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

Chabot Mathematics. §5.4 Definite Integral Apps. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. 5.3. Review §. Any QUESTIONS About §5.3 → Fundamental Theorem and Definite Integration Any QUESTIONS About HomeWork §5.3 → HW-24. §5.4 Learning Goals.

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

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  1. Chabot Mathematics §5.4 DefiniteIntegral Apps Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. 5.3 Review § • Any QUESTIONS About • §5.3 → Fundamental Theorem and Definite Integration • Any QUESTIONS About HomeWork • §5.3 → HW-24

  3. §5.4 Learning Goals • Explore a general procedure for using definite integration in applications • Find area between two curves, and use it to compute net excess profit and distribution of wealth (Lorenz curves) • Derive and apply a formula for the average value of a function • Interpret average value in terms of rate and area

  4. Strip Integration Need for Strip-Like 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 engineering or science 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 Strip Integration • To Improve Accuracy the • TOP of the Strip can Be • Slanted Lines • Trapezoidal Rule • Parabolas • Simpson’s Rule • Higher Order PolyNomials

  6. Strip Integration • Game Plan: Divide Unknown Area into Strips (or boxes), and Add Up • To Improve Accuracy • Increase the Number of strips; i.e., use smaller ∆x • Modify Strip-Tops • Slanted Lines (used most often) • Parabolas • High-Order Polynomials • Hi-No. of Flat-StripsWorks Fine.

  7. 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.

  8. Example  NONconstant∆x • The T-table at Right displays the Data Collected by the PSC Materials Engineer • 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

  9. 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)/2 • Example: Pavg,6 = (P7 + P6)/2 = (147+178)/2 = 162.5 kW = 162.5 kJ/sec

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

  11. % 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

  12. 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

  13. Area Between Two Curves • Let f and g be continuous functions, the area bounded above by y = f (x) and below by y = g(x) on [a, b] is • Provided that • The Areal DifferenceRegion, R, Graphically R a b

  14. Example Area Between Curves • Find the area between functions f & g over the interval x = [0,10] • The Graphsof f & g

  15. Example Area Between Curves • The process Graphically − =

  16. Example Area Between Curves • Do the Math → ≈ 70.20

  17. Example Area Between Curves • ThusAns A = 70.200

  18. % Bruce Mayer, PE % MTH-15 • 25Jun13 % clear; clc; clf; % clf clears figure window % % The Limits xmin = 0; xmax = 10; ymin = 0; ymax = 20; % The FUNCTION x = linspace(xmin,xmax,500); y1 = (-8/25)*(x-5).^2 + 10; y2 = 11*exp(-x/6)+9; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green subplot(1,3,2) area(x,y1,'FaceColor',[1 .8 .4], 'LineWidth', 3),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'),ylabel('\fontsize{14}ylo = (-8/25)*(x-5)^2+10 • yhi = 11e^-^x^/^6+9'),... title(['\fontsize{16}MTH15 • Area Between Curves',]),... annotation('textbox',[.5 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7) hold on set(gca,'XTick',[xmin:2:xmax]); set(gca,'YTick',[ymin:2:ymax]) set(gca,'Layer','top') hold off % subplot(1,3,1) area(x,y2, 'FaceColor',[0 1 0], 'LineWidth', 3),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'),... annotation('textbox',[.15 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7) hold on set(gca,'XTick',[xmin:2:xmax]); set(gca,'YTick',[ymin:2:ymax]) set(gca,'Layer','top') hold off % xn = linspace(xmin, xmax, 500); subplot(1,3,3) fill([xn,fliplr(xn)],[(-8/25)*(xn-5).^2 + 10, fliplr(11*exp(-xn/6)+9)],'m'),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'),... annotation('textbox',[.85 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7) hold on set(gca,'XTick',[xmin:2:xmax]); set(gca,'YTick',[ymin:2:ymax]) set(gca,'Layer','top') hold off % disp('Showing SubPlot - Hit Any Key to Continue') pause % clf fill([xn,fliplr(xn)],[(-8/25)*(xn-5).^2 + 10, fliplr(11*exp(-xn/6)+9)],'m'),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}x'),,ylabel('\fontsize{14}ylo = (-8/25)*(x-5)^2+10 • yhi = 11e^-^x^/^6+9'),... title(['\fontsize{16}MTH15 • Area Between Curves',]),... annotation('textbox',[.6 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7) hold on set(gca,'XTick',[xmin:2:xmax]); set(gca,'YTick',[ymin:2:ymax]) set(gca,'Layer','top') hold off MATLAB Code

  19. MuPAD Code f := 11*exp(-x/6)+9 g := (-8/25)*(x-5)^2+10 fminusg := f-g AntiDeriv := int(fminusg, x) ABC := int(fminusg, x=0..10) float(ABC)

  20. Example  Net Excess Profit • The Net Excess Profit of an investment plan over another is given by • Where dP1/dt & dP2/dt are the rates of profitability of plan-1 & plan-2 • The Net Excess Profit (NEP) gives the total profit gained by plan-1 over plan-2 in a given time interval.

  21. Example  Net Excess Profit • Find the net excess profit during the period from now until plan-1 is no longer increasing faster than plan-2: • Plan-1 is an investment that is currently increasing in value at $500 per day and dP1/dt (P1’) is increasing instantaneously by 1% per day, as compared to plan-2 which is currently increasing in value at $100 per day and dP2/dt (P2’) is increasing instantaneously by 2% per day

  22. Example  Net Excess Profit • SOLUTION: • The functions are each increasing exponentially (instantaneously), with dP1/dt initially 500 and growing exponentially with k = 0.01, so that • Similarly, dP2/dt is initially 100 and growing exponentially with k = 0.02, so that

  23. Example  Net Excess Profit • ReCall theNEP Eqn • where a and b are determined by the time for which plan-1 is increasing faster than plan-2, that is, [a,b] includes those times, t, such that: • Using the Given Data

  24. Example  Net Excess Profit • Dividing Both Sides of the InEquality • Taking the Natural Log of Both Side • Divide both Sides by 0.01 to Solve for t

  25. Example  Net Excess Profit • The plan-1 is greater than plan-2 from day-0 to day 160.94. • Thus after rounding the NEP covers the time interval [0,161]. The the NEP Eqn: • Doing the Calculus

  26. Example  Net Excess Profit • STATE: In the initial 161 days, the Profit from plan-1 exceeded that of plan-2 by approximately $80k

  27. Example  Net Excess Profit • The Profit Rates • The NEP (ABC) Area Between Curves

  28. % Bruce Mayer, PE % MTH-15 • 25Jun13 % clear; clc; clf; % clf clears figure window % xmin = 0; xmax = 161; ymin = 0; ymax = 2.5; % The FUNCTION x = linspace(xmin,xmax,500); y1 = .5*exp(x/100); y2 = .1*exp(x/50); % x in days • y's in $k % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y1, x,y2, 'LineWidth', 4),axis([xminxmaxyminymax]),... grid, xlabel('\fontsize{14}t (days)'), ylabel('\fontsize{14} P_1''= 0.5e^x^/^1^0^0 • P_2'' = 0.5e^x^/^5^0^ ($k)'),... title(['\fontsize{16}MTH15 • Net Excess Profit',]),... annotation('textbox',[.6 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7) hold on set(gca,'XTick',[xmin:20:xmax]); set(gca,'YTick',[ymin:0.5:ymax]) disp('Hit ANY KEY to show Fill') pause % xn = linspace(xmin, xmax, 500); fill([xn,fliplr(xn)],[.5*exp(xn/100), fliplr(.1*exp(x/50))],'m') hold off MATLAB Code

  29. Recall: Average Value of a fcn • Mathematically - If f is integrable on [a, b], then the average value of fover [a, b] is • Example  Find the Avg Value: • Use Average Definition:

  30. Example  GeoTech Engineering • A Model for The rate at which sediment gathers at the delta of a river is given by • Where • t ≡ the length of time (years) since study began • M ≡ the Mass of sediment (tons) accumulated • What is the average rate at which sediment gathers during the first six months of study? • )

  31. Example  GeoTech Engineering • By the Avg Value eqn the average rate at which sediment gathers over the first six months (0.5 years) • No Integration Rule applies so try subsitution. Let

  32. Example  GeoTech Engineering • And • Then the Transformed Integral • Working the Calculus

  33. Example  GeoTech Engineering • The average rate at which sediment was gathering for the first six months was 0.863 tons per year. • dM/dt along with its average value on [0,0.5]: Equal Areas

  34. WhiteBoard Work • Problems From §5.4 • P46 → Worker Productivity • P60 → Cardiac Fluidic Mechanics

  35. All Done for Today DilBertIntegration

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

  37. P5.4-46(b) • Production Rates • Cumulative Difference • Qtot = 184/3 units ABC = 184/3

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