1. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §6.3 Improper Integrals

2. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §6.2 → Numerical Integration  Any QUESTIONS About HomeWork §6.2 → HW-02 6.2

3. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 3 Bruce Mayer, PE Chabot College Mathematics Evaluating Improper Integrals  Use LIMITS to Evaluate Improper Integrals.  Given AntiDerivative:  And assuming M, N are real Numbers

4. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 4 Bruce Mayer, PE Chabot College Mathematics 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 = t calendar − 2000

5. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 5 Bruce Mayer, PE Chabot College Mathematics Example  ImProper Population  Then Calculate and Explain  SOLUTION:  First Convert to From La Grange Notation to the MORE ILLUMINATING Lebniz form

6. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 6 Bruce Mayer, PE Chabot College Mathematics Example  ImProper Population  The first Eqn is a Definite integral of a type calculated many times in MTH15  By the Transitive Property  So

7. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 7 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 8 Bruce Mayer, PE Chabot College Mathematics Example  ImProper Population  Or  INTERPRETATION: If the Model holds for a Long Period of Time then the Tasmanian Devil Population will STABILIZE at about 12.5 k-Devils above theY2k Level

9. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 9 Bruce Mayer, PE Chabot College Mathematics Common ImProper Integ Limit  Negative Exponentials Often Occur in ImProper Integrals.  A useful limit in these Circumstances: For any Power, p, and Positive Number, k

10. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  UseFul Limit  Use  Then the AUC for  Find By Repeated Use of Integration by Parts The Limit:

11. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 11 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 12 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  Double Infinity  Then the “Split” Integral  Now Engage the SubStitution  Then And then the Limits

14. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example  Double Infinity  Making the SubStitution Can Drop ABS bars as 1 & M & N are POS

15. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Double Infinity  Continuing the Reduction  So

16. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 16 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 17 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Double Disaster  Using M=P=N write the Limits  Using above in the Reduction  ReCall (InCompletely) the Limit Property

19. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Double Disaster  Apply the Difference-of-Limits Property  Thus One might be Tempted to Say

20. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 20 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Double Disaster  Deceptive Plot Suggests Net AUC = 0 Infinite AreasNegative Area Positive Area  It looks Like The “Equal but Opposite” areas “Cancel Each Other Out”, adding to ZERO → WRONG!

22. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Semi-Infinity  Consider the same Fcn: z = 0→+∞  By the Same SubStitution  Thus the “Semi-Infinite” ImProper Integral DIVERGES

23. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Semi-Infinity Infinite Area Area_Under_Curve_Hatch_Plot_BMayer_1401.mn

24. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 24 Bruce Mayer, PE Chabot College Mathematics 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. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 25 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §6.3 P34 → Professorial Endowed Chair P42 → Waste Accumulation

26. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 26 Bruce Mayer, PE Chabot College Mathematics All Done for Today Break at Asymptote Location

27. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 27 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

28. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 28 Bruce Mayer, PE Chabot College Mathematics

29. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 29 Bruce Mayer, PE Chabot College Mathematics

30. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 30 Bruce Mayer, PE Chabot College Mathematics

31. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 31 Bruce Mayer, PE Chabot College Mathematics

32. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 32 Bruce Mayer, PE Chabot College Mathematics

33. BMayer@ChabotCollege.edu MTH16_Lec-01_sec_6-1_Integration_by_Parts.pptx 33 Bruce Mayer, PE Chabot College Mathematics 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]) )