1. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §5.1 Integration

2. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §4.4 → Exp & Log Math Models  Any QUESTIONS About HomeWork §4.4 → HW-21 4.4

3. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 3 Bruce Mayer, PE Chabot College Mathematics §5.1 Learning Goals  Define AntiDerivative  Study and compute indefinite integrals  Explore differential equations and Initial/Boundary value problems  Set up and solve Variable- Separable differential equations

4. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 4 Bruce Mayer, PE Chabot College Mathematics Fundamental Theorem of Calculus  The fundamental theorem* of calculus is a theorem that links the concept of the derivative of a function with the concept of the integral. Part-1: Definite Integral (Area Under Curve) Part-2: AntiDerivative * The Proof is Beyond the Scope of MTH15

5. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 5 Bruce Mayer, PE Chabot College Mathematics AntiDifferentiation  Using the 2 nd Part of the Theorem  F(x) is called the AntiDerivative of f(x) Example: Find f(x) when ONE Answer is As Verified by

6. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 6 Bruce Mayer, PE Chabot College Mathematics Fundamental Property of Antiderivs  The Process of Finding an AntiDerivavite is Called: InDefinite Integration  The Fundamental Property of AntiDerivatives: If F(x) is an AntiDerivative of the continuous fcn f(x), then any other AntiDerivative of f(x) has the form G(x) = F(x) + C, for some constant C

7. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 7 Bruce Mayer, PE Chabot College Mathematics Fundamental Property of Antiderivs  Proof of G(x) = F(x) + C  Assertion: both G(x) & F(x)+C are AntiDerivatives of f(x); that is:  Using Derivative Rules Derivative of a Sum Derivative of a Const  Transitive Property

8. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 8 Bruce Mayer, PE Chabot College Mathematics The Indefinite Integral  The family of ALL AntiDerivatives of f(x) is written  The result of ∫f(x)dx is called the indefinite integral of f(x)  Quick Example for: u(x) has in INFINITE NUMBER of Results, Two Possibilities:

9. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 9 Bruce Mayer, PE Chabot College Mathematics The Meaning of “C”  The Constant, C, is the y-axis “Anchor Point” for the “natural Response” fcn F(x) for which C = 0. C is then the y-intercept of F(x)+C; i.e.,  Adding C to F(x) creates a “family” of functions, or curves on the graph, with the SAME SHAPE, but Shifted VERTICALLY on the y-axis

10. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 10 Bruce Mayer, PE Chabot College Mathematics The Meaning of “C” Graphically

11. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 11 Bruce Mayer, PE Chabot College Mathematics MATLAB Code MATLAB Code % Bruce Mayer, PE % MTH-15 20Jul13 % XYfcnGraph6x6BlueGreenBkGndTemplate1306.m % % The Limits xmin = -4; xmax = 4; ymin = -10; ymax = 20; % The FUNCTION x = linspace(xmin,xmax,1000); y = 7*exp(-x/2.5) + 5*x -8; % % The ZERO Lines zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax]; % % the 6x6 Plot axes; set(gca,'FontSize',12); whitebg(['white']) % whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green plot(x,y, x,y+9,x,y-pi,x,y+sqrt(13),x,y-7, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = G(x) = F(x)+C = 7e^-^5^x^/^2 + 5x - 8 + C'),... title(['\fontsize{16}MTH15 Familiy of AntiDerivatives',]),... annotation('textbox',[.71.05.0.1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'B. Mayer 20Jul13','FontSize',7) hold on plot(zxv,zyv, 'k', zxh,zyh, 'k', [-1.4995, -1.4995], [ymin,ymax], '--m', 'LineWidth', 2) set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax])

12. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 12 Bruce Mayer, PE Chabot College Mathematics MuPAD Code MuPAD Code Bruce Mayer, PE MTH15 20Jul13 F(x) = 7*exp(-2*x/5) + 5*x -8 f(x) = int(G, x) G := 7*exp(-2*x/5) + 5*x -8 dgdx := diff(G, x) assume(x > -6): xmin := solve(dgdx, x) xminNo := float(xmin) Gmin := subs(G, x = xmin) GminNo := float(Gmin) plot(G, x=-4..4, GridVisible = TRUE, LineWidth = 0.04*unit::inch)

13. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 13 Bruce Mayer, PE Chabot College Mathematics Evaluating C by Initial/Boundary  A number can be found for C if the situation provides a value for a SINGLE known point for G(x) → (x, G(x)); e.g., (x n, G(x n )) = (73.2, 4.58) For Temporal (Time-Based) problems the known point is called the INITIAL Value –Called Initial Value Problems For Spatial (Distance-Based) problems the known point is called the BOUDARY Value –Called Boundary Value Problems –

14. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 14 Bruce Mayer, PE Chabot College Mathematics Common Fcn Integration Rules 1.Constant Rule: for any constant, k 2.Power Rule: for any n ≠ −1 3.Logarithmic Rule: for any x ≠ 0 4.Exponential Rule: for any constant, k

15. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 15 Bruce Mayer, PE Chabot College Mathematics Integration Algebra Rules 1.Constant Multiple Rule: For any constant, a 2.The Sum or Difference Rule: This often called the Term-by-Term Rule

16. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  Use the Rules  Find the family of AntiDerivatives corresponding to  SOLUTION:  First Term-by-Term → break up each term over addition and subtraction:

17. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Use the Rules  Move out the constant in the 2 nd integral (2), and state sqrt as fractional power  Using the Power Rule  Cleaning Up →

18. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Propensity to Consume  The propensity to consume (PC) is the fraction of income dedicated to spending (as opposed to saving).  A Math Model for the marginal propensity to consume (MPC) for a certain population: Where –MPC is the rate of change in PC –x is the fraction of income that is disposable.

19. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Propensity to Consume  If the propensity to consume is 0.8 when disposable income is 0.92 of total income, find a formula for PC(x)  SOLUTION:  From the Problem Statement that the MPC is a marginal function discern that  Thus the PC fcn is the AntiDerivative of MPC(x)

20. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Propensity to Consume

21. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Propensity to Consume  Use the (x,PC) = (0.92,0.8) Boundary Value to Find a NUMBER for the Constant of Integration, C  With C ≈ 1.4, state the particular solution to this Boundary Value Problem

22. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 22 Bruce Mayer, PE Chabot College Mathematics Differential Equations (DE’s)  A Differential Equation is an equation that involves differentials or derivatives, and a function that satisfies such an equation is called a solution  A Simple Differential Equation is an equation which includes two differentials in the form of a derivative

23. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 23 Bruce Mayer, PE Chabot College Mathematics Differential Equations (DE’s)  For some function f. Such a Simple Differential Equation can be solved by integrating:  In summary the Solution, y, to a Simple DE can be found by the integration

24. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  Simple DE  From the Previous Example  As previously solved for the general solution by Integration:  Then used the Boundary Value, (0.92, 0.8), to find the Particular Solution

25. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 25 Bruce Mayer, PE Chabot College Mathematics Variable-Separable DE’s  A Variable Separable Differential equation is a differential equation of the form For some integrable functions f and g  Such a differential equation can be solved by separating the single-variable functions and integrating:

26. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  Fluid Dynamics  The rate of change in volume (in cubic centimeters) of water in a draining container is proportional to the square root of the depth (in cm) of the water after t seconds, with constant of proportionality 0.044.  Find a model for the volume of water after t seconds, given that initially the container holds 400 cubic centimeters.

27. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 27 Bruce Mayer, PE Chabot College Mathematics Example  Fluid Dynamics  SOLUTION:  First, TRANSLATE the written description into an equation: “rate of change in volume” “is proportional to the square root of volume” “with constant of proportionality equal to 0.044”

28. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 28 Bruce Mayer, PE Chabot College Mathematics Example  Fluid Dynamics  So the (Differential) Equation  Note that the right side does not explicitly depend on t, so we can’t simply integrate with respect to t. Instead move the expression containing V to the left side:  The Variables are now Separated, allowing simple integration

29. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example  Fluid Dynamics  Integrating  Where  Squaring Both Sides Find:

30. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example  Fluid Dynamics  For The particular solution find the a number for C using the Initial Value: when t = 0, V = 400 cc: Sub (0,400) into DE Solution  Thus the volume of water in the Draining Container as a fcn of time:

31. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 31 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §5.1 P58 → Oil Production (not a Gusher…) P73 → Car Stopping Distance

32. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 32 Bruce Mayer, PE Chabot College Mathematics All Done for Today LOTS more on DE’s in MTH25

33. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 33 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

34. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 34 Bruce Mayer, PE Chabot College Mathematics ConCavity Sign Chart abc −−−−−−++++++−−−−−−++++++ x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points InflectionNO Inflection Inflection

35. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 35 Bruce Mayer, PE Chabot College Mathematics

36. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 36 Bruce Mayer, PE Chabot College Mathematics

37. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 37 Bruce Mayer, PE Chabot College Mathematics

38. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 38 Bruce Mayer, PE Chabot College Mathematics

39. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 39 Bruce Mayer, PE Chabot College Mathematics

40. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 40 Bruce Mayer, PE Chabot College Mathematics

41. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 41 Bruce Mayer, PE Chabot College Mathematics

42. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 42 Bruce Mayer, PE Chabot College Mathematics

43. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 43 Bruce Mayer, PE Chabot College Mathematics

44. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 44 Bruce Mayer, PE Chabot College Mathematics

45. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 45 Bruce Mayer, PE Chabot College Mathematics

46. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 46 Bruce Mayer, PE Chabot College Mathematics

47. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 47 Bruce Mayer, PE Chabot College Mathematics

48. BMayer@ChabotCollege.edu MTH15_Lec-22_sec_5-1_Integration.pptx 48 Bruce Mayer, PE Chabot College Mathematics