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

BMayer@ChabotCollege.edu • MTH15_Lec-22_sec_5-1_Integration.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics§5.1

Integration

Review §

Any QUESTIONS About• §4.4 → Exp & Log

Math Models Any QUESTIONS

About HomeWork• §4.4 → HW-21

§5.1 Learning Goals Define AntiDerivative Study and compute

indefinite integrals Explore differential

equations and Initial/Boundary value problems

Set up and solve Variable-Separable differential equations

Fundamental Theorem of Calculus 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

aaFbFdxxf

xfdxxfdxdxF

dxddxxfxF thenif

AntiDifferentiation Using the 2nd Part

of the Theorem F(x) is called the AntiDerivative of f(x)

• Example: Find f(x) when

• ONE Answer is

• As Verified by

dxxfxFxfdxdF or

34xdxxdf

34 4xxdxdxf

Fundamental Property of Antiderivs The Process of Finding an

AntiDerivavite is Called: InDefinite Integration

The Fundamental Property of AntiDerivatives:• If F(x) is an AntiDerivative of the

continuous fcn f(x), then any other AntiDerivative of f(x) has the formG(x) = F(x) + C, for some constant C

Fundamental Property of Antiderivs Proof of G(x) = F(x) + C Assertion: both G(x) & F(x)+C are

AntiDerivatives of f(x); that is:

Using DerivativeRules

CxFdxdxfxG

CxFdxdxG

xfdxdF

dxdGxf

Derivative of a Sum

Derivative of a Const

Transitive Property

The Indefinite Integral The family of ALL AntiDerivatives of f(x)

is written

The result of ∫f(x)dx is called the indefinite integral of f(x)

Quick Example for:• u(x) has in INFINITE

NUMBER of Results, Two Possibilities:

CxFdxxf )( )(

Cxdxxdxxu 43 4

The Meaning of “C” The Constant, C, is the y-axis “Anchor

Point” for the “natural Response” fcn F(x) for which C = 0.• C is then the y-intercept

of F(x)+C; i.e., Adding C to F(x) creates a “family” of

functions, or curves on the graph, with the SAME SHAPE, but Shifted VERTICALLY on the y-axis

CFG 00

The Meaning of “C” Graphically

-4 -3 -2 -1 0 1 2 3 4-10

CMTH15 • Familiy of AntiDerivatives

B. May er • 20Jul13

ode% Bruce Mayer, PE% MTH-15 • 20Jul13% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m%% The Limitsxmin = -4; xmax = 4; ymin = -10; ymax = 20;% The FUNCTIONx = linspace(xmin,xmax,1000); y = 7*exp(-x/2.5) + 5*x -8;% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg(['white']) % whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(x,y, x,y+9,x,y-pi,x,y+sqrt(13),x,y-7, 'LineWidth', 4),axis([xmin xmax ymin ymax]),... grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = G(x) = F(x)+C = 7e^-^5^x^/^2 + 5x - 8 + C'),... title(['\fontsize{16}MTH15 • Familiy of AntiDerivatives',]),... annotation('textbox',[.71 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none', 'String', 'B. Mayer • 20Jul13','FontSize',7)hold onplot(zxv,zyv, 'k', zxh,zyh, 'k', [-1.4995, -1.4995], [ymin,ymax], '--m', 'LineWidth', 2)set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax])

Bruce Mayer, PEMTH15 20Jul13F(x) = 7*exp(-2*x/5) + 5*x -8 f(x) = int(G, x)G := 7*exp(-2*x/5) + 5*x -8dgdx := diff(G, x)assume(x > -6):xmin := solve(dgdx, x)xminNo := float(xmin)Gmin := subs(G, x = xmin)GminNo := float(Gmin)plot(G, x=-4..4, GridVisible = TRUE,LineWidth = 0.04*unit::inch)

Evaluating C by Initial/Boundary A number can be found for C if the

situation provides a value for a SINGLE known point for G(x) → (x, G(x)); e.g., (xn, G(xn)) = (73.2, 4.58)• For Temporal (Time-Based) problems the

known point is called the INITIAL Value– Called Initial Value Problems

• For Spatial (Distance-Based) problems the known point is called the BOUDARY Value– Called Boundary Value Problems–

Common Fcn Integration Rules1. Constant Rule:

for any constant, k

2. Power Rule:for any n ≠ −1

3. Logarithmic Rule:for any x ≠ 0

4. Exponential Rule:for any constant, k

Cxkdxk

Cnxdxxn

dxe kxkx 1

Integration Algebra Rules1. Constant Multiple Rule: For any

constant, a

2. The Sum or Difference Rule:

• This often called the Term-by-Term Rule

dxxuadxxua

dxxvdxxudxxvxu

dxxqdxxpdxxqxp

Example Use the Rules Find the family of

AntiDerivatives corresponding to

SOLUTION: First Term-by-Term → break up each

term over addition and subtraction:

dxxx 122

dxdxxdxxdxxx 1 2 12 22

Example Use the Rules Move out the constant in the 2nd integral

(2), and state sqrt as fractional power

Using the Power Rule

CleaningUp →

dxdxxdxxdxxx 1 2 12 2122

Cxxxdxdxxdxx

1012/12

12 1 2

1012/1122/12

Cxxxdxxx 2/33

Example Propensity to Consume The propensity to consume (PC) is the

fraction of income dedicated to spending (as opposed to saving).

A Math Model for the marginal propensity to consume (MPC) for a certain population:

• Where – MPC is the rate of change in PC – x is the fraction of income that is disposable.

xexMPC 8.0

Example Propensity to Consume If the propensity to consume is 0.8

when disposable income is 0.92 of total income, find a formula for PC(x)

SOLUTION: From the Problem Statement that the

MPC is a marginal function discern that

Thus the PC fcn is the AntiDerivative of MPC(x)

,xPCdxdxMPC

Example Propensity to Consume Find PC by

Integrating

This is satisfactory for a general solution, but need the particular solution so that PC(0.92) = 0.8

dxxMPCxPC

dxe x 8.0

Ce x 8.025.1

Example Propensity to Consume Use the (x,PC) = (0.92,0.8) Boundary

Value to Find a NUMBER for the Constant of Integration, C

With C ≈ 1.4, state the particular solution to this Boundary Value Problem

4.125.1 8.0 xexPC

Differential Equations (DE’s) A Differential Equation is an equation

that involves differentials or derivatives, and a function that satisfies such an equation is called a solution

A Simple Differential Equation is an equation which includestwo differentials in the formof a derivative

)(xfdxdy

Differential Equations (DE’s) For some function f. Such a Simple

Differential Equation can be solved by integrating:

In summary the Solution, y, to a Simple DE can be found by the integration

dxxfdxdxdydxxf

dxdyxf

dxxfydxxfdydxxfdydxxfdy )(1

dxxfy )(

Example Simple DE From the

Previous Example As previously solved for the general

solution by Integration:

Then used the Boundary Value, (0.92, 0.8), to find the Particular Solution

xexPCdxd 8.0

CexPC x 8.025.1

4.125.1 8.0 xexPC

Variable-Separable DE’s A Variable Separable Differential

equation is a differential equation of the form• For some integrable functions f and g

Such a differential equation can be solved by separating the single-variable functions and integrating:

dxxfdyygdxxfdyygygxf

Example Fluid Dynamics The rate of change in volume (in cubic

centimeters) of water in a draining container is proportional to the square root of the depth (in cm) of the water after t seconds, with constant of proportionality 0.044.

Find a model for the volume of water after t seconds, given that initially the container holds 400 cubic centimeters.

Example Fluid Dynamics SOLUTION: First, TRANSLATE the written

description into an equation:• “rate of change

in volume”• “is proportional to the

square root of volume”• “with constant of

proportionality equal to 0.044”

044.0k

Example Fluid Dynamics So the (Differential)

Equation Note that the right side does not

explicitly depend on t, so we can’t simply integrate with respect to t. • Instead move the expression

containing V to the left side: The Variables are now Separated,

allowing simple integration

VdtdV 044.0

Example Fluid Dynamics Integrating

SquaringBoth SidesFind:

1221 CCC

2022.0 CtV

Example Fluid Dynamics For The particular solution find the a

number for C using the Initial Value: when t = 0, V = 400 cc:• Sub (0,400) into

DE Solution

Thus the volume of water in the Draining Container as a fcn of time:

20400 C

220022.0 ttV

WhiteBoard Work Problems From §5.1

• P58 → Oil Production(not a Gusher…)

• P73 → Car StoppingDistance

All Done for Today

LOTS moreon DE’s

in MTH25

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

Appendix

srsrsr 22

ConCavity Sign Chart

−−−−−−++++++ −−−−−− ++++++

ConCavityForm

d2f/dx2 Sign

Critical (Break)Points Inflection NO

InflectionInflection

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

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