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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]PowerPoint Presentation

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

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Review §

- Any QUESTIONS About
- §5.3 → Fundamental Theorem and Definite Integration

- Any QUESTIONS About HomeWork
- §5.3 → HW-24

§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

- 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

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 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 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 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 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

- ∆tn = tn+1 − tn

Example Add Up NONconstant∆x

- The GamePlanGraphically
- Note the VariableWidth, ∆x,of the StripTops

% 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 CodeExample 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 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 Add UpArea Between Curves

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

% 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 CodeMuPAD 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 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 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 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 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 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 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 Add Up Net Excess Profit

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

% 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 CodeRecall: 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 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

- Where
- What is the average rate at which sediment gathers during the first six months of study?
- )

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 Add Up GeoTech Engineering

- And
- Then the Transformed Integral
- Working the Calculus

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 Add Up Work

- Problems From §5.4
- P46 → Worker Productivity
- P60 → Cardiac Fluidic Mechanics

All Done for Today Add Up

DilBertIntegration

Chabot Mathematics Add Up

Appendix

Do On

Wht/BlkBorad

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

–

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