mathematics ii 2011

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Course Title Mathematics II

Course CIC Code Math II

Credit Hours 3 Credit Hours

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1. Text BookThe main text book for the course is calculus (Early TranscendentalFunction) Third edition .

By Smith Minton

2. Course ObjectivesThis course is a transition from a course in elementary Calculus to anotheradvanced course. The main topics in this course are integration (Method of

integration-Application of integration)- Complex Analysis ² Vector andthree-dimensional Analytic Geometry- Linear algebra- Theory of Equations-Numerical Integration

The aims of this course are:

To emphasize of developing an understanding of concepts in advanced

Mathematics rather than learning by rule.To gain experience in the method of evaluating different types of integrations.

To gain an approval of importance of the complex analysis theory, Vectorand three-Dimensional, linear algebra, theory of equations and numericalintegration

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3.Course Description

The main concepts covered in this course include:�Integration Techniques and Application of Integration

�Vectors and three-dimensional Analytic Geometry

�Linear algebra

�Theory of Equations

�Complex Analysis�Numerical Integration

4. Homework ²AssignmentsProblem sets are assigned for the credit and will be graded by

the teaching Assistant (TA).Your Homework assignments should be submitted in class.

There will be four Quizzes done in the lectures

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5. Reading MaterialsThe following Chapters will covered this semester

A. Integration Techniques and Application of Integration� Trigonometric Substitution

� Integration By Parts

� Partial Fractions

� Area of a Region Between two curves

� Volume (The Disc Method ² The Shell Method)

� Improper Integrals

� Arc Length of the Curve

� Surface area

B. Vectors and three-dimensional Analytic Geometry

� Planes and lines

� Surfaces of degree two

� Vectors in three dimensions

c. Linear Algebra

� Determinates

� Matrices

� System of linear equations

�

Eigen values and Eigenvectors5



D. Theory of Equations�Roots Properties

�Relation between coefficients and roots .

�E. Complex Analysis�Complex numbers

�Logarithmic and exponential functions

�Limits and continuity of a complex function

�

Complex integration�F. Numerical Integration�Rectangular Rule

�Trapezoidal rule

�Simpon's rule

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6. Attendance PolicyAttendance at lectures and Tutorial is a must. It will be taken during eachlecture students are not allowed to miss lectures which might contain quizzes.

7.Grading SystemThe grade will be given as follows

Description Grade Note

Final ExamParticipationQuizzesTutorials

50%

8%

32%10%

This course outline issubjects to changes ifnecessary, in which caseyou will be notified

Total 100%

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Integration ² Overview

Integration means the anti-derivative

( ) ( ) f x dx g x c! ´Why ?

Because ( ) ( )d

g x f xd x

!

Example

1

2

1tan

1dx x c

x

! ´ because 1

2

1tan

1

d x

dx x

!

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Rules of Integration

Rule 1

k dx

k x c! ´

If k is a constant

Rule 2

1

11

n

n x x d x c n

n

! {´

In the case 1n ! 1 1 x dx dx Ln x c

x

! ! ´ ´

Why ?

Why ?

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Examples

3

7

18 x x dx

x ´

4 3 / 2 6

84 3 / 2 6

x x x x c

!

3 1/ 2 7 8 x x x dx! ´

5 / 2 13 x dx

x

´

7 / 2

3

7 / 2

x Ln x x c!

21

x dx x

¨ ¸© ¹

ª º´ 2

2

12 x dx

x! ´

3 1

23 1

x x x c

!

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Rule 3

? A? A

1( )

( ) '( ) 1

1

n

n f x f x f x dx c n

n

! {

´Why ?

Examples

2 112 10 (3 7)

6 (3 7)

11

x x x d x c

!

´2 101

6 (3 7)6

x x d x! ´2 10(3 7) x x dx´ 6

6v

2 111 (3 7)

6 11

xc

!

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Rule 4

'( )( )

( )

f xd x n f x c

f x! ´

Examples

2

2

6(3 7)

3 7

xdx Ln x c

x!

´

23 7

xdx

x ´ 2

2

1 6 1(3 7)

6 3 7 6

xd x n x c

x! !

´6

6v

Why ?

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sin sintan

cos cos

cos

x x x d x d x d x

x x

n x c

! !

!

´ ´ ´

coscot

sin

sin

x x dx dx

x

Ln x c

!

!

´ ´

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Rule 5

( ) ( )'( ) f x f xe f x dx e c! ´ Why ?

Examples

tan 2 tansec

x xe x dx e c! ´2 21 12 x x x e d x e c ! ´

2 1 x x e d x´2 2

1 11 12

2 2

x x x e d x e c ! ! ´2

2v

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Rule 6

sin ( ) '( ) cos ( ) f x f x dx f x c! ´W

hy ?

Examples

2 28 sin(4 1) cos(4 1) x x dx x c ! ´

cos ( ) '( ) sin ( ) f x f x dx f x c! ´

2 21cos(3 1) sin(4 1)

6 x x dx x c ! ´

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Examples

(1) 7

sin cos x x dx! ´8sin

8

xc!

7sin cos x x dx´

Another solution

Let sinu x! cosdu x d x!

then7 7

sin cos x x dx u d u!´ ´8 8sin

8 8

u xc c! !

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(2) 2sin x dx´ 11 cos 2

2 x d x! ´

1 1sin 22 2 x x c

¨ ¸! © ¹ª º

Try to calculate 2cos x dx

´ ??

1 1 11 cos 2 sin 2

2 2 2 x dx x x c

¨ ¸! ! © ¹ª º

´

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(3) 1 x x dx´Let 2 1u x!

2u du d x!2

1u x !2 2

1 ( 1) 2 x x dx u u u d u ! ´ ´2 2( 1)2u u d u!

´5 3

2 25 3

u uc!

4 2(2 2 )u u d u! ´

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2 1u x! But 1/ 21u x!

Then

5 3

5 / 2 3 / 2

1 2 25 3

2 21 1

5 3

u u x x d x c

x x c

!

!

´

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(4)2

1

9d x

x´

Let

3

xu !

2

1

9 19

dx x

!¨ ¸

© ¹ª º

´

1

3du d x!

2 2

1 13

9 9 1dx d u

x u! ´ ´1

2

1 1 1tan

3 1 3d u u c

u

! ! ´ 11

tan3 3

xc ¨ ¸! © ¹

ª º

3du d x!

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In general we have

1

2 2

1 1tan

xdx c

a x a a

¨ ¸! © ¹ ª º´

(5)2

1

16

d x

x

´ 2

1

16 116

dx

x

!¨ ¸

© ¹ª º

´

Let

4

xu ! 1

4d u dx! 4 du d x!

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2 2

1 14

16 16 1dx dx

x u

!

´ ´

2

1

1du

u

!

´

In general we have

1

2 2

1sin

xdx c

aa x

¨ ¸! © ¹ª º

´

1 1sin sin4

xu c c ¨ ¸! ! © ¹

ª º

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Also we have

1

2 2

1

2 2

1

2 2

1 1

tanh

1sinh

1 cosh

xd x ca x a a

xd x c

a x a

xd x c

a x a

¨ ¸!

© ¹ ª º

¨ ¸! © ¹

ª º

¨ ¸! © ¹ª º

´´

´

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(6)

2

1

25 1dx

x ´ 2

1

1

25 1 25

d x x

!¨ ¸

© ¹© ¹ª º

´

Let1

5

xu

!

1

5d u dx!

2 2

1 15

25 125 1d x du

u x!

´ ´

5 du d x!

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Definite integrals

´!

!

!

b x

a x

dx x f )(

If f is integr able and f ( x)>0 f or every x in [a,b], then

area under the gr aph of

f ( x) f r om x=a to x=b

dx

f(x)

This thin stri p is going to swee p the gr aph f r om

x=a to x=b (remember dx= ( x0)

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Note

If ( ) ( ) f x dx g x c! ´

Then

? A( ) ( ) ( ) ( )

b

b

a

a

f x d x g x g b g a! ! ´

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Problem

Evaluate the f ollowing integr al: ´

2

1

2)1( d x x x

833.4

833.1667.6

)1(2

)1(

3

)1()2(

2

)2(

3

)2(

23)1(

2323

2

1

2

1

23

2

!

!¼½

»

¬

«

¼½

»

¬

«!

¼½

»¬«

!´ x x x

dx x x

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Example

Calculate

/ 2

0

sin 2 x d x

T

´

Solution/ 2/ 2

00

1sin 2 cos 2

2

1 1cos cos 02 2

1 1( 1) (1) 1

2 2

x d x x

T T

T

« »! ¬ ¼½

« »! ¬ ¼½

« »! !¬ ¼

½

´

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Example

Calculate the area between the curves & 2 y x y x! !

and the x-axis

2 y x! y x!

210

Area

1 2

0 1

2 x dx x dx! ´ ´

1 22 2

0 1

22 2

1 4 1 1 30 (4 ) (2 ) 2 1

2 2 2 2 2

x x x« » « »! ¬ ¼ ¬ ¼

½ ½

« » « »! ! !¬ ¼ ¬ ¼½ ½

Solution

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mathematics ii 2011

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