MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §5.4 Definite Integral Apps

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §5.3 → Fundamental Theorem and Definite Integration  Any QUESTIONS About HomeWork §5.3 → HW

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 3 Bruce Mayer, PE Chabot College Mathematics §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

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 4 Bruce Mayer, PE Chabot College Mathematics Need for Strip-Like Integration  Strip 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

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 5 Bruce Mayer, PE Chabot College Mathematics Strip 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

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 6 Bruce Mayer, PE Chabot College Mathematics Strip Integration  Game Plan: Divide Unknown Area into Strips (or boxes), and Add Up  To Improve Accuracy 1.Increase the Number of strips; i.e., use smaller ∆x 2.Modify Strip-Tops –Slanted Lines (used most often) –Parabolas –High-Order Polynomials  Hi-No. of Flat-Strips Works Fine.

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 7 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.

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 8 Bruce Mayer, PE Chabot College Mathematics 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-integral of the Power.  Use Strip-Integration to find the Total Energy in MJ expended by The Furnace during this process run

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 9 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 )/2 –Example: P avg,6 = (P 7 + P 6 )/2 = ( )/2 = kW = kJ/sec

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  NONconstant ∆x  The GamePlan Graphically Note the Variable Width, ∆x, of the Strip Tops

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 11 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 = [ ] y = [ ] % % The ZERO Lines zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([ ]); % Chg Plot BackGround to Blue-Green % Now make AREA Plot area(x,y,'FaceColor',[ ],'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',[ ], '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')

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 12 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 MegaJoules of Total Energy Expended

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 13 Bruce Mayer, PE Chabot College Mathematics 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 Difference Region, R, Graphically a b R

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example Area Between Curves  Find the area between functions f & g over the interval x = [0,10]  The Graphs of f & g

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Area Between Curves  The process Graphically −=

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  Area Between Curves  Do the Math → ≈ 70.20

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Area Between Curves  Thus Ans A =

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 18 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % 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([ ]); % Chg Plot BackGround to Blue-Green subplot(1,3,2) area(x,y1,'FaceColor',[1.8.4], 'LineWidth', 3),axis([xmin xmax ymin ymax]),... 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',[ ], '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([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'),... annotation('textbox',[ ], '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([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'),... annotation('textbox',[ ], '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([xmin xmax ymin ymax]),... 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',[ ], '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

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 19 Bruce Mayer, PE Chabot College Mathematics MuPAD Code 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)

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Net Excess Profit  The Net Excess Profit of an investment plan over another is given by Where dP 1 /dt & dP 2 /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.

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 21 Bruce Mayer, PE Chabot College Mathematics 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 dP 1 /dt (P 1 ’) is increasing instantaneously by 1% per day, as compared to plan-2 which is currently increasing in value at $100 per day and dP 2 /dt (P 2 ’) is increasing instantaneously by 2% per day

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Net Excess Profit  SOLUTION:  The functions are each increasing exponentially (instantaneously), with dP 1 /dt initially 500 and growing exponentially with k = 0.01, so that  Similarly, dP 2 /dt is initially 100 and growing exponentially with k = 0.02, so that

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Net Excess Profit  ReCall the NEP 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

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 24 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example  Net Excess Profit  The plan-1 is greater than plan-2 from day-0 to day  Thus after rounding the NEP covers the time interval [0,161]. The the NEP Eqn:  Doing the Calculus

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  Net Excess Profit  STATE: In the initial 161 days, the Profit from plan-1 exceeded that of plan-2 by approximately $80k

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 27 Bruce Mayer, PE Chabot College Mathematics Example  Net Excess Profit  The Profit Rates  The NEP (ABC) Area Between Curves

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 28 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % 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 $k % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg([ ]); % Chg Plot BackGround to Blue-Green plot(x,y1, x,y2, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}t (days)'), ylabel('\fontsize{14} P_1''= 0.5e^x^/^1^0^0 P_2'' = 0.5e^x^/^5^0^ ($k)'),... title(['\fontsize{16}MTH15 Net Excess Profit',]),... annotation('textbox',[ ], '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

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 29 Bruce Mayer, PE Chabot College Mathematics Recall: Average Value of a fcn  Mathematically - If f is integrable on [a, b], then the average value of f over [a, b] is  Example  Find the Avg Value:  Use Average Definition:

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 30 Bruce Mayer, PE Chabot College Mathematics 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? ))

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 31 Bruce Mayer, PE Chabot College Mathematics 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

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 32 Bruce Mayer, PE Chabot College Mathematics Example  GeoTech Engineering  And  Then the Transformed Integral  Working the Calculus

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 33 Bruce Mayer, PE Chabot College Mathematics Example  GeoTech Engineering  The average rate at which sediment was gathering for the first six months was tons per year.  dM/dt along with its average value on [0,0.5]: Equal Areas

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 34 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §5.4 P46 → Worker Productivity P60 → Cardiac Fluidic Mechanics

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 35 Bruce Mayer, PE Chabot College Mathematics All Done for Today DilBert Integration

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 36 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 37 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 38 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 39 Bruce Mayer, PE Chabot College Mathematics P5.4-46(b)  Production Rates  Cumulative Difference Q tot = 184/3 units ABC = 184/3

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 40 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 41 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 42 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 43 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 44 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 45 Bruce Mayer, PE Chabot College Mathematics

MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 46 Bruce Mayer, PE Chabot College Mathematics