Upload
mohammad-tawfik
View
1.270
Download
0
Embed Size (px)
DESCRIPTION
Citation preview
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Numerical Integration
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Objectives
• The student should be able to– Understand the need for numerical integration– Derive the trapezoidal rule using linear
interpolation– Apply the trapezoidal rule– Derive Simpson’s rule using parabolic
interpolation– Apply Simpson’s rule
ENEM602 Spring 2007
Dr. 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
ENEM602 Spring 2007
Dr. 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
ENEM602 Spring 2007
Dr. 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
ENEM602 Spring 2007
Dr. 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
ENEM602 Spring 2007
Dr. 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!
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
The Trapezoidal Rule
2
0101
yyxxI
2
2
1212
0101
yyxx
yyxxI
Integrating from x0 to x2:
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
General Trapezoidal Rule
• For all the points equally separated(xi+1-xi=h)
• We may write the equation of the previous slide:
321
2323
1212
22
22
yyyh
yyxx
yyxxI
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
In general
n
n
ii yyy
hI
1
10 2
2
Where n is the number if intervals and h=total interval/n
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example
• Integrate• Using the trapezoidal
rule• Use 2 points and
compare with the result using 3 points
1
0
2dxxI
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Solution
• Using 2 points (n=1), h=(1-0)/(1)=1
• Substituting:
212
1yyI
5.0102
1I
ENEM602 Spring 2007
Dr. 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
ENEM602 Spring 2007
Dr. 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
ENEM602 Spring 2007
Dr. 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
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
After substitutions and manipulation!
210 43
2
0
yyyh
dxxfx
x
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
For 4-Intervals
23210 4243
4
0
yyyyyh
dxxfx
x
ENEM602 Spring 2007
Dr. 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
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Example
• Integrate• Using the Simpson
rule• Use 3 points
1
0
2dxxI
ENEM602 Spring 2007
Dr. 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
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Homework #7
• Chapter 21, pp. 610-612, numbers:21.1, 21.3, 21.5, 21.25, 21.28.
• Due date: Week 8-12 May 2005