1. Chapter 17 Numerical Integration Formulas

3. Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral

4. Use of strips to approximate an integral

5. Numerical Integration Net force against a skyscraper Cross-sectional area and volume flowrate in a river Survey of land area of an irregular lot

6. Water exerting pressure on the upstream face of a dam: (a) side view showing force increasing linearly with depth; (b) front view showing width of dam in meters. Pressure Force on a Dam p =  gh =  h

7. Integration  Weighted sum of functional values at discrete points  Newton-Cotes closed or open formulae -- evenly spaced points  Approximate the function by Lagrange interpolation polynomial  Integration of a simple interpolation polynomial  Guassian Quadratures  Richardson extrapolation and Romberg integration

8. Basic Numerical Integration  Weighted sum of function values x0x0 x1x1 xnxn x n-1 x f(x)f(x)

9. Numerical Integration Idea is to do integral in small parts, like the way you first learned integration - a summation Numerical methods just try to make it faster and more accurate

10. Newton-Cotes formulas - based on idea  Approximate f(x) by a polynomial Numerical integration

11.  f n (x) can be linear  f n (x) can be quadratic

12.  f n (x) can also be cubic or other higher-order polynomials

13.  Polynomial can be piecewise over the data

14. Numerical Integration  Newton-Cotes Closed Formulae -- Use both end points  Trapezoidal Rule : Linear  Simpson’s 1/3-Rule : Quadratic  Simpson’s 3/8-Rule : Cubic  Boole’s Rule : Fourth-order*  Higher-order methods*  Newton-Cotes Open Formulae -- Use only interior points  midpoint rule  Higher-order methods

15. Closed and Open Formulae (a) End points are known (b) Extrapolation

16. Trapezoidal Rule Straight-line approximation x0x0 x1x1 x f(x)f(x) L(x)

17. Trapezoidal Rule Lagrange interpolation

18. Example:Trapezoidal Rule Evaluate the integral Exact solution Trapezoidal Rule

19. Better Numerical Integration  Composite integration  Multiple applications of Newton-Cotes formulae  Composite Trapezoidal Rule  Composite Simpson’s Rule  Richardson Extrapolation  Romberg integration

20. Apply trapezoidal rule to multiple segments over integration limits Two segments Four segmentsMany segments Three segments

21. Multiple Applications of Trapezoidal Rule

22. Composite Trapezoidal Rule x0x0 x1x1 x f(x)f(x) x2x2 hhx3x3 hhx4x4

23. Trapezoidal Rule  Truncation error (single application)  Exact if the function is linear ( f  = 0)  Use multiple applications to reduce the truncation error Approximate error

24. Composite Trapezoidal Rule function f = example1(x) % a = 0, b = pi f=x.^2.*sin(2*x);

25. » a=0; b=pi; dx=(b-a)/100; » x=a:dx:b; y=example1(x); » I=trap('example1',a,b,1) I = -3.7970e-015 » I=trap('example1',a,b,2) I = -1.4239e-015 » I=trap('example1',a,b,4) I = -3.8758 » I=trap('example1',a,b,8) I = -4.6785 » I=trap('example1',a,b,16) I = -4.8712 » I=trap('example1',a,b,32) I = -4.9189 Composite Trapezoidal Rule » I=trap('example1',a,b,64) I = -4.9308 » I=trap('example1',a,b,128) I = -4.9338 » I=trap('example1',a,b,256) I = -4.9346 » I=trap('example1',a,b,512) I = -4.9347 » I=trap('example1',a,b,1024) I = -4.9348 » Q=quad8('example1',a,b) Q = -4.9348 MATLAB function

26. n = 2 I = -1.4239 e-15 Exact = -4. 9348

27. n = 4 I = -3.8758 Exact = -4. 9348

28. n = 8 I = -4.6785 Exact = -4. 9348

29. n = 16 I = -4.8712 Exact = -4. 9348

30. Composite Trapezoidal Rule Evaluate the integral

31. Composite Trapezoidal Rule » x=0:0.04:4; y=example2(x); » x1=0:4:4; y1=example2(x1); » x2=0:2:4; y2=example2(x2); » x3=0:1:4; y3=example2(x3); » x4=0:0.5:4; y4=example2(x4); » H=plot(x,y,x1,y1,'g-*',x2,y2,'r-s',x3,y3,'c-o',x4,y4,'m-d'); » set(H,'LineWidth',3,'MarkerSize',12); » xlabel('x'); ylabel('y'); title('f(x) = x exp(2x)'); » I=trap('example2',0,4,1) I = 2.3848e+004 » I=trap('example2',0,4,2) I = 1.2142e+004 » I=trap('example2',0,4,4) I = 7.2888e+003 » I=trap('example2',0,4,8) I = 5.7648e+003 » I=trap('example2',0,4,16) I = 5.3559e+003

32. Composite Trapezoidal Rule

33. Simpson’s 1/3-Rule Approximate the function by a parabola x0x0 x1x1 x f(x)f(x) x2x2 hh L(x)L(x)

34. Simpson’s 1/3-Rule

36. Composite Simpson’s Rule x0x0 x2x2 x f(x)f(x) x4x4 hhx n-2 hxnxn …... Piecewise Quadratic approximations hx3x3 x1x1 x n-1

37. Composite Simpson’s 1/3 Rule  Applicable only if the number of segments is even

38. Composite Simpson’s 1/3 Rule  Applicable only if the number of segments is even  Substitute Simpson’s 1/3 rule for each integral  For uniform spacing (equal segments)

39. Simpson’s 1/3 Rule  Truncation error (single application)  Exact up to cubic polynomial ( f (4) = 0)  Approximate error for (n/2) multiple applications

40. Composite Simpson’s 1/3 Rule  Evaluate the integral n = 2, h = 2 n = 4, h = 1

41. Simpson’s 3/8-Rule  Approximate by a cubic polynomial x0x0 x1x1 x f(x) x2x2 hh L(x) x3x3 h

42. Simpson’s 3/8-Rule  Truncation error

43. Example: Simpson’s Rules  Evaluate the integral  Simpson’s 1/3-Rule  Simpson’s 3/8-Rule

44. function I = Simp(f, a, b, n) % integral of f using composite Simpson rule % n must be even h = (b - a)/n; S = feval(f,a); for i = 1 : 2 : n-1 x(i) = a + h*i; S = S + 4*feval(f, x(i)); end for i = 2 : 2 : n-2 x(i) = a + h*i; S = S + 2*feval(f, x(i)); end S = S + feval(f, b); I = h*S/3; Composite Simpson’s 1/3 Rule

45. Simpson’s 1/3 Rule Simpson’s 1/3 Rule

46. Composite Simpson’s 1/3 Rule

47. » x=0:0.04:4; y=example(x); » x1=0:2:4; y1=example(x1); » c=Lagrange_coef(x1,y1); p1=Lagrange_eval(x,x1,c); » H=plot(x,y,x1,y1,'r*',x,p1,'r'); » xlabel('x'); ylabel('y'); title('f(x) = x*exp(2x)'); » set(H,'LineWidth',3,'MarkerSize',12); » x2=0:1:4; y2=example(x2); » c=Lagrange_coef(x2,y2); p2=Lagrange_eval(x,x2,c); » H=plot(x,y,x2,y2,'r*',x,p2,'r'); » xlabel('x'); ylabel('y'); title('f(x) = x*exp(2x)'); » set(H,'LineWidth',3,'MarkerSize',12); » » I=Simp('example',0,4,2) I = 8.2404e+003 » I=Simp('example',0,4,4) I = 5.6710e+003 » I=Simp('example',0,4,8) I = 5.2568e+003 » I=Simp('example',0,4,16) I = 5.2197e+003 » Q=Quad8('example',0,4) Q = 5.2169e+003 n = 2 n = 4 n = 8 n = 16 MATLAB fun

48. Multiple applications of Simpson’s rule with odd number of intervals Hybrid Simpson’s 1/3 & 3/8 rules

49. Newton-Cotes Closed Integration Formulae

50. Composite Trapezoidal Rule with Unequal Segments  Evaluate the integral  h 1 = 2, h 2 = 1, h 3 = 0.5, h 4 = 0.5

51. Trapezoidal Rule for Unequally Spaced Data

52. MATLAB Function: trapz » x=[0 1 1.5 2.0 2.5 3.0 3.3 3.6 3.8 3.9 4.0] x = Columns 1 through 7 0 1.0000 1.5000 2.0000 2.5000 3.0000 3.3000 Columns 8 through 11 3.6000 3.8000 3.9000 4.0000 » y=x.*exp(2.*x) y = 1.0e+004 * Columns 1 through 7 0 0.0007 0.0030 0.0109 0.0371 0.1210 0.2426 Columns 8 through 11 0.4822 0.7593 0.9518 1.1924 » integr = trapz(x,y) integr = 5.3651e+003  Z = trapz(x,y)

53. Integral of Unevenly-Spaced Data  Trapezoidal rule  Could also be evaluated with Simpson’s rule for higher accuracy

54. Composite Simpson’s Rule with Unequal Segments Evaluate the integral h 1 = 1.5, h 2 = 0.5

55. Newton-Cotes Open Formula Midpoint Rule (One-point) ab x f(x) xmxm

56. Two-point Newton-Cotes Open Formula  Approximate by a straight line x0x0 x1x1 x f(x)f(x) x2x2 hhx3x3 h

57. Three-point Newton-Cotes Open Formula  Approximate by a parabola x0x0 x1x1 x f(x)f(x) x2x2 hhx3x3 hhx4x4

58. Newton-Cotes Open Integration Formulae

59.  Area under the function surface Double Integral

60.  T(x, y) = 2xy + 2x – x 2 – 2y 2 + 40  Two-segment trapezoidal rule  Exact if using single-segment Simpson’s 1/3 rule (because the function is quadratic in x and y) Double Integral