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

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. Why Differentiate, Integrate?
We encounter differentiation and integration on a Daily Basis
Differentiation: Many Important Physical processes/phenomena are best Described in Derivative form; Some Examples
Newton’s 2nd Law:
Heat Flux:
Typical derivative are SPATIAL (x), or TEMPORAL (t) => Space or time derivatives
Drag on a Parachute:
Capacitor Current:

4. Why Differentiate, Integrate?
Integration: Integration is commonplace in Science and Engineering
Calculation of Geographic Areas
River Channel Cross Section
Wind-Force Loading

5. Review: Integration
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.

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

7. Integral Properties Indefinite Intregral w/ Variable End-Pts
Piecewise Property
Initial/Final Value Formulations a c b
Linearity → for Constants p & q
Linearity =>
Take scaling constants OUTside the integral
Term-by-Term integration for addends

8. Derivative Properties
PRODUCT Rule
Given
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. Why Numerical Methods? Numerical Integration
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. Numerical Integration
Game Plan: Divide Unknown Area into Strips (or boxes), and Add Up
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. dy/dx by Finite Difference Approx.
Derivative at Point-x :
y(x)
y(x-Δx)
y(x+Δx)
Forward Difference
mbkwd
Backward Difference
mfwd

15. dy/dx by Finite Difference Approx.
Central Difference = Average of fwd and bkwd Slopes :
y(x)
y(x-Δx)
y(x+Δx)
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
Forward Difference
Analytical
Note Variable Spacing

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)
Prob9_17_Soln_1204.m * Also DEMO for diff command => u = primes(37) & delU = diff(u)
Time For
Live Demo

22. Cap Voltage – Integrate & Plot
i(t)
+ v(t) -
1.0 mF
Similar to 3e text problem 9.11 ** Prob8_14_Plot_i.m

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
Class Q => Based on top Eqn, what is v( s)? => V * v(t) decays to LINE with slope of 10 V/S, and intercept 3.804V

24. Digression
For More Info on
See pages from

25. PieceWise Integration

26. PieceWise Integration Illustrated

27. Cap Chrg PieceWise Integration
Game Plan
Make Function for i(t)/C
Divide 300 mS interval into 1 mS pieces
Use FOR Loop to collect
Vector for Time-Plot
Use ΔV summation to Create a V-Plotting Vector
Time For
Live Demo
Also: Cap_Charge_Anon_Fcn_Soln_1111.m
File List
Fcn → iOverC_CapCharge.m
Calc & Plot → Cap_Charge_Soln_1111.m

28. File Codes function [Cap_Charge] = iOverC_CapCharge(time)
Cap_Charge = (1/0.001)*( *exp(-5*time).*sin(25*pi*time))/1000;
% Cap Charge for Prob for Chp9 in COULOMBS
% 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
File Codes

29.  Units Analysis Examine the Integrand from The Integrand Units Or
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. Trapezoidal Rule … All Done for Today
Use Trapezoids to approximate the area under the curve:
Trapezoidal Rule
n trapezoids …
a b
Width, Δx =

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

33. dy/dx example
x = [ , , , , , , , , , , , , , , , ]
y = [ ]
plot(x,y),xlabel('x'), ylabel('y'), grid