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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Numerical Integration
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Objectives
• The student should be able to– Understand the need for numerical integration– Derive the trapezoidal rule using geometric
insight– Apply the trapezoidal rule– Apply Simpson’s rule
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Need for Numerical Integration!
6
1101
2
1
3
1
231
1
0
231
0
2
x
xxdxxxI
11
0
1
0
1 eedxeI xx
1
0
2
dxeI x
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Area under the graph!
• Definite integrations always result in the area under the graph (in x-y plane)
• Are we capable of evaluating an approximate value for the area?
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Example
• To perform the definite integration of the function between (x0 & x1), we may assume that the area is equal to that of the trapezium:
0101
2
1
0
xxyy
dxxfx
x
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Adding adjacent areas
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
The Trapezoidal Rule
2
2
1212
0101
yyxx
yyxxI
Integrating from x0 to x2:
2
212112101001 yxxyxxyxxyxxI
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
The Trapezoidal Rule
hxxxx 1201
If the points are equidistant
22110 hyhyhyhy
I
210 22
yyyh
I
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Dividing the whole interval into “n” subintervals
n
n
ii yyy
hI
1
10 2
2
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
The Algorithm
• To integrate f(x) from a to b, determine the number of intervals “n”
• Calculate the interval length h=(b-a)/n• Evaluate the function at the points yi=f(xi)
where xi=x0+i*h• Evaluate the integral by performing the
summation
n
n
ii yyy
hI
1
10 2
2
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Note that
X0=a
Xn=b
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Example
• Integrate• Using the trapezoidal
rule• Use 2,3,&4 points and
compare the results
1
0
2dxxI
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Solution
• Using 2 points (n=1), h=(1-0)/(1)=1
• Substituting:
212
1yyI 5.010
2
1I
XY
00
11
2 points, 1 interval
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Solution
• Using 3 points (n=2), h=(1-0)/(2)=0.5
• Substituting:
321 22
5.0yyyI
375.0125.0*202
5.0I
XY
00
0.50.25
11
3 points, 2 interval
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Solution
• Using 4 points (n=3), h=(1-0)/(3)=0.333
• Substituting:
4321 222
333.0yyyyI
3519.01444.0*2111.0*202
333.0I
XY
00
0.330.111
0.6670.444
11
4 points, 3 interval
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Let’s use Interpolation!
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Interpolation!
• If we have a function that needs to be integrated between two points
• We may use an approximate form of the function to integrate!
• Polynomials are always integrable• Why don’t we use a polynomial to
approximate the function, then evaluate the integral
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Example
• To perform the definite integration of the function between (x0 & x1), we may interpolate the function between the two points as a line.
001
010 xx
xx
yyyxf
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Example
• Performing the integration on the approximate function:
1
0
1
0
001
010
x
x
x
x
dxxxxx
yyydxxfI
1
0
0
2
01
010 2
x
x
xxx
xx
yyxyI
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Example
• Performing the integration on the approximate function:
00
20
01
010010
21
01
0110 22
xxx
xx
yyxyxx
x
xx
yyxyI
2
0101
yyxxI
• Which is equivalent to the area of the trapezium!
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
The Trapezoidal Rule
2
0101
yyxxI
2
2
1212
0101
yyxx
yyxxI
Integrating from x0 to x2:
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Simpson’s Rule
Using a parabola to join three adjacent points!
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Quadratic Interpolation
• If we get to interpolate a quadratic equation between every neighboring 3 points, we may use Newton’s interpolation formula:
103021 xxxxbxxbbxf
10102
3021 xxxxxxbxxbbxf
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Integrating
10102
3021 xxxxxxbxxbbxf
2
0
2
0
10102
3021
x
x
x
x
dxxxxxxxbxxbbdxxf
2
0
2
0
10
2
10
3
30
2
21 232
x
x
x
x
xxxx
xxx
bxxx
bxbdxxf
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
After substitutions and manipulation!
210 43
2
0
yyyh
dxxfx
x
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Working with three points!
210 43
2
0
yyyh
dxxfx
x
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
For 4-Intervals
432210 443
4
0
yyyyyyh
dxxfx
x
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
In General: Simpson’s Rule
n
n
ii
n
ii
x
x
yyyyh
dxxfn 2
,..4,2
1
,..3,10 24
30
NOTE: the number of intervals HAS TO BE even
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Example
• Integrate• Using the Simpson
rule• Use 3 points
1
0
2dxxI
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Solution
• Using 3 points (n=2), h=(1-0)/(2)=0.5
• Substituting:
• Which is the exact solution!
210 43
5.0yyyI
3
1125.0*40
3
5.0I
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ENEM602 Spring 2007Dr. Eng. Mohammad Tawfik
Homework #7
• Chapter 21, p. 610, numbers:21.5, 21.6, 21.10, 21.11.