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

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Chabot Mathematics. §5.4 Definite Integral Apps. Bruce Mayer, PE Licensed Electrical &amp; Mechanical Engineer [email protected] 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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Chabot Mathematics

§5.4 DefiniteIntegral Apps

Bruce Mayer, PE

Licensed Electrical & Mechanical [email protected]

5.3

Review §
• §5.3 → Fundamental Theorem and Definite Integration
• §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
Strip IntegrationNeed 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
• To Improve Accuracy the
• TOP of the Strip can Be
• Slanted Lines
• Trapezoidal Rule
• Parabolas
• Simpson’s Rule
• Higher Order PolyNomials
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.
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.
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
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
Example  NONconstant∆x
• The GamePlanGraphically
• Note the VariableWidth, ∆x,of the StripTops

% 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
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
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

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

=

Example Area Between Curves
• Do the Math →

≈ 70.20

% 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

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
• 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 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 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 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 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 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 Profit
• STATE: In the initial 161 days, the Profit from plan-1 exceeded that of plan-2 by approximately \$80k
Example  Net Excess Profit
• The Profit Rates
• The NEP (ABC)

Area Between Curves

% 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
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:
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?
• )
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
Example  GeoTech Engineering
• And
• Then the Transformed Integral
• Working the Calculus
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

WhiteBoard Work
• Problems From §5.4
• P46 → Worker Productivity
• P60 → Cardiac Fluidic Mechanics
All Done for Today

DilBertIntegration

Chabot Mathematics

Appendix

Do On