1. §5.3 Fundamental Theorem of Calc
Chabot Mathematics
§5.3 Fundamental Theorem of Calc
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 5.2 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
§5.2 → AntiDerivatives by Substitution
Any QUESTIONS About HomeWork
§5.22 → HW-23

3. §5.3 Learning Goals
Show how area under a curve can be expressed as the limit of a sum
Define the definite integral and explore its properties
State the fundamental theorem of calculus, and use it to compute definite integrals
Use the fundamental theorem to solve applied problems involving net change
Provide a geometric justification of the fundamental theorem

4. Area Under the Curve (AUC)
The AUC has many Applications in Business, Science, and Engineering
Calculation of Geographic Areas
River Channel Cross Section
Wind-Force Loading

5. Area Under Function Graph
For a Continuous Function, approximate the area between the Curve and the x-Axis by Summing Vertical Strips
Use Rectangles of Equal Width
Three Possible Forms
Left end points
Right end points
Midpoints
LEFT ends points are the most convenient
Strip Width
(n strips)

6. Example: Strip Sum Approximate the area under the graph of Use
n = 4 (4 strips)
Strip MidPoints

7. Example: Strip Sum GamePlan

8. MATLAB Code % Bruce Mayer, PE % MTH-15 • 24Jul13
% XY_Area_fcn_Graph_6x6_BlueGreen_BkGnd_Template_1306.m %
% The FUNCTION
xmin = 0; xmax = 2; ymin = 0; ymax = 8;
x = linspace(xmin,xmax,20); y = 2*x.^2;
% 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', [1 .8 1] , 'LineWidth', 3), axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 2x^2'),...
title(['\fontsize{16}MTH15 • Area by Strip Addition',]),...
annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 24JUul13','FontSize',7)
hold on
set(gca,'Layer','top')
plot(x,y, 'LineWidth', 3),
MATLAB Code

9. MATLAB Code % Bruce Mayer, PE % MTH-15 • 24Jul13 % % The Limits
xmin = 0; xmax = 2; ymin = 0; ymax = 8;
% The FUNCTION
x = linspace(xmin,xmax,500); y = 2*x.^2;
x1 = [0.25:.5:1.75]; y1 = 2*x1.^2
% 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
plot(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 2x^2'),...
title(['\fontsize{16}MTH15 • Area by Strip Addition',]),...
annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer • 24Jul13','FontSize',7)
hold on
area([(x1(1)-.25)*ones(1,100),(x1(1)+.25)*ones(1,100)],[y1(1)*ones(1,100),y1(1)*ones(1,100)],'FaceColor',[1 .8 1])
area([(x1(2)-.25)*ones(1,100),(x1(2)+.25)*ones(1,100)],[y1(2)*ones(1,100),y1(2)*ones(1,100)],'FaceColor',[1 .8 1])
area([(x1(3)-.25)*ones(1,100),(x1(3)+.25)*ones(1,100)],[y1(3)*ones(1,100),y1(3)*ones(1,100)],'FaceColor',[1 .8 1])
area([(x1(4)-.25)*ones(1,100),(x1(4)+.25)*ones(1,100)],[y1(4)*ones(1,100),y1(4)*ones(1,100)],'FaceColor',[1 .8 1])
plot(x,y, 'LineWidth', 4)
set(gca,'Layer','top')
plot(x1,y1,'g d', 'LineWidth', 4)
plot([x1(1)-.25,x1(1)+.25],[y1(1),y1(1)], 'm', [x1(2)-.25,x1(2)+.25],[y1(2),y1(2)], 'm',...
[x1(3)-.25,x1(3)+.25],[y1(3),y1(3)], 'm', [x1(4)-.25,x1(4)+.25],[y1(4),y1(4)], 'm','LineWidth',2)
plot([x1(1)-.25,x1(1)-.25],[0,y1(1)], 'm',[x1(2)-.25,x1(2)-.25],[0,y1(2)], 'm',...
[x1(3)-.25,x1(3)-.25],[0,y1(3)], 'm', [x1(4)-.25,x1(4)-.25],[0,y1(4)], 'm', 'LineWidth', 2)
set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:1:ymax])
MATLAB Code

10. Example: Strip Sum
The Algebra
midpoints

11. Area under a Curve
Width:
(n strips)
a b
GOAL: find the exact area under the graph of a function; i.e., the curve
PLAN: Use an infinite number of strips of equal width and compute their area with a limit.

12. Area Under a Curve Function, f(x), on interval [a,b] is:
Continuous
NonNegative
Then the Area Under the Curve, A
The x1, x2, …, xn-1,xn are arbitrary, n SubIntervals each with width (b − a)/n
a b

13. Riemann Sum ∑f(xk)·∆x
For a Continuous, NonNeg fcn over [a,b] divided into n-intervals of Equal Width, ∆x = (b−a)/n, The AUC can be approximated by the sum the area of Vertical Strips
Riemann ∑

14. Riemann ∑ → Definite Integral
For a Continuous, NonNeg fcn over [a,b] divided into n-intervals of Equal Width, ∆x = (b−a)/n, The AUC can be calculated EXACTLY by the Riemann sum as the number of strips becomes infinite.
This Process of finding an Infinite Sum is called “Integration”;
"to render (something) whole," from Latin integratus, past participle of integrare "make whole" (an “adding up” process)
integrate (v.) 1630s, "to render (something) whole," from Latin integratus, past participle of integrare "make whole," from integer "whole" (see integer). Meaning "to put together parts or elements and combine them into a whole" is from 1802.

15. Riemann ∑ → Definite Integral
As the No. of Strips increase the AUC Calculation becomes more accurate
The Riemann-Sum to Definite-Integral
Twenty Strips
Fifty Strips

16. MATLAB Code % Bruce Mayer, PE % MTH-15 • 24Jul13 % % The Limits
xmin = 0; xmax = 2; ymin = 0; ymax = 8;
% The FUNCTION
x = linspace(xmin,xmax,500); y = 2*x.^2;
x1 = [1/20:1/10:39/20]; y1 = 2*x1.^2;
% 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
plot(x,y, 'LineWidth', 4),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) = 2x^2'),...
title(['\fontsize{16}MTH15 • Area by Strip Addition',]),...
annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer • 24Jul13','FontSize',7)
hold on
bar(x1,y1, 'BarWidth',1, 'FaceColor', [1 .8 1], 'EdgeColor','b', 'LineWidth', 2)
set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:1:ymax])
set(gca,'Layer','top')
plot(x,y, 'LineWidth', 3)
MATLAB Code

17. Definite Integral Symbology
upper limit of integration
Integrating Factor
Integration
Symbol
integrand
variable of integration
(dummy variable)
lower limit of integration
It is called a dummy variable because the answer does not depend on the symbol chosen; it depends only on a&b

18. Dummy Variable of Integration
Used Primarily with DEFINATE Integrals
3 7 𝑥 2 +5 𝑑𝑥= 𝑥 𝑥 𝑥=3 𝑥=7 = − 9−15 = = 376 3
3 7 𝑧 2 +5 𝑑𝑧= 𝑧 𝑧 𝑧=3 𝑧=7 = − 9−15 = = 376 3

19. Derivative ↔ AntiDerivative
ReCall Notation:
UPPER Case is the AntiDerivative
lower Case is the Derivative

20. Recall Fundamental Theorem
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

21. AUC: Saved by AntiDerivative
Thus to find the AUC by the Fundamental Theorem
Find the AntiDerivative of 𝑓 𝑥 ; 𝐹 𝑥
Plug in NUMBERS, 𝑎&𝑏, into 𝐹 𝑥
Subtract 𝐹 𝑏 −𝐹 𝑎 to obtain a NUMBER for the Riemann Sum whichis the AUC

22. Example: AUC by Fund. Th.
Find AUC for Curve at right: 𝑦=𝑓 𝑥 =5 𝑥 3 − 𝑒 1.1𝑥
AUC = 𝑥 3 − 𝑒 1.1𝑥 ∙ 𝑑𝑥= 𝑥 4 4 − 𝑒 1.1𝑥 = 5 ∙ − 𝑒 1.1∙ − 5 ∙ − 𝑒 1.1∙3 1.1
𝑦=𝑓 𝑥 =5 𝑥 3 − 𝑒 1.1𝑥
AUC = − = square Units

23. MATLAB Code % Bruce Mayer, PE
% MTH-15 • 01Aug13 • Rev 11Sep13 • 01May16
% MTH15_Quick_Plot_BlueGreenBkGnd_ m %
clear; clc; clf; % clf clears figure window
% The Domain Limits
xmin = 0; xmax = 7;
% The FUNCTION **************************************
x = linspace(xmin,xmax,1000); y = 5*x.^3 - exp(1.1*x);
% ***************************************************
% the Plotting Range = 1.05*FcnRange
ymin = min(y); ymax = max(y); % 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([1 1 1]); % Chg Plot BackGround to Blue-Green => [ ]
plot(x,y, 'LineWidth', 4),grid, axis([xmin xmax 0 ypmax]),...
xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)'),...
title(['\fontsize{16}MTH15 • Riemann v. Fundamental Th.',]),...
annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'MTH15 Quick Plot BlueGreenBkGnd m','FontSize',7)
% std textbox location: [ ]
hold on
plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)
x1 = linspace(3,6,1000); y1 = 5*x1.^3 - exp(1.1*x1);
area(x1,y1)
set(gca,'XGrid','on', 'Ygrid', 'on','Layer','top')
hold off
% Area Fill Color set by interactive plot utility (darn!)
MATLAB Code

24. Fundamental Theorem – Part2
Previously we stated that the AntiDerivative of f(x) is F(x), so then
Now consider the definite Integral (AUC) Relationship to the AntiDerivative

25. DefiniteIntegral↔AntiDerivative
That is, The AUC for a continuous Function, f(x), spanning domain [a,b] is the AntiDerivative evaluated at b minus the AntiDerivative evaluated at a.
D. F. Riddle, Calculus and Analytical Geometry, Belmont, CA, Wadsworth, 1974, pp , pg. 770

26. Gives the area since 2x3 is nonnegative on [0, 2].
Example  Find AUC
Find the area under the graph of y = 2x3
Then
Gives the area since 2x3 is nonnegative on [0, 2].
Antiderivative
Fund. Thm. of Calculus

27. Rules for Definite Integrals
Constant Rule: for any constant, k
Sum/Diff Rule:
Zero Width Rule
Domain Reversal Rule

28. Rules for Definite Integrals
SubDivision Rule, for (a<b<c)

29. Example  Eval Definite Integral
Find a Value for
The Reduction using the Term-by-Term rule

30. Example  Def Int by Substitution
Find:
Let:
Then find dx(du) and u(x=0), and u(x=1)
Clarify Limits

31. Example  Def Int by Substitution
SubOut x2+3x, and the Limits
Dividing out the 2x+3
Then
Thus Ans

32. The Average Value of a Function
At y = yavg there at EQUAL AREAS above & below the Avg-Line
Avg Line

33. MATLAB Code % Bruce Mayer, PE % MTH-15 • 24Jul13
% Area_Between_fcn_Graph_BlueGreen_BkGnd_Template_1306.m
% Ref: E. B. Magrab, S. Azarm, B. Balachandran, J. H. Duncan, K. E.
% Herhold, G. C. Gregory, "An Engineer's Guide to MATLAB", ISBN
% , Pearson Higher Ed, 2011, pp %
clc; clear
% The Function
xmin = 0; xmax = 16;
ymin = 0; ymax = 350;
xct = 1000
x = linspace(xmin,xmax,xct);
y1 = .5*x.^3-9*x.^2+11*x+330;
yavg = mean(y1)
y2 = yavg*ones(1,xct)
% Find Zeros
plot(x,y1, x,y2, 'k','LineWidth', 2), axis([0 xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)'),...
title(['\fontsize{16}MTH15 • Meaning of Avg',]),...
annotation('textbox',[ ], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'Bruce Mayer, PE • 24Jul13','FontSize',7)
display('Showing 2Fcn Plot; hit ANY KEY to Continue')
% "hold" = Retain current graph when adding new graphs
hold on
nct = 500
xn = linspace(xmin, xmax, nct);
fill([xn,fliplr(xn)],[.5*xn.^3-9*xn.^2+11*xn+330, fliplr(yavg*ones(1,nct))],'m'),grid
plot(x,y1), grid
MATLAB Code

34. The Average Value of a Function
Mathematically - If f is integrable on [a, b], then the average value of f over [a, b] is
Example  Find the Avg Value:
Use Average Definition:

35. Net Change
If the Rate of Change (RoC), dQ/dx = Q’(x) is continuous over the interval [a,b], then the NET CHANGE in Q(x) is Given by

36. Example  Find Net Change
A small importer of Gladiator merchandise has modeled her monthly profits since the company was created on January 1, 1997 by the formula
Where
P ≡ $-Profit in 100’s of Dollars ($h or hector-$)
t ≡ year of operation for the company

37. Example  Find Net Change
What is the importer’s net change in profit between the beginning of the years 2000 and 2003?
SOLUTION:
Recall t is in years after 1997, Thus
Year 2000 corresponds to t = 3
Year 2003 corresponds to t = 6
Then in this case the Net Change in Profit over [3,6] →

38. Example  Find Net Change
Thus Her monthly profits increased by about $1, between 2000 & 2003

39. WhiteBoard Work Problems From §5.3 P74 → Water Consumption
P80 → Distance Traveled

40. Students Should Calc All Done for Today
Google: “third derivative name”

41. Fundamental Theorem Part-1
All Done for Today
Fundamental Theorem Part-1
Google: “third derivative name”

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

43. Fundamental Theorem Proof
Let area under the curve from a to x.
(“a” is a constant)
Then:

44. max f
The area of a rectangle drawn under the curve would be less than the actual area under the curve.
min f
The area of a rectangle drawn above the curve would be more than the actual area under the curve. h

45. As h gets smaller, min f and max f get closer together.
This is the definition of derivative!
initial value
Take the anti-derivative of both sides to find an explicit formula for area.

46. As h gets smaller, min f and max f get closer together.
Area under curve from a to x = antiderivative at x minus antiderivative at a.

47. Critical (Break) Points
ConCavity Sign Chart
ConCavity Form
d2f/dx2 Sign
++++++
−−−−−−
−−−−−−
++++++ x
Critical (Break) Points a b c
Inflection
NO Inflection
Inflection

53. Fundamental Theorem – Part2
Previously we stated that the AntiDerivative of f(x) is F(x), so then
Now consider the definite Integral (AUC) Relationship to the AntiDerivative