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

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Engr/Math/Physics 25. Chp9: Integration &amp; Differentiation. Bruce Mayer, PE Licensed Electrical &amp; Mechanical Engineer BMayer@ChabotCollege.edu. Learning Goals. Demonstrate Geometrically the Concepts of Numerical Integ. &amp; Diff. Integrals → Trapezoidal, Simpson’s, and Higher-order rules

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Engr/Math/Physics 25

Chp9: Integration

& Differentiation

Bruce Mayer, PE

Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

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

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:
Why Differentiate, Integrate?

Calculation of Geographic Areas

River ChannelCross Section

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
Indefinite Intregral w/ Variable End-PtsIntegral Properties
• Piecewise Property
• Initial/Final Value Formulations

a

c

b

• Linearity → for Constants p & q
PRODUCT Rule

Given

Derivative Properties
• QUOTIENT Rule
• Given
• Then
• Then
Alternative Quotient Rule
• Restate Quotient as rational Exponent, then apply Product rule; to whit:
• Then
• Putting 2nd term over common denom
Numerical IntegrationWhy 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
Numerical Integration
• To Improve Accuracy the TOP of the Strip can Be
• Slanted Lines
• Trapezoidal Rule
• Parabolas
• Simpson’s Rule
• Higher Order PolyNomials
Strip-Top Effect
• Trapezoidal Form
• Parabolic (Simpson’s) Form
• Higher-Order-Polynomial Tops Lead to increased, but diminishing, accuracy.
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

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

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

dy/dx by Discrete-Point Difference
• From Previous LET
• The FORWARD Difference Calc
dy/dx by Discrete-Point Difference
• The BACKWARD Difference Calc
• The CENTRAL Difference Calc
Finite Difference Example

ForwardDifference

Analytical

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

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
Digression
• See pages 333-335 from
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

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

Examine the Integrand fromUnits 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

All Done for Today

Use Trapezoids to approximate the area under the curve:

TrapezoidalRule

n trapezoids

a b

Width, Δx=

Engr/Math/Physics 25

Appendix

Time For

Live Demo

Bruce Mayer, PE

Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

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