08 numerical integration 2
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This document discusses numerical integration techniques, including: - The need to approximate definite integrals numerically when exact solutions cannot be found. - The trapezoidal rule, which approximates the area under a curve as a trapezoid, and generalizes this to multiple trapezoids to improve accuracy. - Simpson's rule, which uses a parabolic interpolation between three points to better approximate curved areas. - Examples are provided to demonstrate applying these rules to calculate integrals of simple functions.
![ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Working with three points!
( ) [ ]210 4
3
2
0
yyy
h
dxxf
x
x
++≈∫](https://image.slidesharecdn.com/08numericalintegration-2-101108031240-phpapp01/85/08-numerical-integration-2-17-320.jpg)
![ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
For 4-Intervals
( ) [ ]432210 44
3
4
0
yyyyyy
h
dxxf
x
x
+++++≈∫](https://image.slidesharecdn.com/08numericalintegration-2-101108031240-phpapp01/85/08-numerical-integration-2-18-320.jpg)
08 numerical integration 2
- 1. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Numerical Integration
- 2. 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 geometric insight – Apply the trapezoidal rule – Apply Simpson’s rule
- 3. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Need for Numerical Integration! ( ) 6 11 01 2 1 3 1 23 1 1 0 231 0 2 =− ++= ++=++= ∫ x xx dxxxI ( ) 11 0 1 0 1 −−− −=−== ∫ eedxeI xx ∫ − = 1 0 2 dxeI x
- 4. ENEM602 Spring 2007 Dr. 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?
- 5. ENEM602 Spring 2007 Dr. 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: ( ) ( )01 01 2 1 0 xx yy dxxf x x − + ≈∫
- 6. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Adding adjacent areas
- 7. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik The Trapezoidal Rule ( ) ( ) ( ) ( ) 2 2 12 12 01 01 yy xx yy xxI + −+ + −≈ Integrating from x0 to x2: ( ) ( ) ( ) ( ) 2 212112101001 yxxyxxyxxyxx I −+−+−+− ≈
- 8. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik The Trapezoidal Rule ( ) ( ) hxxxx =−=− 1201 If the points are equidistant 2 2110 hyhyhyhy I +++ ≈ ( )210 2 2 yyy h I ++≈
- 9. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Dividing the whole interval into “n” subintervals ++≈ ∑ − = n n i i yyy h I 1 1 0 2 2
- 10. ENEM602 Spring 2007 Dr. 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 i i yyy h I 1 1 0 2 2
- 11. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Note that X0=a Xn=b
- 12. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Example • Integrate • Using the trapezoidal rule • Use 2,3,&4 points and compare the results ∫= 1 0 2 dxxI
- 13. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Solution • Using 2 points (n=1), h=(1-0)/(1)=1 • Substituting: ( )21 2 1 yyI +≈ ( ) 5.010 2 1 =+≈I X Y 0 0 1 1 2 points, 1 interval
- 14. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Solution • Using 3 points (n=2), h=(1-0)/(2)=0.5 • Substituting: ( )321 2 2 5.0 yyyI ++≈ ( ) 375.0125.0*20 2 5.0 =++≈I X Y 0 0 0.5 0.25 1 1 3 points, 2 interval
- 15. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Solution • Using 4 points (n=3), h=(1-0)/(3)=0.333 • Substituting: ( )4321 22 2 333.0 yyyyI +++≈ ( ) 3519.01444.0*2111.0*20 2 333.0 =+++≈I X Y 0 0 0.33 0.111 0.667 0.444 1 1 4 points, 3 interval
- 16. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Simpson’s Rule Using a parabola to join three adjacent points!
- 17. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Working with three points! ( ) [ ]210 4 3 2 0 yyy h dxxf x x ++≈∫
- 18. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik For 4-Intervals ( ) [ ]432210 44 3 4 0 yyyyyy h dxxf x x +++++≈∫
- 19. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik In General: Simpson’s Rule ( ) +++≈ ∑∑∫ − = − = n n i i n i i x x yyyy h dxxf n 2 ,..4,2 1 ,..3,1 0 24 30 NOTE: the number of intervals HAS TO BE even
- 20. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Example • Integrate • Using the Simpson rule • Use 3 points ∫= 1 0 2 dxxI
- 21. 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 4 3 5.0 yyyI ++≈ ( ) 3 1 125.0*40 3 5.0 =++≈I
- 22. 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 15-19 May 2005