Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School...
Transcript of Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School...
![Page 1: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/1.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Numerical Integration Numerical Integration FormulasFormulas
Prof. Prof. HaeHae--Jin ChoiJin Choi
[email protected]@cau.ac.kr
Part 5Part 5Chapter 17Chapter 17
1
![Page 2: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/2.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Chapter ObjectivesChapter Objectives
2
l Recognizing that Newton-Cotes integration formulas are based on the strategy of replacing a complicated function or tabulated data with a polynomial that is easy to integrate.
l Knowing how to implement the following single application Newton-Cotes formulas:§ Trapezoidal rule§ Simpson’s 1/3 rule§ Simpson’s 3/8 rule
l Knowing how to implement the following composite Newton-Cotes formulas:§ Trapezoidal rule§ Simpson’s 3/8 rule
l Recognizing that even-segment-odd-point formulas like Simpson’s 1/3 rule achieve higher than expected accuracy.
l Knowing how to use the trapezoidal rule to integrate unequally spaced data.
l Understanding the difference between open and closed integration formulas.
![Page 3: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/3.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
IntegrationIntegration
3
l Integration:
is the total value, or summation, of f(x) dx over the range from a to b:
l I represents the area under the curve f(x) between x= a and b.
�
I = f x( )a
bò dx
![Page 4: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/4.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Integration in Engineering and ScienceIntegration in Engineering and Science
4
l This integral can be evaluated over a line, an area, or a volume.
l For example the total mass of gas contained in a volume is given as the product of the density and the volume. However, suppose that the density varies from location to location within a volume, it is necessary to sum the product
l For a continuous case, the integration is expressed by
l There is strong analogy between summation and integration
à Basis of numerical integration
1
n
i ii
mass Vr=
= Då
( , , )mass x y z dxdydzr= òòò ( )V
mass V dVr= òòò
![Page 5: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/5.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
NewtonNewton--Cotes FormulasCotes Formulas
5
l The Newton-Cotes formulas are the most common numerical integration schemes.
l Generally, they are based on replacing a complicated function or tabulated data with a polynomial that is easy to integrate:
where fn(x) is an nth order interpolating polynomial.
�
I = f x( )a
bò dx @ fn x( )a
bò dx
nn
nnn xaxaxaaxf ++++= -
-1
110)( L
First-orderpolynomial(line)
Parabola
![Page 6: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/6.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
The Trapezoidal RuleThe Trapezoidal Rule
6
l The integral can be approximated using a series of polynomials applied piecewise to the function or data over segments of constant length.
l For example, three straight line segments are used to approximate the integral. Higher-order polynomial can be used for the same purpose.
![Page 7: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/7.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
The Trapezoidal RuleThe Trapezoidal Rule
7
l The trapezoidal rule is the first of the Newton-Cotes closed integration formulas; it uses a straight-line approximation for the function:
( )( ) ( ) ( )
( ) ( ) ( )
( )
( )(average height)2
b
na
b
a
I f x dx
f b f aI f a x a dx
b a
f a f bI b a I b a
=
é ù-= + -ê ú-ë û
+= - ® = -
ò
ò
![Page 8: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/8.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Error of the Trapezoidal RuleError of the Trapezoidal Rule
8
l An estimate for the local truncation error of a single application of the trapezoidal rule is:
where x is somewhere between a and b.
l This formula indicates that the error is dependent upon the curvature of the actual function as well as the distance between the points.
l Error can thus be reduced by breaking the curve into parts.
�
Et = -1
12¢ ¢ f x( ) b - a( )3
![Page 9: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/9.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Example 17.1 (1/2)Example 17.1 (1/2)
9
l Q. Use the trapezoidal rule to numerically integrate the following equation from a = 0 to b = 0.8. The true solution is 1.640533.
2 3 4 5( ) 0.2 2.5 200 675 900 400f x x x x x x= + - + - +
Sol.)
® Et = 1.640533 – 0.1728 = 1.467733 ® et = 89.5%
1728.02
232.02.0)08.0( =+
-=I
![Page 10: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/10.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Example 17.1 (2/2)Example 17.1 (2/2)
10
Approximate error:32 000,8800,10050,4400)( xxxxf +-+-=¢¢
6008.0
)000,8800,10050,4400()(
8.0
032
-=-
+-+-=¢¢ ò dxxxx
xf
56.2)8.0)(60(121 3 =--=aE
note : this value is of the same order of magnitude and sign as the true error. Average second derivative is not an accurate approximation of f’’(ξ), so a discrepancy existsand Ea rather Et.
![Page 11: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/11.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
fnabEa ¢¢-
-= 2
3
12)(
Composite Trapezoidal RuleComposite Trapezoidal Rule
11
l One way to improve the accuracy is to divide the integration interval from a to b into a number of segments and apply the method to each segment.
l Assuming n+1 data points are evenly spaced, there will be n intervals over which to integrate.
l The total integral can be calculated by integrating each subinterval and then adding them together:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
1 2
0 0 1 1
0 1 1 2 11 0 2 1 1
1
011
01 Width
Average height
2 2 2
( ) 2 ( ) ( )2 ( )
2 2
n n
n
x x x x
n n n nx x x x
n nn n
n
i nni
i ni
I f x dx f x dx f x dx f x dx
f x f x f x f x f x f xI x x x x x x
f x f x f xhI f x f x f x I b a
n
-
--
-
-=
=
= = + + +
+ + += - + - + + -
+ +é ù
= + + ® = -ê úë û
ò ò ò ò
åå
L
L
123 14444244443
å=
¢¢--=
n
iit ξf
nabE
13
3
)(12
)(
fnξfn
ii ¢¢@¢¢å
=1)(
n
ξff
n
iiå
=
¢¢@¢¢ 1
)(
If the number of segments is doubled, the error will be quartered
If the number of segments is doubled, the error will be quartered
![Page 12: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/12.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
MATLAB ProgramMATLAB Program
12
![Page 13: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/13.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Example 17.2 (1/2)Example 17.2 (1/2)
13
l Q. Use the two-segment and composite trapezoidal rule to estimate the integral of the function from a = 0 to b = 0.8. The exact value is 1.640533.
2 3 4 5( ) 0.2 2.5 200 675 900 400f x x x x x x= + - + - +
232.0)8.0( 456.2)4.0( 2.0)0( === fff
0688.14
232.0)456.2(22.08.0 =++
=I
%9.34 57173.00688.1640533.1 ==-= tt εE
64.0)60()2(12
8.02
2=--=aE
Sol.For n = 2 (h = 0.4)
![Page 14: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/14.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Example 17.2 (2/2)Example 17.2 (2/2)
14
<Results for the composite trapezoidal rule to estimate the integral of f(x) = 0.2 + 25x – 200x2 + 675x3 – 900x4
from x = 0 to 0.8. The exact value is 1.640533>à As the number of segments increases, the error decreases.
n h I et (%)23456789
10
0.40.2667
0.20.16
0.13330.1143
0.10.0889
0.08
1.06881.36951.48481.53991.57031.58871.60081.60911.6150
34.916.59.56.14.33.22.41.91.6
![Page 15: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/15.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Simpson’s RulesSimpson’s Rules
15
l One drawback of the trapezoidal rule is that the error is related to the second derivative of the function.
l More complicated approximation formulas can improve the accuracy for curves - these include using (a) 2nd and (b) 3rd order polynomials.
l The formulas that result from taking the integrals under these polynomials are called Simpson’s rules.
![Page 16: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/16.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Simpson’s 1/3 RuleSimpson’s 1/3 Rule
16
l Simpson’s 1/3 rule corresponds to using second-order polynomials. Using the Lagrange form for a quadratic fit of three points:
l Integration over the three points simplifies to:
( )
( ) ( ) ( )
2
0
0 1 20 1 2
( ) 4 ( ) ( )4 ( )3 6
x
nxI f x dx
f x f x f xhI f x f x f x I b a
=
+ +é ù= + + = = -ë û
ò�
fn x( )=x - x1( )x0 - x1( )
x - x2( )x0 - x2( )
f x0( )+x - x0( )x1 - x0( )
x - x2( )x1 - x2( )
f x1( )+x - x0( )x2 - x0( )
x - x1( )x2 - x1( )
f x2( )
Where, h = (b – a)/2, a = x0, b = x2, and x1 = (a + b)/2
![Page 17: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/17.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Error of Simpson’s 1/3 RuleError of Simpson’s 1/3 Rule
17
l An estimate for the local truncation error of a single application of Simpson’s 1/3 rule is:
where again x is somewhere between a and b.l This formula indicates that the error is dependent upon the fourth-
derivative of the actual function as well as the distance between the points.
l Note that the error is dependent on the fifth power of the step size (rather than the third for the trapezoidal rule).
l Error can thus be reduced by breaking the curve into parts.
�
Et = -1
2880f 4( ) x( ) b - a( )5
![Page 18: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/18.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Example 17.3Example 17.3
18
l Q. Use Simpson 1/3 rule to integrate the following equation from a = 0 to b = 0.8. Exact solution is 1.640533.
2 3 4 5( ) 0.2 2.5 200 675 900 400f x x x x x x= + - + - +
sol) for n = 2 (h = 0.4)에 대해서
® This shows improved results compared to the trapezoidal rule.
232.0)8.0( 456.2)4.0( 2.0)0( === fff
367467.16
232.0)456.2(42.08.0 =++
=I
%6.16 2730667.0367467.1640533.1 ==-= tt εE
2730667.0)2400(2880
8.0 5
=--=aE
![Page 19: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/19.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Composite Simpson’s 1/3 RuleComposite Simpson’s 1/3 Rule
19
l For improved results, Simpson’s 1/3 rule can be used on a set of subintervals in much the same way the trapezoidal rule was, except there must be an odd number of points..
l Because of the heavy weighting of the internal points, the formula is a little more complicated than for the trapezoidal rule:
�
I = fn x( )x0
xnò dx = fn x( )x0
x2ò dx + fn x( )x2
x4ò dx +L+ fn x( )xn-2
xnò dx
I =h3
f x0( )+ 4 f x1( )+ f x2( )[ ]+h3
f x2( )+ 4 f x3( )+ f x4( )[ ]+L+h3
f xn-2( )+ 4 f xn-1( )+ f xn( )[ ]
I =h3
f x0( )+ 4 f xi( )i=1i, odd
n-1
å + 2 f xi( )j=2j , even
n-2
å + f xn( )é
ë
ê ê ê
ù
û
ú ú ú
)4(4
5
180)( f
nabEa
--=
![Page 20: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/20.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Example 17.4Example 17.4
20
l Q. Use composite Simpson 1/3 rule to integrate the following equation from a = 0 to b = 0.8. Exact solution is 1.640533.
2 3 4 5( ) 0.2 2.5 200 675 900 400f x x x x x x= + - + - +
Sol.) for n = 4 (h = 0.2)
232.0)8.0(464.3)6.0( 456.2)4.0(288.1)2.0( 2.0)0(
=====
fffff
623467.112
232.0)456.2(2)464.3288.1(42.08.0 =++++
=I
%04.1 017067.0623467.1640533.1 ==-= tt εE
017067.0)2400()4(180
8.04
5
=--=aEEstimated error:
![Page 21: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/21.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Example 17.4Example 17.4
21
l The composite version of Simpson 1/3 rule is superior to the trapezoidal rule for most applications.
l It is limited to cases where the values are equispaced.,even number of segments, and odd number of points.
l Odd segment and even point formula is known as Simpson 3/8 formula.
![Page 22: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/22.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2 22
Simpson’s 3/8 RuleSimpson’s 3/8 Rulel Simpson’s 3/8 rule corresponds to
using third-order polynomials to fit four points. Integration over the four points simplifies to:
l Simpson’s 3/8 rule is generally used in concert with Simpson’s 1/3 rule when the number of segments is odd.
�
I = fn x( )x0
x3ò dx
I =3h8
f x0( )+ 3 f x1( )+ 3 f x2( )+ f x3( )[ ]
![Page 23: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/23.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
HigherHigher--Order FormulasOrder Formulas
23
l Higher-order Newton-Cotes formulas may also be used - in general, the higher the order of the polynomial used, the higher the derivative of the function in the error estimate and the higher the power of the step size.
l As in Simpson’s 1/3 and 3/8 rule, the even-segment-odd-point formulas have truncation errors that are the same order as formulas adding one more point. For this reason, the even-segment-odd-point formulas are usually the methods of preference.
![Page 24: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/24.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Example 17.5 (1/2)Example 17.5 (1/2)
24
l Q. (a) Use Simpson 3/8 to integrate from a = 0 to b = 0.8. (b) Use it in conjunction with Simpson 1/3 for five segment integration.
2 3 4 5( ) 0.2 2.5 200 675 900 400f x x x x x x= + - + - +
Sol.) (a) For n = 3 (h = 0.2667)
232.0)8.0( 487177.3)5333.0(432724.1)2667.0( 2.0)0(
====
ffff
51970.18
232.0)487177.3432724.1(32.08.0 =+++
=I
(b) For n = 5 (h = 0.16)
232.0)80.0( 181929.3)64.0(186015.3)48.0( 743393.1)32.0(296919.1)16.0( 2.0)0(
======
ffffff
![Page 25: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/25.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Example 17.5 (2/2)Example 17.5 (2/2)
25
The integral for the first two segments using Simpson 1/3
3803237.06
743393.1)296919.1(42.032.0 =++
=I
For the last three segments, the Simpson 3/8
264754.18
232.0)181929.3186015.3(3743393.148.0 =+++
=I
The total integral is by summing the two results.
645077.1264754.13803237.0 =+=I
![Page 26: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/26.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
SummarySummary
26
n points name 공 식 절단오차
1 2 Trapezoidal rull
2 3 Simpson 1/3
3 4 Simpson 3/8
4 5 Boole’s rull
5 6
<Newton-Cotes integration: h = (b – a)/n>
2)()(
)( 10 xfxfab
+- ( ) )(12/1 3 x¢¢- fh
6)()(4)(
)( 210 xfxfxfab
++- ( ) )(90/1 )4(5 x- fh
8)()(3)(3)(
)( 3210 xfxfxfxfab
+++- ( ) )(80/3 )4(5 x- fh
90)(7)(32)(12)(32)(7
)( 43210 xfxfxfxfxfab ++++- ( ) )(945/8 )6(7 x- fh
288)(19)(75)(50)(50)(75)(19)( 543210 xfxfxfxfxfxfab +++++
- ( ) )(096,12/275 )6(7 x- fh
![Page 27: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/27.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Integration with Unequal SegmentsIntegration with Unequal Segments
27
l Previous formulas were simplified based on equispaced data points - though this is not always the case.
l The trapezoidal rule may be used with data containing unequal segments:
�
I = fn x( )x0
xnò dx = fn x( )x0
x1ò dx + fn x( )x1
x2ò dx +L+ fn x( )xn-1
xnò dx
I = x1 - x0( ) f x0( )+ f x1( )2
+ x2 - x1( ) f x1( )+ f x2( )2
+L+ xn - xn-1( ) f xn-1( )+ f xn( )2
![Page 28: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/28.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Integration Code for Unequal Integration Code for Unequal SegmentsSegments
28
![Page 29: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/29.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
MATLAB Functions (option)MATLAB Functions (option)
29
l MATLAB has built-in functions to evaluate integrals based on the trapezoidal rule
l z = trapz(y)z = trapz(x, y)produces the integral of y with respect to x. If x is omitted, the program assumes h=1.
>> x = [0 .12 .22 .32 .36 .4 .44 .54 .64 .7 .8];
>> y = 0.2 + 25*x - 200*x.^2 + 675*x.^3 - 900*x.^4 + 400*x.^5;
>> trapz(x,y)
ans =
1.5948
![Page 30: Part 5 Chapter 17 - CAUisdl.cau.ac.kr/.../numerical.analysis/Lecture15.pdfPart 5 Chapter 17 1 School of Mechanical Engineering Chung-AngUniversity Numerical Methods 2010-2 Chapter](https://reader034.fdocuments.in/reader034/viewer/2022042313/5edc976cad6a402d66675243/html5/thumbnails/30.jpg)
School of Mechanical EngineeringChung-Ang UniversityNumerical Methods 2010-2
Multiple IntegralsMultiple Integrals
30
• Multiple integrals can be determined numerically by first integrating in one dimension, then a second, and so on for all dimensions of the problem.