[email protected] MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 1 Bruce Mayer, PE Chabot...

[email protected] • MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§5.4 DefiniteIntegral

Apps



Review §

Any QUESTIONS About• §5.3 → Fundamental Theorem and

Definite Integration

Any QUESTIONS About HomeWork• §5.3 → HW-24

5.3

http://www.google.com/url?sa=i&rct=j&q=definite+integration&source=images&cd=&cad=rja&docid=CZSu6JkaLdUSsM&tbnid=qsr0KwSrzI2XxM:&ved=0CAUQjRw&url=http://ouchmath.wordpress.com/2011/02/10/the-definite-integral/&ei=ZU_xUdOLFqH9igLWnYGYCA&bvm=bv.49784469,d.cGE&psig=AFQjCNHlQPo-Up3tfg4Ih-0K3bKYezdV3Q&ust=1374855357618384



§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

• 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

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 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-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.

0 50 100 150 200 250 300 3500

50

100

150

200

250

time, t (min)

Pow

er C

onsu

mpt

ion,

P (

kW)

Furnace Power Consumption




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 theTotal Energy in MJ expended byThe Furnace during this processrun

Time (min)

Power (kW)

0 47 24 107 45 104 74 146 90 126

118 178 134 147 169 211 180 151 218 233 229 184 265 222 287 180 340 247




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

Time (min)

Power (kW)

0 47 24 107 45 104 74 146 90 126

118 178 134 147 169 211 180 151 218 233 229 184 265 222 287 180 340 247




The GamePlan Graphically• Note the

VariableWidth, ∆x,of the StripTops

t (minutes)

P (

kW)

MTH15 • Variable-Width Strip-Integration

0 50 100 150 200 250 300 3500

25

50

75

100

125

150

175

200

225Bruce May er, PE • 25Jul13

4x

9x



MA

TL

AB

Co

de

% Bruce Mayer, PE% MTH-15 • 25Jul13% XY_Area_fcn_Graph_6x6_BlueGreen_BkGnd_Template_1306.m%clear; clc; clf; % clf is clear figure%% The FUNCTIONxmin = 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 Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green% Now make AREA Plotarea(x,y,'FaceColor',[1 0.6 1],'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',[.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')




The NONconstant Strip-Width Integration is conveniently done in an Excel SpreadSheet

The 13 ∆Q strips Add up to 3456.69 MegaJoules of Total Energy Expended

n Time, t Power ∆t = 60*(tn+1-tn) Pavg=(Pn+1−Pn)/2 ∆Q= Pavg*∆t

(cnt) (min) (kW) (Sec) (kW) (kJ)

1 0 47

1 1440 77 110880

2 24 107

2 1260 105.5 132930

3 45 104

3 1740 125 217500

4 74 146

4 960 136 130560

5 90 126

5 1680 152 255360

6 118 178

6 960 162.5 156000

7 134 147

7 2100 179 375900

8 169 211

8 660 181 119460

9 180 151

9 2280 192 437760

10 218 233

10 660 208.5 137610

11 229 184

11 2160 203 43848012 265 22212 1320 201 26532013 287 18013 3180 213.5 67893014 340 247

3456.69Total Energy in MJ = (∑∆Q)/1000 =



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

( ) ( )b

af x g x dx

( ) ( ) on , .f x g x a b ( )y g x

( )y f x

a b

R

x

y



Example Area Between Curves

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

The Graphsof f & g

10

25

58and

911

2

6

xxg

exf x

0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

x

ylo

= (

-8/2

5)*

(x-5

)2 +1

0 •

yh

i = 1

1e-x

/6+

9

MTH15 • Area Between Curves

Bruce May er, PE • 25Jul13

911 6 xexf

10

25

58 2

x

xg




The process Graphically

x

ylo

= (

-8/2

5)*

(x-5

)2+

10

• y

hi =

11

e-x/

6 +9


0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20

x0 2 4 6 8 10

0

2

4

6

8

10

12

14

16

18

20

x0 2 4 6 8 10

0

2

4

6

8

10

12

14

16

18

20

Bruce May er, PE • 25Jul13Bruce May er, PE • 25Jul13 Bruce May er, PE • 25Jul13

− = 10

0

6 911 xe x dx

x

10

0

2

25

5810

10

0dxxgxf




Do the Math → 10

0dxxgxf

xf xg

xgxf

dxxgxf

10

0dxxgxf

≈ 70.20




ThusAns

x

ylo

= (

-8/2

5)*

(x-5

)2+

10

• y

hi =

11

e-x/

6 +9


0 2 4 6 8 100

2

4

6

8

10

12

14

16

18

20


A = 70.200



MA

TL

AB

Co

de

% Bruce Mayer, PE% MTH-15 • 25Jun13%clear; clc; clf; % clf clears figure window%% The Limitsxmin = 0; xmax = 10; ymin = 0; ymax = 20;% The FUNCTIONx = linspace(xmin,xmax,500); y1 = (-8/25)*(x-5).^2 + 10; y2 = 11*exp(-x/6)+9;% % the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greensubplot(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',[.5 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7)hold onset(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',[.15 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7)hold onset(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',[.85 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7)hold onset(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%clffill([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',[.6 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7)hold onset(gca,'XTick',[xmin:2:xmax]); set(gca,'YTick',[ymin:2:ymax])set(gca,'Layer','top')hold off



MuPAD Code

f := 11*exp(-x/6)+9g := (-8/25)*(x-5)^2+10fminusg := f-gAntiDeriv := 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.

b

a

b

adtdtdPdtdPdttPtP 2121 ''




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




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

tedt

dP 01.01 500

tedt

dP 02.02 100




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

b

adtdtdPdtdPNEP 21

tt dt

dP

dt

dP 21

tt

tt

eedt

dP

dt

dP 02.001.021 100500




Dividing Both Sides of the InEquality

Taking the Natural Log of Both Side

Divide both Sides by 0.01 to Solve for t

tttt

tt

eee

ee 01.001.002.001.0

02.001.0

5100

100500

te t 01.05ln5ln 01.0

94.16001.0

5ln

01.0

5ln tt




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

dteeNEP tt days 161

days 0

02.001.0 100500

161

0

02.001.0161

0

02.001.0

02.0

100

01.0

500100500

tttt eedtee




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

)0(02.0)0(01.0)161(02.0)161(01.0

02.0

100

01.0

500

02.0

100

01.0

500eeee

96.999,79



Example Net Excess Profit The Profit Rates The NEP (ABC)

0 20 40 60 80 100 120 140 1600

0.5

1

1.5

2

2.5

t (days)

P1'=

0.5

ex/

100 •

P2' =

0.5

ex/

50 (

$k)

MTH15 • Net Excess Profit


0 20 40 60 80 100 120 140 1600

0.5

1

1.5

2

2.5

t (days)

P1'=

0.5

ex/

100

• P

2' = 0

.5e

x/50

($

k)

MTH15 • Net Excess Profit


dtdP1

dtdP2Area Between Curves



MA

TL

AB

Co

de

% Bruce Mayer, PE% MTH-15 • 25Jun13%clear; clc; clf; % clf clears figure window%xmin = 0; xmax = 161; ymin = 0; ymax = 2.5;% The FUNCTIONx = linspace(xmin,xmax,500); y1 = .5*exp(x/100); y2 = .1*exp(x/50);% x in days • y's in $k%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(x,y1, x,y2, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}t (days)'), ylabel('\fontsize{14} P_1''= 0.5e^x^/^1^0^0 • P_2'' = 0.5e^x^/^5^0^ ($k)'),... title(['\fontsize{16}MTH15 • Net Excess Profit',]),... annotation('textbox',[.6 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7)hold onset(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



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:

1( )

b

af x dx

b a

3/ 2( ) over 0,9 .f x x

9 3/ 2

0

1

9 0x dx

9

5/ 2

0

1 2

9 5

x

5/ 229

45 54

5



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?

)

32

3

tdt

dM




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

5.0

0

32

3

05.0

1

1dt

tVdttf

abV

b

a

32 tu

22

2232du

dtdt

dt

du

dt

dutu

dt

d




And

Then the Transformed Integral

Working the Calculus

43135.025.0

330302032

u

utu

4

3

5.0

0 2

3

05.0

1

32

3

05.0

1 u

u

t

t

du

uVdt

tV

434

3

4

3ln3

2

32

2

32 u

u

du

u

duV

8630.02877.033

4ln33ln4ln3

V




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

http://www.google.com/url?sa=i&rct=j&q=worker%20productivity&source=images&cd=&cad=rja&docid=QWg0k44HmdljJM&tbnid=wLWORVShiJuTyM:&ved=0CAUQjRw&url=http://www.bloomberg.com/news/2010-08-10/worker-productivity-in-u-s-unexpectedly-fell-in-second-quarter.html&ei=AOvyUdKTL4XJiQLJ9ICQBg&bvm=bv.49784469,d.cGE&psig=AFQjCNGPzwhqa2SoeXD2n6E8y-vfGrre8A&ust=1374960688058023

http://www.google.com/url?sa=i&rct=j&q=heart%20blood%20flow%20diagram&source=images&cd=&cad=rja&docid=qKBCcK9ZbAwohM&tbnid=_rFjcd5U9Ji2DM:&ved=0CAUQjRw&url=http://seys-science.wikispaces.com/4+How+does+blood+flow+through+our+heart+and+body?&ei=RuvyUfyYJOH8iwLF7ICoBQ&bvm=bv.49784469,d.cGE&psig=AFQjCNGzsLrzcq_wkPdEjJbqLPd-YQ_h3g&ust=1374960809391721



All Done for Today

DilBertIntegration

http://www.google.com/url?sa=i&rct=j&q=strip%20integration&source=images&cd=&cad=rja&docid=Rj-z7w1aXpSnHM&tbnid=tP3pYcxNl4oHWM:&ved=0CAUQjRw&url=http://dilbert.com/strips/comic/2011-03-23/&ei=CuryUfq4JuKWiAKX8oGoCg&bvm=bv.49784469,d.cGE&psig=AFQjCNEBLLbc8d8yQPj0hZyTvLuGlvV6kg&ust=1374960449859507



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22



http://www.google.com/url?sa=i&rct=j&q=worker%20productivity&source=images&cd=&cad=rja&docid=eCv46ZG02z7LSM&tbnid=nUaEZsuFOy_oiM:&ved=0CAUQjRw&url=http://www.hellotrade.com/blog/small-business-channels/small-business-labor-hr/&ei=HvTxUb6GOqmLjALp3IGICw&bvm=bv.49784469,d.cGE&psig=AFQjCNGmJqqTgVN9uSD_0XabrChpTnXJQw&ust=1374897516863763





P5.4-46(b) Production Rates Cumulative Difference

• Qtot = 184/3 units

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

70

t (Hrs after 8am)

Q1'=

60

-2(t

-1)2 •

Q2' =

50

-5t

(un

its/h

r)

MTH15 • P5.4-46


0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

70

t (Hrs after 8am)

Q1'=

60

-2(t

-1)2 •

Q2' =

50

-5t

(un

its/h

r)

MTH15 • P5.4-46


ABC = 184/3



http://www.google.com/url?sa=i&rct=j&q=systole&source=images&cd=&cad=rja&docid=xx2ZX1h9guKF3M&tbnid=UmV_OKZK95B9hM:&ved=0CAUQjRw&url=http://faculty.stcc.edu/AandP/AP/AP2pages/Units18to20/heart/cardiac.htm&ei=ugDyUcWQM4j7iwLFhICwDw&bvm=bv.49784469,d.cGE&psig=AFQjCNE-xqAFCJ20DoI0JwSKkpU_E1-udg&ust=1374900785575829













http://www.google.com/url?sa=i&rct=j&q=derivative+power+rule&source=images&cd=&cad=rja&docid=NjbfKj_JZBSzwM&tbnid=jJkxz7X095T-NM:&ved=0CAUQjRw&url=http://mathworld.wolfram.com/Derivative.html&ei=TfPWUe_nA6akiQKZ64DIDw&bvm=bv.48705608,d.cGE&psig=AFQjCNGw-AM0Kc2tARyUGs-zPM7iK-qH3g&ust=1373127702659906

http://www.google.com/url?sa=i&rct=j&q=related+rates+calculus&source=images&cd=&cad=rja&docid=nuuUeFKHM5pqjM&tbnid=x7raEhpCGJ8edM:&ved=0CAUQjRw&url=http://designatedderiver.wikispaces.com/Games+and+fun+stuff&ei=IzPcUYPxBaOLiwLjs4GYAQ&bvm=bv.48705608,d.cGE&psig=AFQjCNH42r6LfHpjrPyJXK2DJrvdwHWZIQ&ust=1373471717102765

http://www.google.com/url?sa=i&rct=j&q=systole&source=images&cd=&cad=rja&docid=xx2ZX1h9guKF3M&tbnid=UmV_OKZK95B9hM:&ved=0CAUQjRw&url=http://faculty.stcc.edu/AandP/AP/AP2pages/Units18to20/heart/cardiac.htm&ei=ugDyUcWQM4j7iwLFhICwDw&bvm=bv.49784469,d.cGE&psig=AFQjCNE-xqAFCJ20DoI0JwSKkpU_E1-udg&ust=1374900785575829

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[email protected] MTH15_Lec-25_sec_5-4_Definite_Integral_Apps.pptx 1 Bruce Mayer, PE Chabot...

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