x

x

xfxfxxdxxfn

(2)

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� Geometric Interpretation the Trapezoidal Rule� The trapezoidal rule can also be derived

geometrically.� The trapezoidal rule is equivalent to

approximating the area of the trapezoid under the straight line connecting f(xi) andf(xi+1) as shown in the following figure:

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� Geometric Interpretation the Trapezoidal Rule

x

f(x)f(xi+1)

f(xi)

xi xi+1

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� Geometric Interpretation the Trapezoidal Rule� Recall from geometry that the formula for

computing the area of a trapezoid is the height times the average of the bases as shown in the figure.

� Therefore, the integral estimate can be represented as

height average width ×≈I

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� Geometric Interpretation the Trapezoidal Rule

Height

Base

Base

HeightHeight

Width

( ) ( ) ( )[ ]iiii

x

x

xfxfxxdxxfi

i

+−= ++∫

+

11

2

1

height average width ×≈I

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� Example 2Using the trapezoidal rule , estimate the area under the curve, that is

for the following function given in tabulated form:

3-3-13f(xi)4321xi

4321i

( )∫=4

1

dxxfI

21

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� Example 2 (cont�d)Using Eq. 2,

( ) ( ) ( ) ( )∑∫−

=

++

+−≈1

1

11 2

1

n

i

iiii

x

x

xfxfxxdxxfn

( ) ( )[ ] ( )[ ] ( )[ ]

1 0(-2)1

2)3(334

2)3(123

2)1(312

4

1

−=++=

−+−+−+−−+−+−≈∫ dxxf

3-3-13f(xi)4321xi

4321i

3-3-13f(xi)4321xi

4321i

Download - CHAPTER 7c. DIFFERENTIATION AND INTEGRATION ' Assakkaf Slide No. 75 Ł A. J. Clark School of Engineering Ł Department of Civil and Environmental Engineering ENCE 203 Œ CHAPTER 7c

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