[email protected] MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 1 Bruce Mayer, PE Chabot...

[email protected] • MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§8.3 TrigIntegral

Apps



Review §

Any QUESTIONS About• §8. → Trigonometric

Derivatives

Any QUESTIONS About HomeWork• §8.2 → HW-11

8.2



§8.3 Learning Goals

Derive and use integration formulas for trigonometric functions

Apply integrals of periodic functions



Trigonometric AntiDerivatives

Recall the Trig Derivs

Then the Trig AntiDerivatives

uudu

duu

du

dsincoscossin

uuudu

duu

du

dsectansecsectan 2

CuduuCuduu cossinsincos

CuduuuCuduu secsectantansec2



Quick Example Trig AnitDeriv

FindAntiDerivative:

SOLUTION:• There is no formula available for the

immediate AntiDifferentiation of this function, but we observe that the argument of the secant function (i.e., the expression 1/t) has a derivative which is present in the integrand. – This makes SUBSTITUTION a likely choice

dt

ttR t

sec2

12




For the Substitution, let: Next Isolate dt

tu 1

tdt

du

dt

d

tu

dt

d

tu

111

dtt

dudu

tdt

du

tdt

du222

1

1

11

dtduttdtt

du

22

2

1




Substitute for t & dt then Take AntiDerivative

dutt

udt

tt 2

2

2

2

12 sec

sec

duu sec2

Cu tan

Ct 1tan



Example Cyclical Sales

A product is initially quite popular and then settles into cyclical demand. The demand now changes at an instantaneous rate of

• Where– R is the Sales Rate in kUnits per week– t is time in the number of weeks after Product

Introduction

112.0sin1

3

tt

tR




Use the Model to determine How many units are sold in the second month after release (assuming 4.5-week months)

SOLUTION: To find an expression for the total sales

during the second month, find the value of the definite integral over Month-2

9

5.4

2 dttRtS

9

5.4

112.0sin1

3dtt

t




Integrate Term-by-Term

Use TWO Separate Substitutions

9

5.4

9

5.4

9

5.4

1 12.0sin 1

3dtdttdt

t

9

5.4

112.0sin1

32 dtt

tS

dtdudt

dutu 11

dtdv

dt

dvtv

12.012.012.0



Example Cyclical Sale

Then

Performing the Integrations

9 5.4

9

5.4

9

5.4

12.0

sin

32 tdv

vdu

uS

t

t

t

t

5.4912.0

cosln32

9

5.4

9

5.4

t

t

t

t

vuS

5.4

12.0

12.0cos1ln32

9

5.4

9

5.4

t

t

t

t

ttS

5.45.412.0cos912.0cos12.0

115.4ln19ln3



Example Cyclical Sale

Doing the Calculations

So Finally

Thus During the second month, approximately 9,513 items are sold

5.48577.04713.012.0

1705.1302.232 S

513.9 29

5.4

dttRS



Check by MATLAB MuPADIntegrand := 3/(t+1) + sin(12*t/100) + 1

S_of_t := int(Integrand, t)

Snum := numeric::int(3/(t+1) + sin(0.12*t) + 1, t=4.5..9)

Plot the AREA under the Integrand Curve fArea := plot::Function2d(Integrand, t = 4.5..9, GridVisible = TRUE):plot(plot::Hatch(fArea), fArea, Width = 320*unit::mm, Height = 180*unit::mm,AxesTitleFont = ["sans-serif", 24], TicksLabelFont=["sans-serif", 16],LineWidth = 0.04*unit::inch,BackgroundColor = RGB::colorName([0.8, 1, 1]) )



Exponential·Trigonometric

Integration formulas for the Products of Exponentials and Sinusoids:

Cbubbuaba

edubue

auau

cossinsin

22

Cbubbunaba

edubue

auau

sincoscos

22



Example Periodic-Fund F.V.

A study suggests that investment in equity funds varies in part according to the effects of Seasonal Affect Disorder.

ttI

6cos4

A model for the continuous rate of

Investment in a particular market

Where• I(t) ≡ investment

rate in $M/year• t ≡ time in years

after the Spring of Calendar Year 2010




For this Fund Model find the future value of the market’s investments after 10 years for a prevailing interest rate of 4%

SOLUTION: The future value of a continuous income

stream f(t) invested for T years at an annual rate-of-return, r :

Tt

t

rtrT dtetfeTFV0




For T = 10 and r = 0.04 (4%)

10

0

04.0)10(04.0 6

cos410t

t

t dteteFV

10

0

04.004.04.0 6

cos4 dtteee tt

10

0

26

2

04.004.04.0

6sin

66cos04.0

)04.0(04.0

4

tt

eee

tt




Continuing the Calculation

Doing the Arithmetic find:• Thus After 10 years of continuous

investment, the market will accrue about $47,682,000 (compared to the ~$38.3M of its own money that was invested).

10

6sin

610

6cos04.0

)04.0(04.0

42

62

)10(04.0)10(04.04.0

eee

0

6sin

60

6cos04.0

)04.0(04.0

42

62

)0(04.0)0(04.04.0

eee

682.47FV



WhiteBoard Work

Problems From §8.3• P8.3-51 →

Heating Degree Days



All Done for Today

Trig Anti

Derivs2

2

1. sin cos

2. cos sin

3. tan ln cos

4. cot ln sin

5. sec ln sec tan

6. csc ln csc cot

7. sec tan

8. sec tan sec

9. cot csc csc

10. csc cot

udu u c

udu u c

udu u c

udu u c

udu u u c

udu u u c

udu u c

u udu u c

u udu u c

udu u c

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=36QefxUlMAjl9M&tbnid=I89vUC-fN_VV9M:&ved=0CAUQjRw&url=http://www.artofmanliness.com/2012/11/19/how-throw-a-spiral-football/&ei=Io7VUrvACY-hogSr_IKoDA&bvm=bv.59378465,d.cGU&psig=AFQjCNFpdAGRelnFsLiPrVTkxg_YeEefqw&ust=1389813543017809



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

–

srsrsr 22

a2 b2



Plot FunctionHoft := 25 + 22*cos(2*PI*(t-35)/365) plot(Hoft, t =0..365, GridVisible = TRUE,LineWidth = 0.04*unit::inch, Width = 320*unit::mm, Height = 180*unit::mm,AxesTitleFont = ["sans-serif", 24],TicksLabelFont=["sans-serif", 16])



Verify Average Calculation Hoft := 25 + 22*cos(2*PI*(t-35)/365)

Have := int(Hoft, t=0..90)/90

Havenum := float(Have) Plot the H(t) Function over 0→365 daysplot(Hoft, t =0..365, GridVisible = TRUE,LineWidth = 0.04*unit::inch, Width = 320*unit::mm, Height = 180*unit::mm,AxesTitleFont = ["sans-serif", 24],TicksLabelFont=["sans-serif", 16])

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