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

1. Engr/Math/Physics 25 Chp9: Integration & Differentiation Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. Learning Goals • Demonstrate Geometrically the Concepts of Numerical Integ. & Diff. • Integrals → Trapezoidal, Simpson’s, and Higher-order rules • Derivative → Finite Difference Methods • Use MATLAB to Numerically Evaluate Math/Data Integrals • Use MATLAB to Numerically Evaluate Math/Data Derivatives

3. We encounter differentiation and integration on a Daily Basis Differentiation: Many Important Physical processes/phenomena are best Described in Derivative form; Some Examples Why Differentiate, Integrate? • Newton’s 2nd Law: • Heat Flux: • Drag on a Parachute: • Capacitor Current:

4. Integration: Integration is commonplace in Science and Engineering Why Differentiate, Integrate? Calculation of Geographic Areas River ChannelCross Section Wind-ForceLoading

5. Integration: the area under the curve described by the function f(x) with respect to the independent variable x, evaluated between the limits x = a to x = b. Review: Integration

6. Differentiation: rate of change of a dependent variable with respect to an independent variable. Review: Differentiation

7. Indefinite Intregral w/ Variable End-Pts Integral Properties • Piecewise Property • Initial/Final Value Formulations a c b • Linearity → for Constants p & q

8. PRODUCT Rule Given Derivative Properties • QUOTIENT Rule • Given • Then • Then

9. Alternative Quotient Rule • Restate Quotient as rational Exponent, then apply Product rule; to whit: • Then • Putting 2nd term over common denom

10. Numerical Integration Why Numerical Methods? • Very often, the function f(x) to differentiate, or the integrand to integrate, is TOO COMPLEX to yield exact analytical solutions. • In most cases in engineering testing, the function f(x) is only available in a TABULATED form with values known only at DISCRETE POINTS

11. Game Plan: Divide Unknown Area into Strips (or boxes), and Add Up Numerical Integration • To Improve Accuracy the TOP of the Strip can Be • Slanted Lines • Trapezoidal Rule • Parabolas • Simpson’s Rule • Higher Order PolyNomials

12. Strip-Top Effect • Trapezoidal Form • Parabolic (Simpson’s) Form • Higher-Order-Polynomial Tops Lead to increased, but diminishing, accuracy.

13. Strip-Count Effect 10 Strips 20 Strips • Adaptive Integration → INCREASE the strip-Count in Regions with Large SLOPES • More Strips of Constant Width Tends to work just as well

14. y(x) y(x-Δx) y(x) y(x) y(x+Δx) dy/dx by Finite Difference Approx. • Derivative at Point-x : • Forward Difference mbkwd • Backward Difference mfwd

15. y(x) y(x-Δx) y(x) y(x) y(x+Δx) dy/dx by Finite Difference Approx. • Central Difference = Average of fwd and bkwd Slopes : mcent

16. dy/dx by Discrete-Point Difference • From Previous LET • The FORWARD Difference Calc

17. dy/dx by Discrete-Point Difference • The BACKWARD Difference Calc • The CENTRAL Difference Calc

18. Finite Difference Example ForwardDifference Analytical

19. Discrete Point dy/dx

20. Compare Fwd, Bkwd, Cent Diffs

21. Finite Difference Fence-Post Errors • If we have data vectors for x & f(x) we can calc m = df(x)/dx by the Fwd, Bkwd or Central Difference methods • If there are 1 to n Data points then can NOT calc • mfwd for pt-n (cannot extend fwd beyond n-1) • mbk for pt-1 (cannot extend bkwd beyond 1) • mcnt for pt-1 and pt-n (cannot extend bk beyond 1, cannot extend fwd beyond n) Time For Live Demo

22. 1.0 mF + v(t) - i(t) Cap Voltage – Integrate & Plot

23. Cap Charging • The Current can Be integrated Analytically to find v(t), but it’s Painful • Let’s Tackle The Problem Numerically • Use the PieceWise Property

24. Digression • For More Info on • See pages 333-335 from

25. PieceWise Integration

26. PieceWise Integration Illustrated

27. Cap ChrgPieceWise Integration • Game Plan • Make Function for i(t)/C • Divide 300 mS interval into 1 mS pieces • Use 1-300 FOR Loop to collect • Vector for Time-Plot • Use ΔV summation to Create a V-Plotting Vector Time For Live Demo • File List • Fcn → iOverC_CapCharge.m • Calc& Plot → Cap_Charge_Soln_1111.m

28. function [Cap_Charge] = iOverC_CapCharge(time) Cap_Charge = (1/0.001)*(10 + 300*exp(-5*time).*sin(25*pi*time))/1000; % Cap Charge for Prob for Chp9 in COULOMBS File Codes % B. Mayer 08Nov11 % Cap Charging: Piecewise Ingegration % Cap_Charge_Soln_1111.m % % use 500 pts using LinSpace % => Ask user for max time tmax = input('Enter Max Time in Sec = ') tmin = 0; n = 500; t = linspace(tmin,tmax,n); % in Sec TimePts =length(t) % 2X check number of time points % % Initalize the Vminus1 & Plotting Vectors Vminus1 = 0; Vplot = 0; tplot = 0; % % Use FOR Loop with Lobratto Integrating quadl function on Cap Charge % Function for k = 1:n-1 tplot(k) = t(k); del_v(k) = quadl('iOverC_CapCharge', t(k), t(k+1)); % The Incremental Area Under the Curve; can be + or - Vplot(k) = Vminus1 + del_v(k); Vminus1 = Vplot(k); end plot(1000*tplot, del_v), xlabel('time (mS)'), ylabel('DelV (V)'),... title('Capacitor Voltage PieceWise Integral'), grid disp('Showing del_v PLOT - hit any key to show V(t) plot') pause plot(1000*tplot, Vplot), xlabel('time (mS)'), ylabel('Cap Potential (V)'),... title('Capacitor Voltage'), grid

29. Examine the Integrand from Units Analysis • A → A (a base unit) • S → S (a base unit) • F → m−2•kg−1•S4•A2 • V → m2•kg•S−3•A−1 • The Integrand Units • Or • Recall From ENGR10 A, S, & F in SI Base Units • But 

30. Result

31. All Done for Today Use Trapezoids to approximate the area under the curve: TrapezoidalRule n trapezoids … a b Width, Δx=

32. Engr/Math/Physics 25 Appendix Time For Live Demo Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

33. dy/dx example x = [1.215994, 2.263081, 3.031708, 4.061534, 5.122477, 6.12396, 7.099754, 8.070701, 9.215382, 10.04629, 11.16794, 12.22816, 13.02504, 14.13544, 15.20385, 16.01526] y = [0.381713355 1.350058777 1.537968679 2.093069052 1.002924647 1.123878013 7.781303297 14.2596343 13.96413795 4.352973409 51.45863097 22.85918559 100.8729773 106.5041434 34.15277499 134.2488143] plot(x,y),xlabel('x'), ylabel('y'), grid