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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

Chabot Mathematics

§5.4 DefiniteIntegral Apps

Bruce Mayer, PE

Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu


Review

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
§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


Need for strip like integration

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


Strip integration

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


Strip integration1
Strip Integration Add Up

  • 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.


Example nonconstant x
Example Add Up 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.


Example nonconstant x1
Example Add Up 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


Example nonconstant x2
Example Add Up 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


Example nonconstant x3
Example Add Up NONconstant∆x

  • The GamePlanGraphically

    • Note the VariableWidth, ∆x,of the StripTops


Matlab code

% Bruce Mayer, PE Add Up

% 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


Example nonconstant x4
Example Add Up 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


Area between two curves
Area Between Two Curves Add Up

  • 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


Example area between curves
Example Add UpArea Between Curves

  • Find the area between functions f & g over the interval x = [0,10]

  • The Graphsof f & g


Example area between curves1
Example Add UpArea Between Curves

  • The process Graphically

=


Example area between curves2
Example Add UpArea Between Curves

  • Do the Math →

≈ 70.20


Example area between curves3
Example Add UpArea Between Curves

  • ThusAns

A = 70.200


Matlab code1

% Add Up 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


Mupad code
MuPAD Add Up 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)


Example net excess profit
Example Add Up 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.


Example net excess profit1
Example Add Up 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


Example net excess profit2
Example Add Up 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


Example net excess profit3
Example Add Up 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


Example net excess profit4
Example Add Up 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


Example net excess profit5
Example Add Up 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


Example net excess profit6
Example Add Up Net Excess Profit

  • STATE: In the initial 161 days, the Profit from plan-1 exceeded that of plan-2 by approximately $80k


Example net excess profit7
Example Add Up Net Excess Profit

  • The Profit Rates

  • The NEP (ABC)

Area Between Curves


Matlab code2

% Bruce Mayer, PE Add Up

% 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


Recall average value of a fcn
Recall: Average Value of a Add Upfcn

  • 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:


Example geotech engineering
Example Add Up 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?

  • )


Example geotech engineering1
Example Add Up 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


Example geotech engineering2
Example Add Up GeoTech Engineering

  • And

  • Then the Transformed Integral

  • Working the Calculus


Example geotech engineering3
Example Add Up 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


Whiteboard work
WhiteBoard Add Up Work

  • Problems From §5.4

    • P46 → Worker Productivity

    • P60 → Cardiac Fluidic Mechanics


All done for today
All Done for Today Add Up

DilBertIntegration


Bruce mayer pe licensed electrical mechanical engineer bmayer chabotcollege

Chabot Mathematics Add Up

Appendix

Do On

Wht/BlkBorad

Bruce Mayer, PE

Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu


P5 4 46 b
P5.4-46(b) Add Up

  • Production Rates

  • Cumulative Difference

    • Qtot = 184/3 units

ABC = 184/3