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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

Chabot Mathematics. §6.2 Numerical Integration. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. 6.1. Review §. Any QUESTIONS About §6.1 → Integration by Parts, Use of Integral Tables Any QUESTIONS About HomeWork §6.1 → HW-01. §6.2 Learning Goals.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

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  1. Chabot Mathematics §6.2 NumericalIntegration Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. 6.1 Review § • Any QUESTIONS About • §6.1 → Integration by Parts, Use of Integral Tables • Any QUESTIONS About HomeWork • §6.1 → HW-01

  3. §6.2 Learning Goals • Explore the trapezoidal rule and Simpson’s rule for numerical integration • Use error bounds for numerical integration • Interpret data using numerical integration

  4. Numerical Integration Why Numerical Methods? • Very often, the function f(x) to differentiate, or the integrand to integrate, is TOO COMPLEX to yield exact analytical solutions. • In most cases in Real World testing, the function f(x) is only available in a TABULATED form with values known only at DISCRETE POINTS

  5. Game Plan: Divide Unknown Area into Strips (or boxes), and Add Up Numerical Integration • To Improve Accuracy the TOP of the Strip can Be • Slanted Lines • Trapezoidal Rule • Parabolas • Simpson’s Rule • Higher Order PolyNomials

  6. Strip-Top Effect • Trapezoidal Form • Parabolic (Simpson’s) Form • Higher-Order-Polynomial Tops Lead to increased, but diminishing, accuracy.

  7. Strip-Count Effect 10 Strips 20 Strips • Adaptive Integration → INCREASE the strip-Count in Regions with Large SLOPES • More Strips of Constant Width Tends to work just as well

  8. AUC by Flat Tops

  9. Trapezoidal Area • By the Diagram at Right • Side Heights: • Width: • Now “Stack Up” for 2A • Then • or

  10. AUC by Trapezoids

  11. The Trapezoidal Rule • To Find the APPROXIMATE Area Under the Curve given by y = f(x), and divided into vertical strips of equal width, Δx • Where:

  12. Trapezoidal Rule Error • AUC by the Trapezoidal Approximation incurs error in the amount of • Where • n ≡ the strip count • K ≡ the maximum value of |d2y/dx2|

  13. Trapezoidal Rule Error Example • The Function does NOT have a Closed Form, Analytical Solution • Calculate the Area Under the Curve for this function between x=1 & x=3 using a 10-strip Trapezoidal Calculation

  14. Trapezoidal Rule Error Example • CalculateΔx • Then make Fcn T-Table using • ThenTheT-Table • Then theApproximation

  15. Trapezoidal Rule Error Example • ReCall from Error Equation • Taking the Derivative Twice • Plot d2y/dx2to EyeBallMax Value • Maximumat x = 3

  16. Trapezoidal Rule Error Example • Then • Thus, to 5 Sig-Figs: • Finally the Maximum 10-Strip, Trapezoidal Error

  17. Simpson’s Rule • The Simpson Method tops TWO Strips with successive 3-pt Curve-Fit Parabolas • A Parabola can befit EXACTLY to ANY 3 (x,y) points

  18. Simpson’s Rule • Since 3-pts defines 2-strips Simpson’s Rule requires an EVEN Strip-Count • Then for an Even Counting Number, n • if M = max(|d4y/dx4|)then the Error

  19. Simpson’s Rule Example • Use Simpson’s rule with n = 10 strips to approximate: • SOLUTION • From the Trapezoidal example Δx = 0.2 • Now the SideWays T-Table

  20. Find Precise Value by MuPAD • The Integrand Function • fOFx := E^x/x • Plot the AREA under the Integrand Curve • fArea := plot::Function2d(fOFx, x = 1..3):plot(plot::Hatch(fArea), fArea) • The Precise Value • AUCn = numeric::int(fOFx, x=1..3)

  21. Simpson’s Error • Find Fourth Derivative by MuPAD • d4fdx4 := diff(fOFx, x $ 4) • Then the 4th Derivative Plot • plot(d4fdx4, x=1..3, GridVisible = TRUE) • Max at x=1

  22. Simpson’s Error • Then the Error Calc • The Error comparing to MuPAD Value • Thus the TextBook Formula is Conservative

  23. NO Equation Functions • Often in REAL LIFE “functions” disguise themselves as “Data Tables” • When I was Research Tech at Lawrence Berkeley Lab (1978) I made Ventilation-Duct Volume-Flow measurements. A typical Data Set

  24. NO-Equation Functions • I then had to Calculate the Duct Volume Flow, Q, from the Data Table using the integration • This type of Integration OccursFrequently in the Physical, Life, and Social Sciences, aswell as in the Business world

  25. NO-Eqn Integration Example • The Cylindrical Tank shown at right has a bottom area of 130 ft2 . The tank is initially EMPTY. To Fill the Tank, Water Flows into the top at varying rates as given in the Tank-Table below.

  26. NO-Eqn Integration Example • For this situation determine the water height ,H, at t = 18 minutes • SOLUTION • Use the TRAPEZOIDAL Rule to Integrate the WaterFlow to arrive at the the Total Water VOLUME • Use the Max No. of strips permitted by Data

  27. NO-Eqn Integration Example • Make ΔV Calcsfor the 10 strips • Then by GeoMetry • So Finally

  28. NO-Eqn Integration Example • Note that in this case Δx is NON-constant • 10 Strips of Varying Width • Thus SIMPSON’s Rule Can NOT be Used • Simpson’s RuleRequires constant Δx

  29. % Bruce Mayer, PE % MTH-15 • 01Aug13 • Rev 11Sep13 % MTH15_Quick_Plot_BlueGreenBkGnd_130911.m % clear; clc; clf; % clf clears figure window % % The Domain Limits xmin = -6; xmax = 6; % The FUNCTION ************************************** x = [0 1 3 5 6 9 11 12 13 15 18]; y = [0 80 130 150 150 160 165 170 160 140 120]; % *************************************************** % the Plotting Range = 1.05*FcnRange ymin = min(y); ymax = max(y); % the Range Limits xmin = min(x); xmax = max(x); % the Range Limits R = ymax - ymin; ymid = (ymax + ymin)/2; ypmin = ymid - 1.025*R/2; ypmax = ymid + 1.025*R/2 % % The ZERO Lines zxh = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ypmin*1.05 ypmax*1.05]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y, '-d', 'LineWidth', 4),grid, axis([xminxmaxypminypmax]),... xlabel('\fontsize{14}t (min)'), ylabel('\fontsize{14}Q = (ft^3/min)'),... title(['\fontsize{16}MTH16 • Bruce Mayer, PE',]),... annotation('textbox',[.53 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 Quick Plot BlueGreenBkGnd 130911.m','FontSize',7) hold on plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) stem(x,y, '-r.', 'LineWidth', 2) hold off MatLab Code

  30. WhiteBoard Work • Problems From §6.2 • P40 → Consumer’s Surplus

  31. All Done for Today TrackingTrapezoids

  32. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

  33. P6.2-40 MatLAB Code % Bruce Mayer, PE % MTH-16 • 11Jan14 % MTH15_Quick_Plot_BlueGreenBkGnd_130911.m % clear; clc; clf; % clf clears figure window % % The FUNCTION ************************************** x = [0:4:24]; y = [49.12 42.9 31.32 19.83 13.87 10.58 7.25]; % *************************************************** % the Plotting Range = 1.05*FcnRange ymin = min(y); ymax = max(y); % the Range Limits xmin = min(x); xmax = max(x); % the Range Limits R = ymax - ymin; ymid = (ymax + ymin)/2; ypmin = ymid - 1.025*R/2; ypmax = ymid + 1.025*R/2 ypmin =0 % % The ZERO Lines zxh = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ypmin*1.05 ypmax*1.05]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([1 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y, '-d', 'LineWidth', 4),grid, axis([xminxmaxypminypmax]),... xlabel('\fontsize{14}q (kUnits)'), ylabel('\fontsize{14}p ($/Unit)'),... title(['\fontsize{16}MTH16 • Bruce Mayer, PE',]),... annotation('textbox',[.53 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 Quick Plot BlueGreenBkGnd 130911.m','FontSize',7) hold on plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2) stem(x,y, '-r.', 'LineWidth', 2) plot([xmin, xmax], [7.25 7.25], '-.m', 'LineWidth', 3) hold off x = [0 4 8 12 16 20 24] y = [49.1200 42.9000 31.3200 19.8300 13.8700 10.5800 7.2500] ps = y-ymin M = [1 4 2 4 2 4 1] CS1 = ps.*M CS2 = (4/3)*CS1 CS3 = sum(CS2) CS4 = sum(CS1) CStot = (4/3)*CS4

  34. Example  NONconstant∆x • Pacific Steel Casting Company (PSC) in Berkeley CA, uses huge amounts of electricity during the metal-melting process. • The PSC Materials Engineer measures the power, P, of a certain melting furnace over 340 minutes as shown in the table at right. See Data Plot at Right.

  35. Example  NONconstant∆x • The T-table at Right displays the Data Collected by the PSC Materials Enginer • Recall from Physics that Energy(or Heat), Q, is the time-integralof the Power. • Use Strip-Integration to find theTotal Energy in MJ expended byThe Furnace during this processrun

  36. Example  NONconstant∆x • GamePlan for Strip Integration • Use a Forward Difference approach • ∆tn = tn+1 − tn • Example: ∆t6 = t7 − t6 = 134 − 118 = 16min → 16min·(60sec/min) = 960sec • Over this ∆t assume the P(t) is constant at Pavg,n =(Pn+1 − Pn ) • Example: Pavg,6 = (P7 − P6)/2 = (147+178)/2 = 162.5 kW = 162.5 kJ/sec

  37. Example  NONconstant∆x • The GamePlanGraphically • Note the VariableWidth, ∆x,of the StripTops

  38. % Bruce Mayer, PE % MTH-15 • 25Jul13 % XY_Area_fcn_Graph_6x6_BlueGreen_BkGnd_Template_1306.m % clear; clc; clf; % clf is clear figure % % The FUNCTION xmin = 0; xmax = 350; ymin = 0; ymax = 225; x = [0 24 24 45 45 74 74 90 90 118 118 134 134 169 169 180 180 218 218 229 229 265 265 287 287 340] y = [77 77 105.5 105.5 125 125 136 136 152 152 162.5 162.5 179 179 181 181 192 192 208.5 208.5 203 203 201 201 213.5 213.5] % % The ZERO Lines zxh = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green % Now make AREA Plot area(x,y,'FaceColor',[1 0.6 1],'LineWidth', 3),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}t (minutes)'), ylabel('\fontsize{14}P (kW)'),... title(['\fontsize{16}MTH15 • Variable-Width Strip-Integration',]),... annotation('textbox',[.15 .82 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7) set(gca,'XTick',[xmin:50:xmax]); set(gca,'YTick',[ymin:25:ymax]) set(gca,'Layer','top') MATLAB Code

  39. Example  NONconstant∆x • The NONconstant Strip-Width Integration is conveniently done in an Excel SpreadSheet • The 13 ∆Q strips Add up to 3456.69 MegaJoulesof Total Energy Expended

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