Reviewer - Elementary Analysis II

32

Elementary Analysis II

6. Integration Techniques

6.1. Integration by Parts

Suppose we want to evaluate an integral of

the form ( ) ( ) , assuming that is

differentiable and ( ) is an antiderivative

of ( )

o

, ( ) ( )- ( ) ( )

( ) ( )

o Deriving the equation, ( ) ( )

, ( ) ( )- ( ) ( )

o Integrating both sides,

( ) ( ) ( ) ( )

( ) ( )

Theorem:

By letting ( ) ( ) and

( ) ( ), then

, which is integration by parts

Note:

As a rule of thumb, the order of choosing the

term is: Logarithmic, Inverse

trigonometric, Algebraic, Trigonometric, and

Exponential

Tabular Integration by Parts

Given ( ) ( ) , tabular integration by

parts can be used if one of the functions is

finitely differentiable and the other function

is integrable

( ) ( )

∑ ( ) ( ) ( ) | ( )

The consequence of tabular integration by

parts is that it cannot be used when the first

function is infinitely differentiable

Integration by Parts of Definite Integrals

The definite integral can be solved with

integration by parts, provided that the

functions satisfy its conditions

|

6.2. Trigonometric Integrals

1. Integrating powers of sine and cosine

2. Integrating products of sine and cosine

o If is odd,

Split off a factor of

Use the Pythagorean identity

Let

o If is odd,



Let

o If both and are even,

Use

Use

( ) ( )

o and ,

Use

, ( ) ( )-

o ,

Use

, ( ) ( )-

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33

o ,

Use

, ( ) ( )-

3. Integrating powers of tangent and secant

4. Integrating products of tangent and secant

o If is odd,



Let

o If is even,



Let

o If is even and is odd,


Use the reduction formula

for powers of

Note:

For powers of sine and cosine, should be a

positive integer

For powers of tangent and secant, should

be greater than 1

To evaluate integrals of cosecant and

cotangent, use the formulae for tangent and

secant, and substitute the corresponding

cofunctions

6.3. Trigonometric Substitution

Substitutions are used if the following

expressions are found in the integrand:

o √

o √

o √

Steps in Integration using Trigonometric

Substitution

1. Substitute the values for and

2. Integrate

3. Return the variables to its original form

6.4. Integration by Partial Fractions

Linear Factor Rule

o Factors of the form ( ) in the

denominator of a proper rational

functions will contribute to terms

of partial fractions; that is,

( )

∑

( ) * +

Quadratic Factor Rule

o For each factor of the form

( ) , the partial

fraction decomposition contributes

to terms of partial fractions that is,

( )

∑

( )

Note:

If the degree of the numerator is greater

than or equal to the degree of the

denominator, then long division must first be

carried out before advanced to partial

fraction decomposition

Partial fraction decomposition gives way to

the easier use of simple integration



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6.5. Improper Integrals

Improper integrals are definite integrals

whose limit of integration reaches infinity, is

a value of which makes the graph of the

function infinitely discontinuous, or a

combination of both

Improper Integrals with Infinite Integration

Intervals

Consider ( )

o ( )

o The area of the region bounded by

( ) and , - is

o

Theorem:

( )

( )

( )

( )

( )

( )

( )

Improper Integrals with Infinite Discontinuity

Consider the same function

o It has an infinite discontinuity at

o By first inverting the interval such

that ( -, the new area of the

region bounded by the function and

the interval is

o

Theorem:

If is continuous on , -, except at and

infinite discontinuity at , then the improper

integral of over , - is ( )

( )

If is continuous on , -, except at and

infinite discontinuity at , then the improper

integral of over , - is ( )

( )

If is continuous on , -, except at an

infinite discontinuity at ( ), then the

improper integral of over , - is

( )

( )

( )

6.6. Review on Separable Differential Equations

and Applications

A differential equation is an equation

involving the derivative/s of an unknown

function

A first order separable differential equation

is an equation of the form ( )

( )



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Some Applications of SODEs

Malthusian Population Model

o Let ( ) be time, population

at given time, birth rate, and death

rate, respectively

o

( )

Integrating both sides,

( ) ( )

*( ) +

o The initial population will be the

population at time zero, that is,

( ) ( ) *(

) +

( )

( )

Verhulstian Population Model

o Let ( ) .

/ be time,

population at given time, carrying

capacity, and per capita income

increase, respectively

o

.

/

Integrating both sides,

|

| ( )

* +

* +

o The initial population will be

( )

* +

( )( * + )

( )

( )

6.7. Orthogonal Trajectories

Two curves are said to be orthogonal if their

tangent lines are perpendicular at every

point of intersection

Two families of curves are said to be

orthogonal trajectories of each other if each

member of one family is orthogonal to each

member of the other

Recall:



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Note:

If

and

, then has a horizontal

tangent line

If

and

, then has a vertical

tangent line

If

, then has a singular point

Higher Derivatives

Let ( ) and ( ) be a pair of

parametric equations, then

The second order derivative,

, can be

expressed as

In general, the nth derivative of is

7.4. Arc Length of Parametric Curves

Let be the parametric curve defined by

( ) ( )

If no segment of is traced more than once

from to , then the arc length of

from to is

√, - , -

, given that the pair

of parametric equations is differentiable

over , -

7.5. Polar Coordinates

A point ( ) on the polar coordinate system

can be determined by its distance from the

pole , and the angle of the radial line with

respect to the polar axis

Conversion of Polar and Rectangular Coordinates

Polar to Rectangular -

( ) ( )

Rectangular to Polar -

( ) .√

/

Remarks:

( ) ( )

( ) ( )

7.6. Graphs of Polar Equations

A polar equation is an equation of the form

( )

Theorem:

A polar curve is symmetric about the x-axis if

( ) ( )

A polar curve is symmetric about the y-axis if

( ) ( )

A polar curve is symmetric about the origin if

( ) ( ) or if negating the equation

will still produce an equivalent equation



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8. The Real Space

8.1. Three Dimensional Coordinate System

A point in is defined by an ordered

triple, ( )

To locate in , find first the point ( )

in the -plane then move the point units

up if , or down if

Distance Between Two Points

Let ( ) and ( ) be points

in

In the -plane,

√| | | |

√( ) ( )

Suppose another plane exists where and

lies,

√ | |

√( ) ( ) ( )

Midpoint Formula in

The midpoint of the points ( ) and

( ) is ( )|

* +

8.2. Surfaces

Cylindrical Surfaces

An equation that contains only two of the

variables represents a cylindrical

surface in

The system can be obtained by the equation

in the corrdinate plane of the two variables

that appear in the equation and then

translating that graph parallel to the axis of

the missing variable

Quadric Surfaces

Ellipsoid

o

Hyperboloid

o One sheet

o Two sheets

Elliptic Paraboloid

o

o

o

Elliptic Cone

o

o

o

Hyperbolic Paraboloid

o

o

o

Note:

If (if occurs in the quadric), then

a circle will occur in at least one cross-

section plane



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o From the reference point, define an

arbitrary direction along the curve as

the positive direction; and the

oppositve direction as the negative

direction

o All points in the positive direction

are said to have positive arc lengths

o All points in the negative direction

are said to have negative arc lengths

Define ‖

‖

as the arc length

parametrization of with reference point

Properties:

a. ‖

‖

b. ‖

‖

Note:

If is an A.L.P., then , where is a

vector-valued function of

The A.L.P. is a function dependent on arc

length , which finds the vector ( ) that is

units along from the reference point

9.5. Unit Tangent, Normal, and Binormal Vectors

Let ( ) be a smooth function

o Define ( ) ( )

‖ ( )‖, called the unit

tangent vector

o Define ( ) ( )

‖ ( )‖

( )

‖ ( )‖,

called the unit normal vector

o Define ( ) ( ) ( ), called

the unit binormal vector

Remark:

The unit tangent, normal, and binormal

vectors make up of what is known as the

moving trihedral

Let be an A.L.P.

o ( ) ( )

o ( ) ( )

‖ ( )‖

( )

‖ ( )‖

o

‖ ‖

The unit normal vector points to the

concavity of

9.6. Curvature

Let be a smooth curve

o The sharpness of bend of is

measured by its curvature

o The curvature of a curve is

defined by as ( ) ‖

‖

‖ ( )‖

o Other forms for are ( )

{

‖ ( )‖

‖ ( )‖

‖ ( ) ( )‖

‖ ( )‖



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Radius of Curvature

Let be a circle with radius , then ( )

is defined to be the radius of curvature

Note:

( ) ( )

9.7. Curvilinear Motion

Let ( ), a smooth vector-valued function,

be the position function of partical moving in

space

o The unit tangent vector points to

the direction of motion of a particle

o

, the rate of change of the arc

length with respect to time, is the

speed of the particle

o The velocity vector is defined as

( )

( )

Distance and Displacement

Define as the displacement vector of a

particle travelling from to , then

( ) ( )

Let be the distance travelled of a particle

from to

o ‖

‖

‖ ( )‖

Normal and Tangential Componenents of

Acceleration

Recall: ( )

( )

The acceleration vector may be derived from

the velocity vector

o ( ) ( )

. ( )

/

( )

o Since ( ) ( )

‖ ( )

‖

and

‖ ( )

‖, then

( ), which

implies ( )

( )

o With the above, ( )

.

/ ( )

( )

Define

as the tangential scalar

component of acceleration, and

.

/

as the normal scalar component of

acceleration

Remarks:

Other formulas for the scalar components

include

‖ ‖,

‖ ‖

‖ ‖,

‖ ‖

‖ ‖

9.8. Projectile Motion

Let ( ) ⟨ ⟩, where is the

acceleration due to gravity

o ( ) ( ) ⟨ ⟩

Let ( ) and be the initial position and

velocity of the particle, respectively



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o ( ) , then ( )

⟨ ⟩

o ( ) ( ) ⟨

⟩

o ( ) ⟨ ⟩, then ( )

⟨

⟩

Parametric Equations of Projectile Motion

Let ⟨ ⟩; then

‖ ‖ ‖ ‖

( ) ⟨‖ ‖

‖ ‖ ⟩

{ ‖ ‖

‖ ‖



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10. Multivariate Differential Calculus

10.1. Multivariate Functions

A function of 2 variables, and , is a rule

that assigns a unique real number for each

ordered pair ( )

A function of 3 variables, , , and , is a rule

that assigns a unique real number for each

ordered triple ( )

In general, a function of variables,

, is a rule that assigns a

unique real number for each ordered -tuple

( )

The domain of is defined, and strictly

follwed, as

* ( )| ( ) +

Level Curves and Surfaces

Let ( ), then the projection of the

trace of on the plane , onto

the -plane is called the level curve of at

Let ( ), then the graph of

( ) , is called a surface of

at

10.2. Limits and Continuity

Let be a smooth parametric curve defined

by ( ) ( ) ( ), and

suppose that at ( )

( ) ( )

o Let ( ), then

( ) ( ) ( )

( ( ) ( ) ( ))

In , if is defined by ( ) and

( ), and if ( ), then

( ) ( ) ( )

( ( ) ( ))

o Unlike the limits in , infinitely

many points are being approached

to ( ) in infinitely many paths,

in which, the well defined curve C is

required

o To get the limit, the points of are

projected onto the surface

A function ( ) is said to be continuous at

( ) if and only if

o ( )

o ( ) ( ) ( )

o ( ) ( ) ( ) ( )

Theorem:

If ( ) is continuous at , and if ( ) is

continuous at , then ( ) ( ) ( )

is continuous at ( )

If ( ) is continuous at ( ) and if

( ) is continuous at ( ), then

( ( )) is continuous at ( )



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Remarks:

If ( ) ( ) ( ) , then

( ) ( ) ( )

| |

( ) ( ) ( )

( ) ( ) ( ), then the limit

does not exist

The sum/difference/product of two

continuous functions is also continuous

The quotient of two continuous functions is

continuous, except at those points where

the denominator is zero

10.3. Partial Derivatives

Let ( ) be a continuous function

o The partial derivative of with

respect to at ( ) is defined as

( ) ( )

o Similarly, the partial derivative of

with respect to at ( ) is

defined as

( ) ( )

A partial derivative can be interpreted as the

slope of the tangent line at the cross section

of the surface at ( )

Notations include

for -partial

derivatives, and

for -partial

derivatives

Higher-Order Partial Derivatives

Let ( ); then

.

/,

.

/,

.

/, and

.

/

Theorem: Clairaut’s Theorem

If the partial derivatives and are both

continuous and defined on ( ), then

( ) ( )

10.4. Implicit Partial Differentiation

Suppose a function ( ) is expressed

in a general form ( )

o The equation may be solved by

explicitly solving for the partial

derivatives of

o If the equation cannot be expressed

simply as ( ), implicit partial

differentiation is used

Assumptions in Implicit Partial Differentiation

1. Treat the variable as a partially

differentiable function of and

2. Since equal functions have the same

derivative on both sides, partially

differentiate both sides of the equation

3. Solve for the partial derivative



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10.5. Local Linear Approximation

Let ( ) ( )

o Define the local linear approximation

of at point ( ) as

( ) ( )

( )

For those points ( ) that are very close to

( ), then ( )

In general, if ( ), then

∑

10.6. Differentiability

A function ( ) is said to be

differentiable at ( ) if

( ) ( ) ( )

√( ) ( )

In general, a function ( )

( ) is said to be differentiable at

point if

( )

√∑ ( )

Note:

( ) is the error in the approximation if

the local linear approximation is used

10.7. Differentials

Let ( )

o Define the total differential of at

( ) as ( )

( )

Define ( ) ( )

o

o If and , then

( ) ( )

10.8. Multivariate Chain Rule

Let ( ) ( ) ( )

o If the end-function of the general

function is univariate, then define

In general, if ( ) ( ),

then define

∑

, provided that

the end-function is univariate

Remark:

To aid in MCR, a tree diagram may be used

o The use of the tree diagram exhausts

all possible paths from the most

general function to the most specific

function with the variable of interest

Note:

( ) and ( ) can be substituted directly

after ( ) and proceed with

univariate differentiation

If at least one of the functions that define

is multivariate, then MCR produces a partial

derivative

A combination of univariate and multivariate

end-functions is possible

o If such happens, for as long as the

end-function is univariate, a

univariate derivative is multiplied;

else, a partial derivative is multiplied



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