1. §6.2 Numerical Integration
Chabot Mathematics
§6.2 Numerical Integration
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 6.1 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
§6.1 → Integration by Parts, Use of Integral Tables
Any QUESTIONS About HomeWork
§6.1 → HW-01

3. §6.2 Learning Goals
Explore the trapezoidal rule and Simpson’s rule for numerical integration
Use error bounds for numerical integration
Interpret data using numerical integration

4. 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 Real World testing, the function f(x) is only available in a TABULATED form with values known only at DISCRETE POINTS

5. 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

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

7. 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

8. AUC by Flat Tops

9. Trapezoidal Area By the Diagram at Right
Side Heights:
Width:
Now “Stack Up” for rectangle of area 2A
Then or
The area of a straight-sided Trapezoid = [Average Height]x[Width]

10. AUC by Trapezoids

11. The Trapezoidal Rule
To Find the APPROXIMATE Area Under the Curve given by y = f(x), and have divided the area in question into vertical strips of equal width, Δx
Where:

12. Trapezoidal Rule Error
AUC by the Trapezoidal Approximation incurs error in the amount of
Where
n ≡ the strip count
K ≡ the maximum value of |d2y/dx2|

13. Trapezoidal Rule Error Example
This Function does NOT have a Closed Form, Analytical Solution
Calculate the Area Under the Curve for this function between x=1 & x=3 using a 10-strip Trapezoidal Calculation

14. Trapezoidal Rule Error Example
Calculate Δx
Then make Fcn T-Table using
Then The T- Table
Then the Approximation

15. Trapezoidal Rule Error Example
ReCall from Error Equation
Taking the Derivative Twice
Plot d2y/dx2 to EyeBall Max Value
Maximum at x = 3

16. Trapezoidal Rule Error Example
Then
Thus, to 5 Sig-Figs:
Finally the Maximum 10-Strip, Trapezoidal Error

17. Simpson’s Rule
The Simpson Method tops TWO Strips with successive 3-pt Curve-Fit Parabolas
A Parabola can be fit EXACTLY to ANY 3 (x,y) points

18. Simpson’s Rule
Since 3-pts defines 2-strips Simpson’s Rule requires an EVEN Strip-Count
Then for an Even Counting Number, n
if M = max(|d4y/dx4|) then the Error

19. Simpson’s Rule Example
Use Simpson’s rule with n = 10 strips to approximate:
SOLUTION
From the Trapezoidal example Δx = 0.2
Now the SideWays T-Table

20. Find Precise Value by MuPAD
The Integrand Function
fOFx := E^x/x
Plot the AREA under the Integrand Curve
fArea := plot::Function2d(fOFx, x = 1..3):plot(plot::Hatch(fArea), fArea)
The Precise Value
AUCn = numeric::int(fOFx, x=1..3)

21. Simpson’s Error Find Fourth Derivative by MuPAD
d4fdx4 := diff(fOFx, x $ 4)
Then the 4th Derivative Plot
plot(d4fdx4, x=1..3, GridVisible = TRUE)
Max at x=1

22. Simpson’s Error Then the Error Calc The Error comparing to MuPAD Value
Thus the TextBook Formula is Conservative as 𝐸 𝑛 𝑏𝑘 > 𝐸 𝑛 𝑀𝑢𝑃𝑎𝑑

23. NO Equation Functions
Often in REAL LIFE “functions” disguise themselves as “Data Tables”
When I was Research Tech at Lawrence Berkeley Lab (1978) I made Ventilation-Duct Volume-Flow measurements. A typical Data Set

24. NO-Equation Functions
I then had to Calculate the Duct Volume Flow, Q, from the Data Table using the integration
This type of Integration Occurs Frequently in the Physical, Life, and Social Sciences, as well as in the Business world

25. NO-Eqn Integration Example
The Cylindrical Tank shown at right has a bottom area of 130 ft2 . The tank is initially EMPTY. To Fill the Tank, Water Flows into the top at varying rates as given in the Tank-Table below.
Time (min) 1 3 5 6 9 11 12 13 15 18
FlowRate (ft3/min) 80
130
150
160
165
170
140
120

26. NO-Eqn Integration Example
For this situation determine the water height ,H, at t = 18 minutes
SOLUTION
Use the TRAPEZOIDAL Rule to Integrate the WaterFlow to arrive at the the Total Water VOLUME
Use the Max No. of strips permitted by Data

27. NO-Eqn Integration Example
Make ΔV Calcs for the 10 strips
Then by GeoMetry
So Finally

28. NO-Eqn Integration Example
Note that in this case Δx is NON-constant
10 Strips of Varying Width
Thus SIMPSON’s Rule Can NOT be Used
Simpson’s Rule Requires constant Δx

29. MatLab Code % Bruce Mayer, PE % MTH-15 • 01Aug13 • Rev 11Sep13
% MTH15_Quick_Plot_BlueGreenBkGnd_ m %
clear; clc; clf; % clf clears figure window
% The Domain Limits
xmin = -6; xmax = 6;
% The FUNCTION **************************************
x = [ ];
y = [ ];
% ***************************************************
% the Plotting Range = 1.05*FcnRange
ymin = min(y); ymax = max(y); % the Range Limits
xmin = min(x); xmax = max(x); % the Range Limits
R = ymax - ymin; ymid = (ymax + ymin)/2;
ypmin = ymid *R/2; ypmax = ymid *R/2
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ypmin*1.05 ypmax*1.05];
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([ ]); % Chg Plot BackGround to Blue-Green
plot(x,y, '-d', 'LineWidth', 4),grid, axis([xmin xmax ypmin ypmax]),...
xlabel('\fontsize{14}t (min)'), ylabel('\fontsize{14}Q = (ft^3/min)'),...
title(['\fontsize{16}MTH16 • Bruce Mayer, PE',]),...
annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 Quick Plot BlueGreenBkGnd m','FontSize',7)
hold on
plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)
stem(x,y, '-r.', 'LineWidth', 2)
hold off
MatLab Code

30. WhiteBoard Work
Problems From §6.2
P40 → Consumer’s Surplus

31. All Done for Today
Tracking Trapezoids
Google: “third derivative name”

32. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Wht/Blk Borad
Do On
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –

35. P MATLAB Code
% Bruce Mayer, PE
% MTH-16 • 11Jan14
% MTH15_Quick_Plot_BlueGreenBkGnd_ m %
clear; clc; clf; % clf clears figure window
% The FUNCTION **************************************
x = [0:4:24];
y = [ ];
% ***************************************************
% the Plotting Range = 1.05*FcnRange
ymin = min(y); ymax = max(y); % the Range Limits
xmin = min(x); xmax = max(x); % the Range Limits
R = ymax - ymin; ymid = (ymax + ymin)/2;
ypmin = ymid *R/2; ypmax = ymid *R/2
ypmin =0
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ypmin*1.05 ypmax*1.05];
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([1 1 1]); % Chg Plot BackGround to Blue-Green
plot(x,y, '-d', 'LineWidth', 4),grid, axis([xmin xmax ypmin ypmax]),...
xlabel('\fontsize{14}q (kUnits)'), ylabel('\fontsize{14}p ($/Unit)'),...
title(['\fontsize{16}MTH16 • Bruce Mayer, PE',]),...
annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 Quick Plot BlueGreenBkGnd m','FontSize',7)
hold on
plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)
stem(x,y, '-r.', 'LineWidth', 2)
plot([xmin, xmax], [ ], '-.m', 'LineWidth', 3)
hold off
x = [ ]
y = [ ]
ps = y-ymin
M = [ ]
CS1 = ps.*M
CS2 = (4/3)*CS1
CS3 = sum(CS2)
CS4 = sum(CS1)
CStot = (4/3)*CS4

36. Example  NONconstant ∆x
Pacific Steel Casting Company (PSC) in Berkeley CA, uses huge amounts of electricity during the metal-melting process.
The PSC Materials Engineer measures the power, P, of a certain melting furnace over 340 minutes as shown in the table at right. See Data Plot at Right.
Ref: ENGR-25_Final_Exam_Fa12_Solution_ doc

37. Example  NONconstant ∆x
The T-table at Right displays the Data Collected by the PSC Materials Enginer
Recall from Physics that Energy (or Heat), Q, is the time-integral of the Power.
Use Strip-Integration to find the Total Energy in MJ expended by The Furnace during this process run

38. Example  NONconstant ∆x
GamePlan for Strip Integration
Use a Forward Difference approach
∆tn = tn+1 − tn
Example: ∆t6 = t7 − t6 = 134 − 118 = 16min → 16min·(60sec/min) = 960sec
Over this ∆t assume the P(t) is constant at Pavg,n =(Pn+1 + Pn )/2
Example: Pavg,6 = (P7 + P6)/2 = ( )/2 = kW = kJ/sec

39. Example  NONconstant ∆x
The GamePlan Graphically
Note the Variable Width, ∆x, of the Strip Tops

40. MATLAB Code % Bruce Mayer, PE % MTH-15 • 25Jul13
% XY_Area_fcn_Graph_6x6_BlueGreen_BkGnd_Template_1306.m %
clear; clc; clf; % clf is clear figure
% The FUNCTION
xmin = 0; xmax = 350; ymin = 0; ymax = 225;
x = [ ]
y = [ ]
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([ ]); % Chg Plot BackGround to Blue-Green
% Now make AREA Plot
area(x,y,'FaceColor',[ ],'LineWidth', 3),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}t (minutes)'), ylabel('\fontsize{14}P (kW)'),...
title(['\fontsize{16}MTH15 • Variable-Width Strip-Integration',]),...
annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 25Jul13','FontSize',7)
set(gca,'XTick',[xmin:50:xmax]); set(gca,'YTick',[ymin:25:ymax])
set(gca,'Layer','top')
MATLAB Code

41. Example  NONconstant ∆x
The NONconstant Strip-Width Integration is conveniently done in an Excel SpreadSheet
The 13 ∆Q strips Add up to MegaJoules of Total Energy Expended
MTH15_NONconst_delX_Strip-Integration_BMayer_1307.xlsx