1. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §6.2 Numerical Integration

2. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §6.1 → Integration by Parts, Use of Integral Tables  Any QUESTIONS About HomeWork §6.1 → HW-01 6.1

3. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 3 Bruce Mayer, PE Chabot College Mathematics §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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 4 Bruce Mayer, PE Chabot College Mathematics Why Numerical Methods?  Numerical Integration 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 5 Bruce Mayer, PE Chabot College Mathematics Numerical Integration  Game Plan: Divide Unknown Area into Strips (or boxes), and Add Up  To Improve Accuracy the TOP of the Strip can Be Slanted Lines –Trapezoidal Rule Parabolas –Simpson’s Rule Higher Order PolyNomials

6. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 6 Bruce Mayer, PE Chabot College Mathematics Strip-Top Effect  Parabolic (Simpson’s) Form  Trapezoidal Form Higher-Order-Polynomial Tops Lead to increased, but diminishing, accuracy.

7. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 7 Bruce Mayer, PE Chabot College Mathematics Strip-Count Effect  Adaptive Integration → INCREASE the strip- Count in Regions with Large SLOPES More Strips of Constant Width Tends to work just as well 10 Strips20 Strips

8. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 8 Bruce Mayer, PE Chabot College Mathematics AUC by Flat Tops

9. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 9 Bruce Mayer, PE Chabot College Mathematics Trapezoidal Area  By the Diagram at Right Side Heights: Width:  Now “Stack Up” for 2A  Then  or

10. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 10 Bruce Mayer, PE Chabot College Mathematics AUC by Trapezoids

11. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 11 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 12 Bruce Mayer, PE Chabot College Mathematics Trapezoidal Rule Error  AUC by the Trapezoidal Approximation incurs error in the amount of  Where n ≡ the strip count K ≡ the maximum value of |d 2 y/dx 2 |

13. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 13 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 14 Bruce Mayer, PE Chabot College Mathematics Trapezoidal Rule Error Example  Calculate Δx  Then make Fcn T-Table using  Then The T- Table  Then the Approximation

15. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 15 Bruce Mayer, PE Chabot College Mathematics Trapezoidal Rule Error Example  ReCall from Error Equation  Taking the Derivative Twice  Plot d 2 y/dx 2 to EyeBall Max Value  Maximum at x = 3

16. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 16 Bruce Mayer, PE Chabot College Mathematics Trapezoidal Rule Error Example  Then  Thus, to 5 Sig-Figs:  Finally the Maximum 10-Strip, Trapezoidal Error

17. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 17 Bruce Mayer, PE Chabot College Mathematics Simpson’s Rule  The Simpson Method tops TWO Strips with successive 3-pt Curve-Fit Parabolas  A Parabola can be fit EXACTLY to ANY 3 (x,y) points

18. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 18 Bruce Mayer, PE Chabot College Mathematics 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(|d 4 y/dx 4 |) then the Error

19. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 19 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 20 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 21 Bruce Mayer, PE Chabot College Mathematics Simpson’s Error  Find Fourth Derivative by MuPAD d4fdx4 := diff(fOFx, x $ 4)  Then the 4 th Derivative Plot plot(d4fdx4, x=1..3, GridVisible = TRUE) Max at x=1

22. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 22 Bruce Mayer, PE Chabot College Mathematics Simpson’s Error  Then the Error Calc  The Error comparing to MuPAD Value Thus the TextBook Formula is Conservative

23. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 23 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 24 Bruce Mayer, PE Chabot College Mathematics NO-Equation Functions  I then had to Calculate the Duct Volume Flow, Q, from the Data Table using the integration  This type of Integration Occurs Frequently in the Physical, Life, and Social Sciences, as well as in the Business world

25. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 25 Bruce Mayer, PE Chabot College Mathematics NO-Eqn Integration Example  The Cylindrical Tank shown at right has a bottom area of 130 ft 2. The tank is initially EMPTY. To Fill the Tank, Water Flows into the top at varying rates as given in the Tank-Table below. Time (min) 0135691112131518 FlowRate (ft 3 /min) 080130150 160165170160140120

26. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 26 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 27 Bruce Mayer, PE Chabot College Mathematics NO-Eqn Integration Example  Make ΔV Calcs for the 10 strips  Then by GeoMetry  So Finally

28. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 28 Bruce Mayer, PE Chabot College Mathematics 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 Rule Requires constant Δx

29. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 29 Bruce Mayer, PE Chabot College Mathematics MatLab Code % 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 = [xmin xmax]; 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([xmin xmax ypmin ypmax]),... 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

30. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 30 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §6.2 P40 → Consumer’s Surplus

31. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 31 Bruce Mayer, PE Chabot College Mathematics All Done for Today Tracking Trapezoids

32. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 32 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

33. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 33 Bruce Mayer, PE Chabot College Mathematics

34. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 34 Bruce Mayer, PE Chabot College Mathematics

35. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 35 Bruce Mayer, PE Chabot College Mathematics P6.2-40 MatLAB Code 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 % 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 = [xmin xmax]; 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([xmin xmax ypmin ypmax]),... 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

36. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 36 Bruce Mayer, PE Chabot College Mathematics 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.

37. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 37 Bruce Mayer, PE Chabot College Mathematics 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-integral of the Power.  Use Strip-Integration to find the Total Energy in MJ expended by The Furnace during this process run

38. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 38 Bruce Mayer, PE Chabot College Mathematics Example  NONconstant ∆x  GamePlan for Strip Integration  Use a Forward Difference approach ∆t n = t n+1 − t n –Example: ∆t 6 = t 7 − t 6 = 134 − 118 = 16min → 16min·(60sec/min) = 960sec Over this ∆t assume the P(t) is constant at P avg,n =(P n+1 − P n ) –Example: P avg,6 = (P 7 − P 6 )/2 = (147+178)/2 = 162.5 kW = 162.5 kJ/sec

39. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 39 Bruce Mayer, PE Chabot College Mathematics Example  NONconstant ∆x  The GamePlan Graphically Note the Variable Width, ∆x, of the Strip Tops

40. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 40 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % 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 = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax]; % % 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([xmin xmax ymin ymax]),... 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')

41. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 41 Bruce Mayer, PE Chabot College Mathematics Example  NONconstant ∆x  The NONconstant Strip-Width Integration is conveniently done in an Excel SpreadSheet  The 13 ∆Q strips Add up to 3456.69 MegaJoules of Total Energy Expended