09 numerical integration

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09 numerical integration

  1. 1. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Numerical Integration
  2. 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. 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. 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. 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. 6. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Adding adjacent areas
  7. 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. 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. 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. 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. 11. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Note that X0=a Xn=b
  12. 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. 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. 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. 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. 16. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Let’s use Interpolation!
  17. 17. 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
  18. 18. 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. ( ) ( )0 01 01 0 xx xx yy yxf − − − +≈
  19. 19. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Example • Performing the integration on the approximate function: ( ) ( )∫∫       − − − +≈= 1 0 1 0 0 01 01 0 x x x x dxxx xx yy ydxxfI 1 0 0 2 01 01 0 2 x x xx x xx yy xyI               − − − +≈
  20. 20. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Example • Performing the integration on the approximate function:                 − − − +−                 − − − +≈ 00 2 0 01 01 0010 2 1 01 01 10 22 xx x xx yy xyxx x xx yy xyI ( ) ( ) 2 01 01 yy xxI + −≈ • Which is equivalent to the area of the trapezium!
  21. 21. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik The Trapezoidal Rule ( ) ( ) 2 01 01 yy xxI + −≈ ( ) ( ) ( ) ( ) 2 2 12 12 01 01 yy xx yy xxI + −+ + −≈ Integrating from x0 to x2:
  22. 22. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Simpson’s Rule Using a parabola to join three adjacent points!
  23. 23. 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 −−+−+≈ ( ) ( ) ( )( )1010 2 3021 xxxxxxbxxbbxf ++−+−+≈
  24. 24. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Integrating ( ) ( ) ( )( )1010 2 3021 xxxxxxbxxbbxf ++−+−+≈ ( ) ( ) ( )( )∫∫ ++−+−+≈ 2 0 2 0 1010 2 3021 x x x x dxxxxxxxbxxbbdxxf ( ) ( ) 2 0 2 0 10 2 10 3 30 2 21 232 x x x x xxx x xx x bxx x bxbdxxf             ++−+      −+≈∫
  25. 25. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik After substitutions and manipulation! ( ) [ ]210 4 3 2 0 yyy h dxxf x x ++≈∫
  26. 26. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Working with three points! ( ) [ ]210 4 3 2 0 yyy h dxxf x x ++≈∫
  27. 27. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik For 4-Intervals ( ) [ ]432210 44 3 4 0 yyyyyy h dxxf x x +++++≈∫
  28. 28. 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
  29. 29. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Example • Integrate • Using the Simpson rule • Use 3 points ∫= 1 0 2 dxxI
  30. 30. 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
  31. 31. ENEM602 Spring 2007 Dr. Eng. Mohammad Tawfik Homework #7 • Chapter 21, p. 610, numbers: 21.5, 21.6, 21.10, 21.11.

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