Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §8.3 TrigIntegral Apps Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 8.2 Review § • Any QUESTIONS About • §8. → TrigonometricDerivatives • Any QUESTIONS About HomeWork • §8.2 → HW-11

3. §8.3 Learning Goals • Derive and use integration formulas for trigonometric functions • Apply integrals of periodic functions

4. Trigonometric AntiDerivatives • Recall the Trig Derivs • Then the Trig AntiDerivatives

5. Quick Example  Trig AnitDeriv • FindAntiDerivative: • SOLUTION: • There is no formula available for the immediate AntiDifferentiationof 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

6. Quick Example  Trig AnitDeriv • For the Substitution, let: • Next Isolate dt

7. Quick Example  Trig AnitDeriv • Substitute for t & dt then Take AntiDerivative

8. 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

9. 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

10. Example  Cyclical Sales • Integrate Term-by-Term • Use TWO Separate Substitutions

11. Example  Cyclical Sale • Then • Performing the Integrations

12. Example  Cyclical Sale • Doing the Calculations • So Finally • Thus During the second month, approximately 9,513 items are sold

13. 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 = 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]) )

14. Exponential·Trigonometric • Integration formulas for the Products of Exponentials and Sinusoids:

15. 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

16. 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 :

17. Example  Periodic-Fund F.V. • For T = 10 and r = 0.04 (4%)

18. 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).

19. WhiteBoard Work • Problems From §8.3 • P8.3-51 → Heating Degree Days

20. All Done for Today Trig AntiDerivs

21. Chabot Mathematics Appendix Do On Wht/BlkBorad Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

29. Plot Function Hoft := 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])

30. 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 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])