MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §8.3 Trig Integral Apps

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §8. → Trigonometric Derivatives  Any QUESTIONS About HomeWork §8.2 → HW

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 3 Bruce Mayer, PE Chabot College Mathematics §8.3 Learning Goals  Derive and use integration formulas for trigonometric functions  Apply integrals of periodic functions

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 4 Bruce Mayer, PE Chabot College Mathematics Trigonometric AntiDerivatives  Recall the Trig Derivs  Then the Trig AntiDerivatives

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 5 Bruce Mayer, PE Chabot College Mathematics Quick Example  Trig AnitDeriv  Find AntiDerivative:  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

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 6 Bruce Mayer, PE Chabot College Mathematics Quick Example  Trig AnitDeriv  For the Substitution, let:  Next Isolate dt

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 7 Bruce Mayer, PE Chabot College Mathematics Quick Example  Trig AnitDeriv  Substitute for t & dt then Take AntiDerivative

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 8 Bruce Mayer, PE Chabot College Mathematics 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

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  Cyclical Sales  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

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  Cyclical Sales  Integrate Term-by-Term  Use TWO Separate Substitutions

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 11 Bruce Mayer, PE Chabot College Mathematics Example  Cyclical Sale  Then  Performing the Integrations

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  Cyclical Sale  Doing the Calculations  So Finally  Thus During the second month, approximately 9,513 items are sold

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 13 Bruce Mayer, PE Chabot College Mathematics Check by MATLAB MuPAD Integrand := 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 = , 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]) )

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 14 Bruce Mayer, PE Chabot College Mathematics Exponential·Trigonometric  Integration formulas for the Products of Exponentials and Sinusoids:

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Periodic-Fund F.V.  A study suggests that investment in equity funds varies in part according to the effects of Seasonal Affect Disorder.  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

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  Periodic-Fund F.V.  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 :

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Periodic-Fund F.V.  For T = 10 and r = 0.04 (4%)

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Periodic-Fund F.V.  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).

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 19 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §8.3 P → Heating Degree Days

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 20 Bruce Mayer, PE Chabot College Mathematics All Done for Today Trig Anti Derivs

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 21 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 22 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 23 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 24 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 25 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 26 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 27 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 28 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 29 Bruce Mayer, PE Chabot College Mathematics Plot Function Hoft := *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])

MTH16_Lec-12_sec_8-3_TrigFcn_Integrals.pptx 30 Bruce Mayer, PE Chabot College Mathematics Verify Average Calculation Hoft := *cos(2*PI*(t-35)/365) Have := int(Hoft, t=0..90)/90 Havenum := float(Have) Plot the H(t) Function over 0→365 days 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])