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

1. Chabot Mathematics §6.3 ImproperIntegrals Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 6.2 Review § • Any QUESTIONS About • §6.2 → Numerical Integration • Any QUESTIONS About HomeWork • §6.2 → HW-02

3. Evaluating Improper Integrals • Use LIMITS to Evaluate Improper Integrals. • Given AntiDerivative: • And assuming M, N are real Numbers

4. Example  ImProper Population • The Tasmanian Devil Population on The Isolated Australian Island of Tasmania CHANGES according to the Model • Where • P ≡ the Tasmanian Devil Population in k-Devils • t ≡ the Number Years after Calendar year 2000; i.e; t = tcalendar − 2000

5. Example  ImProper Population • Then Calculate and Explain • SOLUTION: • First Convert to From La Grange Notation to the MORE ILLUMINATING Lebniz form

6. Example  ImProper Population • The first Eqn is a Definite integral of a type calculated many times in MTH15 • By the Transitive Property • So

7. Example  ImProper Population • INTERPRETATION: In the Yr 2001 the TD population will have Increased by about 961 Devils compared to Yr 2000 • Now, assuming the Model Holds over Loooong Time-Scales • AntiDerivating

8. Example  ImProper Population • Or • INTERPRETATION:If the Model holdsfor a Long Periodof Time then theTasmanian Devil Population willSTABILIZE at about 12.5 k-Devils above theY2k Level

9. Common ImProperInteg Limit • Negative Exponentials Often Occur in ImProper Integrals. • A useful limit in these Circumstances: For any Power, p, and Positive Number, k

10. Example  UseFul Limit • Use • Then the AUC for • Find By Repeated Use of • Integration by Parts • The Limit:

11. ImProper Integral Divergence • ImProper Integration Often Times FAILS to Return a Finite Value, that is: • Example: Find the AUC for • Thus this ImProper Integral DIVERGES

12. Example  Double Infinity • Find the value of this ImProper Integral • SOLUTION: • The integral can be divided into TWO separate integrals, EACH containing ONE infinite limit of integration. • From the definition, we choose middle-Limit c = 0 for convenience • Note that c Can be ANY RealNumber

13. Example  Double Infinity • Then the “Split” Integral • Now Engage the SubStitution • Then • And thenthe Limits

14. Example  Double Infinity • Making the SubStitution Can Drop ABS bars as 1 & M & N are POS

15. Example  Double Infinity • Continuing the Reduction • So

16. Example  Double Infinity • But BOTH of the Limits DIVERGE • Since the ∞ is NOT a Number, then the subtraction, ∞ − ∞, is MEANINGLESS • Thus This Expression has NO Number Value

17. Example  Double Disaster • Be CareFul in this Case – It’s Easy to Make a Disastrous Mistake • From the Previous Reduction • Now since M & N are DUMMY Variables it can be written that

18. Example  Double Disaster • Using M=P=N write the Limits • Using above in the Reduction • ReCall (InCompletely) the Limit Property

19. Example  Double Disaster • Apply the Difference-of-Limits Property • Thus One might be Tempted to Say

20. Example  Double Disaster • Q) What’s WRONG with Assessment? • A) ReCall From Limits Properties the Qualifying Statement • IF these Limit EXIST • (only) THEN • In the current case, BOTH individual Limits did NOT exist

21. Example  Double Disaster • Deceptive Plot Suggests Net AUC = 0 • It looks Like The “Equal but Opposite” areas “Cancel Each Other Out”, adding to ZERO →WRONG! PositiveArea NegativeArea Infinite Areas

22. Example  Semi-Infinity • Consider the same Fcn: z = 0→+∞ • By the Same SubStitution • Thus the “Semi-Infinite” ImProper Integral DIVERGES

23. Example  Semi-Infinity Infinite Area Area_Under_Curve_Hatch_Plot_BMayer_1401.mn

24. MuPAD Code fOFx := fOFx := x^3/(x^4 + 1) plot(fOFx, x=-50..50, GridVisible = TRUE,LineWidth = 0.04*unit::inch, Width = 320*unit::mm, Height = 180*unit::mm,AxesTitleFont = ["sans-serif", 24],TicksLabelFont=["sans-serif", 16],BackgroundColor = RGB::colorName([0.8, 1, 1])) Plot the AREA under the Integrand Curve (a very cool plot don't you think...) Need to CHECK Graph Box: Scene2D→BackGroundColor fArea := plot::Function2d(fOFx, x = 0..50, GridVisible = TRUE):plot(plot::Hatch(fArea), fArea, Width = 320*unit::mm, Height = 180*unit::mm,AxesTitleFont = ["sans-serif", 24], TicksLabelFont=["sans-serif", 12],LineWidth = 0.04*unit::inch,BackgroundColor = RGB::colorName([0.8, 1, 1]) )

25. WhiteBoard Work • Problems From §6.3 • P34 → Professorial Endowed Chair • P42 → Waste Accumulation

26. All Done for Today Break atAsymptoteLocation

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

33. Bruce Mayer, PE MTH15 • 13Jan14 Area_Under_Curve_Hatch_Plot_BMayer_1401.mn Plot the Integrand fOFx := ln(x)/x^2 int(ln(x)/x^2, x) AUC = int(ln(x)/x^2, x=1..infinity) AUC = int(ln(x)/x^2, x=1..1E6) Plot the AREA under the Integrand Curve • CHECK Graph Box: Scene2D→BackGroundColor fArea := plot::Function2d(fOFx, x = 1..20, 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]) )