MAT 1235Calculus II

4.1, 4.2 Part I

The Definite Integral

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Homework

WebAssign HW 4.2 I

Major Themes in Calculus I

Abstract World

The Tangent Problem

h

afhafh

)()(lim

0

( )y f x

x a

Real World

The Velocity Problem2t

( )y f t

t a

h

afhafh

)()(lim

0

Major Themes in Calculus I

Abstract World

The Tangent Problem

h

afhafh

)()(lim

0

( )y f x

x a

We do not like to use the definition

Develop techniques to deal with different functions

Major Themes in Calculus II

The Area Problem

( )

( ) 0 on [ , ]

y f x

f x a b

Abstract World

1

lim ( )n

ini

A f x x

The Energy Problem

( )y f x

( )f x

Real World

Major Themes in Calculus II

We do not like to use the definition

Develop techniques to deal with different functions

1

lim ( )n

ini

A f x x

The Area Problem

( )

( ) 0 on [ , ]

y f x

f x a b

Abstract World

Preview

Look at the definition of the definite integral on

Look at its relationship with the area between the graph and the -axis on

Properties of Definite Integrals

Example 0

]5,1[on )( 2xxf

Example 0 ]5,1[on )( 2xxf

)1(f

)5.1(f

)4(f

)5.4(f

)2(f

Use left hand end points to get an estimation

Example 0 ]5,1[on )( 2xxf

)5.2(f

)5.1(f

)5(f

)5.4(f

)2(f

Use right hand end points to get an estimation

Example 0 Observation:

What happen to the estimation if we increase the number of subintervals?

In General

ith subinterval

ix

sample point

)( ixf

In General

Suppose is a continuous function defined on , we divide the interval into subintervals of equal width

b ax

n

The area of the rectangle is

xxf i )(

In General

ith subinterval sample point

xxf i )(

In General

Sum of the area of the rectangles is

n

ii

n

xxf

xxfxxfxxfxxf

1

321

)(

)()()()(

Riemann Sum

n

ii

n

xxf

xxfxxfxxfxxf

1

321

)(

)()()()(

In General

Sum of the area of the rectangles is

Sigma Notation for summation

n

ii

n

xxf

xxfxxfxxfxxf

1

321

)(

)()()()(

In General

Sum of the area of the rectangles is

IndexInitial value (lower limit)

Final value (upper limit)

In General

Sum of the area of the rectangles is

As we increase , we get better and better estimations.

n

ii

n

xxf

xxfxxfxxfxxf

1

321

)(

)()()()(

Definition

The Definite Integral of from to

n

ii

n

b

axxfdxxf

1

)(lim)(

Definition

The Definite Integral of from to

n

ii

n

b

axxfdxxf

1

)(lim)(

upper limit

lower limit

integrand

Definition

The Definite Integral of from to

n

ii

n

b

axxfdxxf

1

)(lim)(

Integration : Process of computing integrals

Example 1

Express the limit as a definite integral on the given interval.

n

ii

n

b

axxfdxxf

1

)(lim)(

1

lim cos( ) , [0, ]n

i in

i

x x x

?, ?

( ) ?

a b

f x

Example 1

Express the limit as a definite integral on the given interval.

n

ii

n

b

axxfdxxf

1

)(lim)(

1

lim cos( ) , [0, ]n

i in

i

x x x

?, ?

( ) ?

a b

f x

1

( )

lim cos( ) n

i in

i

a

b

f x

x x x

Remarks

We are not going to use this limit definition to compute definite integrals.

In section 4.3, we are going to use antiderivative (indefinite integral) to compute definite integrals.

We will use this limit definition to derive important properties for definite integrals.

More Remarks

b

adxxf )(

If on , then under"" Area )( fdxxfb

a

More Remarks

If on , then under"" Area )( fdxxfb

a

If on , then above"" Area )( fdxxfb

a

b

adxxf )(

b

adxxf )( (

)if x

( )i iA f x x

More Remarks

b

adxxf )(

( )i iA f x x

(

)if x

1 1

lim lim ( )n n

i in n

i i

Area A f x x

Example 2

43a b c

)(xfy

( )

( )

b

a

c

b

f x dx

f x dx

Example 3

Compute by interpreting it in terms of area

2

1)1( dxx

21

1xy1

2

1( 1)x dx

Example 4

Compute 3

3

29 dxx

Properties

The follow properties are labeled according to the textbook.

Property (a)

are called the dummy variables

( ) ( )b b

a af d fx tx dt

Example 5

2

1( 1)x dx

21

1xy

1

x

21

1y t

1

t

2

1( 1)dt t

Property (b)

The definition of definite integral is well-defined even if

upper limit < lower limit And

b

a

a

bdxxfdxxf )()(

1

3. . ( )e g f t dt

Property (b)

The definition of definite integral is well-defined even if

upper limit < lower limit And

b

a

a

bdxxfdxxf )()(

b ax

n

a b

xn

1

lim ( )n

ini

xf x

1

3. . ( )e g f t dt

Example 6

21

1xy

2

1)1(

2

1 dxx 1

x

1

2( 1)x dx

Note: If lower limit > upper limit, the integral has no obvious geometric meaning

Example 7

If , what is ?4)(3

1 dxxf

1

3)( dttf

Example 7

If , what is ?

Q1: What is the answer?

Q2: How many steps are needed to clearly demonstrate the solutions?

4)(3

1 dxxf

1

3)( dttf

Property (c)

0)( a

adxxf

Example 8

1

1

3 2

3

( 1)

(sin tan 4 )

x dx

x x x dx

Classwork

2 persons per group. Work with your partner and your partner ONLY.

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