Chapter 15 - Methods and Applications of Integration

INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences

2007 Pearson Education Asia

Chapter 15 Chapter 15 Methods and Applications of IntegrationMethods and Applications of Integration


INTRODUCTORY MATHEMATICAL ANALYSIS

0. Review of Algebra

1. Applications and More Algebra

2. Functions and Graphs

3. Lines, Parabolas, and Systems

4. Exponential and Logarithmic Functions

5. Mathematics of Finance

6. Matrix Algebra

7. Linear Programming

8. Introduction to Probability and Statistics


9. Additional Topics in Probability10. Limits and Continuity11. Differentiation12. Additional Differentiation Topics13. Curve Sketching14. Integration

15. Methods and Applications of Integration16. Continuous Random Variables17. Multivariable Calculus

INTRODUCTORY MATHEMATICAL ANALYSIS


• To develop and apply the formula for integration by parts.

• To show how to integrate a proper rational function.

• To illustrate the use of the table of integrals.

• To develop the concept of the average value of a function.

• To solve a differential equation by using the method of separation of variables.

• To develop the logistic function as a solution of a differential equation.

• To define and evaluate improper integrals.

Chapter 15: Methods and Applications of Integration

Chapter ObjectivesChapter Objectives


Integration by Parts

Integration by Partial Fractions

Integration by Tables

Average Value of a Function

Differential Equations

More Applications of Differential Equations

Improper Integrals

15.1)

15.2)

15.3)


Chapter OutlineChapter Outline

15.4)

15.5)

15.6)

15.7)



15.1 Integration by Parts15.1 Integration by Parts

Example 1 – Integration by Parts

Formula for Integration by Parts

Find by integration by parts.

Solution: Let and

Thus,

duvuvdvu

dxxx ln

Cxx

dxx

xxxdxxx

2ln2

122ln ln 2/1

xu ln dxx

dv 1

dxx

du 1 2/12/1 2 xdxxv


Chapter 15: Methods and Applications of Integration15.1 Integration by Parts

Example 3 – Integration by Parts where u is the Entire IntegrandDetermine

Solution: Let and

Thus,

. ln dyy

yvdydv

CyyCyyy

dyy

yyydyy

1lnln

1ln ln

dyy

du

yu

1

ln


Chapter 15: Methods and Applications of Integration15.1 Integration by Parts

Example 5 – Applying Integration by Parts TwiceDetermine

Solution: Let and

Thus,

. 122 dxex x

dxxduxu

2

2

2/

12

12

x

x

evdxedv

dxxeex

dxxeexdxex

xx

xxx

2

)2(22

12122

12122122

1

1212

121212

42

22

Cexe

dxexedxxe

xx

xxx


Chapter 15: Methods and Applications of Integration15.1 Integration by PartsExample 5 – Applying Integration by Parts Twice

Solution (cont’d):

Cxxe

Cexeexdxex

x

xxxx

21

2

422

212

1212122122



15.2 Integration by Partial Fractions15.2 Integration by Partial Fractions

Example 1 – Distinct Linear Factors

• Express the integrand as partial fractions

Determine by using partial fractions.

Solution: Write the integral as

Partial fractions:

Thus,

dxx

x 27312

2

. 912

31

2 dxx

x

65

67

2

,3 if and ,3 If3333

12912

AxBxx

Bx

Axx

xx

x

Cxx

xdx

xdx

dxx

x

3ln673ln

65

31

3331

27312 6

765

2


Chapter 15: Methods and Applications of Integration15.2 Integration by Partial Fractions

Example 3 – An Integral with a Distinct Irreducible Quadratic FactorDetermine by using partial fractions.Solution: Partial fractions:

Equating coefficients of like powers of x, we have

Thus,

dxxxx

x

4223

xCBxxxAx

xxCBx

xA

xxxx

)()1(42

1142

2

22

2 ,4 ,4 CBA

Cxxx

Cxxx

dxxx

xx

dxxxCBx

xA

4

22

2

22

1ln

1ln2ln4

1244

1


Chapter 15: Methods and Applications of Integration15.2 Integration by Partial Fractions

Example 5 – An Integral Not Requiring Partial FractionsFind

Solution: This integral has the form

Thus,

. 13

322 dx

xxx

Cxxdxxx

x

13ln 13

32 22

. 1 duu



15.3 Integration by Tables15.3 Integration by Tables

Example 1 – Integration by Tables

• In the examples, the formula numbers refer to the Table of Selected Integrals given in Appendix B of the book.

Find

Solution: Formula 7 states

Thus,

.

32

2 xdxx

C

buaabua

bbuaduu

ln1 22

C

xxdx

xx

32232ln

91

32 2


Chapter 15: Methods and Applications of Integration15.3 Integration by Tables

Example 3 – Integration by Tables

Find


Let u = 4x and a = √3, then du = 4 dx.

.316 2

xxdx

Cu

aauaauu

du

22

22ln1

Cx

xxx

dx

4

3316ln31

316

2

2



Example 5 – Integration by TablesFind


If we let u = 4x, then du = 4 dx. Hence,

. 4ln7 2 dxxx

C

nu

nuuduuu

nnn

2

11

11lnln

Cxx

Cxxx

dxxxdxxx

14ln39

7

94

34ln4

647

44ln447 4ln7

3

33

2

32



Example 7 – Finding a Definite Integral by Using TablesEvaluate


Letting u = 2x and a2 = 2, we have du = 2 dx.

Thus,

.24

4

12/32

xdx

Caua

uau

du

2222/322

Caua

uau

du

2222/322

621

662

2221

221

24

8

22

4

12/32

4

12/32

u

uu

dux

dx



15.4 Average Value of a Function15.4 Average Value of a Function

Example 1 – Average Value of a Function

• The average value of a function f (x) is given by

Find the average value of the function f(x)=x2 over the interval [1, 2].

Solution:

dxxfab

fb

a

1

37

3121

1

2

1

32

1

2

xdxx

dxxfab

fb

a



15.5 Differential Equations15.5 Differential Equations

Example 1 – Separation of Variables

• We will use separation of variables to solve differential equations.

Solve

Solution: Writing y’ as dy/dx, separating variables and integrating,

.0, if ' yxxyy

xCy

dxx

dyy

xy

dxdy

lnln

11

1


Chapter 15: Methods and Applications of IntegrationExample 1 – Separation of Variables


0,

ln

ln

1

1

xCxCy

eey

ey

x

C

xC


Chapter 15: Methods and Applications of Integration15.5 Differential Equations

Example 3 – Finding the Decay Constant and Half-LifeIf 60% of a radioactive substance remains after 50 days, find the decay constant and the half-life of the element.Solution: Let N be the size of the population at time t, tλeNN 0

01022.050

6.0ln

6.0

6.0 and 50 When50

00

0

λ

eNN

NNtλ

days. 82.672ln is life half the and Thus, 01022.00

λeNN t



15.6 More Applications of Differential Equations15.6 More Applications of Differential Equations

Logistic Function

• The function

is called the logistic function or the Verhulst–Pearl logistic function.

Alternative Form of Logistic Function

ctbeMN

1

tbCMN

1


Chapter 15: Methods and Applications of Integration15.6 More Applications of Differential Equations

Example 1 – Logistic Growth of Club MembershipSuppose the membership in a new country club is to be a maximum of 800 persons, due to limitations of the physical plant. One year ago the initial membership was 50 persons, and now there are 200. Provided that enrollment follows a logistic function, how many members will there be three years from now?


Chapter 15: Methods and Applications of Integration15.6 More Applications of Differential EquationsExample 1 – Logistic Growth of Club Membership

Solution: Let N be the number of members enrolled in t years,

Thus,

1511

800501

,0 and 800 When

bbbC

MN

tM

t

5lnln151800200

,200 and 1 When

51

ce

Nt

c

781

151800

451

N


Chapter 15: Methods and Applications of Integration15.6 More Applications of Differential Equations

Example 3 – Time of MurderA wealthy industrialist was found murdered in his home. Police arrived on the scene at 11:00 P.M. The temperature of the body at that time was 31◦C, and one hour later it was 30◦C. The temperature of the room in which the body was found was 22◦C. Estimate the time at which the murder occurred.

Solution: Let t = no. of hours after the body was discovered andT(t) = temperature of the body at time t.By Newton’s law of cooling,

22 TkdtdTaTk

dtdT


Chapter 15: Methods and Applications of Integration15.6 More Applications of Differential EquationsExample 3 – Time of Murder


CktT

dtkT

dT

22ln

22

9ln02231ln,0 and 31 When

CCktT

98ln9ln12230ln

,1 and 30 When

kk

tT

ktTInktT

9229ln22ln Hence,


Chapter 15: Methods and Applications of Integration15.6 More Applications of Differential EquationsExample 3 - Time of Murder


Accordingly, the murder occurred about 4.34 hours before the time of discovery of the body (11:00 P.M.). The industrialist was murdered at about 6:40 P.M.

34.4

9/8ln9/15ln

98ln2237ln

, 37 When

tt

T



15.7 Improper Integrals15.7 Improper Integrals

• The improper integral is defined as

• The improper integral is defined as

dxxfa

dxxfdxxfr

ar

a

lim

dxxfdxxfdxxf 0

0

dxxf


Chapter 15: Methods and Applications of Integration 15.7 Improper Integrals

Example 1 – Improper IntegralsDetermine whether the following improper integrals are convergent or divergent. For any convergent integral, determine its value.

21

210

2lim lim 1 a.

1

2

1

3

13

r

r

r

r

xdxxdxx

1lim lim b.0

00

r

x

rr

x

r

x edxedxe

r

r

r

rxdxxdx

x 12/1

1

2/1

1

2lim lim 1 c.


Chapter 15: Methods and Applications of Integration15.7 Improper Integrals

Example 3 – Density FunctionIn statistics, a function f is called a density function if f(x) ≥ 0 and .

Suppose is a density function. Find k.

Solution:

1

dxxf

elsewhere 00 for xkexf

x

11lim1lim

101

00

00

0

kkedxke

dxkedxxfdxxf

rx

r

rx

r

x

Chapter 15 - Methods and Applications of Integration

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Transcript of Chapter 15 - Methods and Applications of Integration