The Definite Integral在线视频

Today we study the Calculus (I)

The topic is the Definite Integral

Maybe you can ask yourself a question

how to compute

the area of the curve trapezoid

Now we look at the figure

means the area

We have known the given idea

If a rectangular is given

the corresponding height f(x) equals h

At this time

the rectangular area formula

equals base time height

In this figure

the corresponding base means

b minus a

the corresponding height is h

so we have known

the corresponding formula

Now the rectangle idea means

rectangle area is approximate

to replace the curve trapezoid area

How to replace

So now look at

if we have five small rectangles

The corresponding area is

approximate to the trapezoid area

If there are ten small rectangles

one two three four five

until ten

At this time

the sum of the areas of the ten small rectangles

is approximate to replace the curve trapezoid area

How to do this idea

So now look at the following idea

If we divide the curve trapezoid

into a number of small rectangles

This means

the number of small rectangles is infinity

At this time

the rectangle's width gradually reduces

then how to find the area of A

Now look at the figure

Yes

maybe we have infinite small rectangles

At this time

means the number of rectangles is infinity

So now look at the figure

If we divide so many small rectangles

we can get

the corresponding area ΔAi

And do the summation

of the small rectangles

How to get it

There are the following four steps

Look at Step 1

means segmentation

Segmentation means

the beginning point a equals x0

the end point b is equal to xn

at this time

divided x1, x2 and xn-1

into the closed interval

So now become

this a, b closed interval become n intervals

And then

how to do

We can get the corresponding interval

distance Δxi

equals the right point xi

minus the left point xi-1

Step 2

how to do

Yes

the approximation

At this time

the corresponding area of the rectangle ΔAi

is approximate to

the area of the rectangle

Look at this

classical rectangle

the corresponding base means

xi minus xi-1

means corresponding our base

Look at the height

any fixed point ξi

ξi is given

We can take the value f(ξi)

means our height

At this time

the area means

base time the height

How to do

And then

Step 3

do the summation

The summation of this area

is approximate

to the summation f(ξi) time Δxi

Why not equal

Yes

if the i from 1 to n

if n is enough large

then think

the area value is

approximate to our real area

So do the step 4

getting the limit

how to get the limit

The limit A equals the limit

λ tends to 0

summation i from 1 to n

if n is enough large

This means

λ equals

the maximum value of

the interval distances Δx1 Δx2 until Δxn

if λ tends to 0

At this time

n tends to positive infinity

means there are some tendency

If there is some tendency

means the limit exist

It is a real number

The real number is our area of the curve trapezoid

So try to understand the 4 steps

So now look at

the concrete definition

If the region A is bounded by

bounded by y equals f(x)

larger than 0

means above the x-axis

And x equals a

x equals b

and y equals 0

bounded by the four curves

Some are lines

And how to find the area of A

The area of A is denoted as I

We have the following idea

means the definite integral

from a to b

f(x) dx is our integral

Integral equals

the limit of summation f(ξi) time Δxi

Summation

pay attention to

i from 1 to n

Look at the tendency

λ tends to 0

Pay attention to

the corresponding signs

Look at a

a is called our lower limit

because it is low

So means lower limit

How about b

Corresponding

upper limit

so means upper limit

Look at f(x)

Yes it is a function

in this signs

means integral function

How about x

It is variable

it is called integral variable

Look at the sum

The sum means integral sum

So try to remember

the signs and understand the formula

If we try to find the area of

the curve trapezoid

how to change the question

into the integral

means definite integral

So now according to the idea

we will do some examples

Look at Example 1

Try to find

the area of the region

he region bounded by

y equals x square

and x equals 1

and x-axis

How to do

Do you remember

the formula definite integral

This means the area of the region

Try to find

some value of definite integral

So this means x square

our integral function

corresponding

lower limit is 0

the upper limit is 1

x is our integral variable

This means the area equals

the value of definite integral

So we have the following concrete steps

Let the beginning point x0 equals 0

and end point xn equals 1

Divide

means x1, x2,…,xn-1

So we can get

he corresponding figure

y equals x square

and x equals 0

x equals 1

Above x-axis

means the corresponding region

the region

how to do

Segmentation

the first step

ok

So

we have the corresponding

so many small rectangles

Look at the classical rectangles

Taking the point xi equals i over n

Why

Look at the first

The end point means 0 over n

means number 0

This point 1 over n

the last point means n over n

So corresponding

in this interval

the point is i over n

So taking xi equals i over n

corresponding Δxi

means the distance of the interval

equals 1 over n

Why

Because between 0 and 1

the distance is 1

How many parts

Yes

n parts

So the distance of every interval is

1 over n

n is fixed ok

n is sometimes some variable

i from 1 2 until n

We let ξi equals xi

ξi means some fixed point

into the given interval

You can select

left point or right point

or any point between the end points

So now we take

ξi equals right point xi

i from 1, 2 until n

And then we have

the sum integral

We have summation i form 1 to n

f(ξi) Δxi

Thinking of the question

Look at

what is our f(ξi)

Yes

ξi equals xi

f(ξi) what

Yes

f means x square

variable square

So this part means ξi square

How about Δxi

Δxi

yes

the interval distance means

1 over n

So we have f(ξi)

means the variable square

So ξi square time Δxi

Taking the given values

xi means corresponding

i over n

Δxi 1 over n

Taking the numbers

We can get ok

summation i from 1 to n

bracket i over n square

time 1 over n

Pay attention to

in the summation formula

n is fixed

What is changeable

i

pay attention to i

so i changeable

Go on compute this expression

taking the concrete number n cube

Summation

summation what

Summation i

why

i is changeable

So now summation i square

So we only need to compute

summation i square

According to the basic knowledge

we know the summation i square

Do you remember the basic formula in your high school

y over n cube

summation i square equals

equals n time bracket n plus 1

time bracket double n plus 1 over 6

So now we have got

the integral summation

How to do

Try to get the limit

So now we can get

1 over 6 is a constant taken out

So in the first term means 1 over 6

Now look at

the term n cube

n cube

n,n delete 1 power

n plus 1 over n

equals 1 plus 1 over n

This term

double n plus 1 over n

equals 2 plus 1 over n

So now the integral sum is

a function of n

If λ tends to 0

do you remember λ

Yes

λ is the maximum value of

the interval distance

So λ tends to 0

This means

n tends to positive infinity

So this gives you idea

how to compute the limit

So the last step

The area of the region means

equals the value of definite integral

from 0 to 1

x square dx

This equals

do you remember the last step

Step 4

equals the limit summation

λ tends to 0

Now look at the summation

Yes

the summation is a function of n

So taking the expression of n

λ tends to 0

means n tends to positive infinity

Why you write it n

Because in this expression

the variable is n

So we need to write it n how to change

So n tends to positive infinity

So how to compute

Yes

equals 1 by 3

Why

Yes

1 over 6 time 1 plus 0 time 2 plus 0

so 2 over 6

together equals 1 over 3

It is our final value

means it is our area of the region

From the question we can find

the area equals this expression

It is a real number

means concrete number

The integral from 0 to 1

is approximate to this expression

Why

Do you remember this is integral sum

The sum means sum expression of n

n is changeable

so not equal

means is close to

not equal

so equals this expression

n how to change

n tends to positive infinity

Why

Because if n takes different values

at this time

the area means definite integral

has different approximation accuracies

n is enough large

At this time

we can get

the limit is our definite integral

means it is our area

Try to understand the idea

So now we look at Example 2

Try to evaluate

the definite integral

by using a regular partition

Do you know partition

Yes

partition means our step 1

segmentation

So now look at the idea

We taking in example 1

Δx equals 1 over n

In this question

Δx equals 5 over n

Why

Can you tell me

Why is 5

Yes

you should pay attention to

the upper limit

and lower limit values

Upper limit equals 3

lower limit equals minus 1

How about the distance of the interval

3 minus minus 2 means 3 plus 2

equals 5

So at this time

the given closed interval

is divided by n intervals

Every interval distance is 5 over n

Taking ξi

the given point equals xi

as our sample point

means some fixed point

in the closed interval

Try to understand 5 over n and xi

Now look at the figure ok

y equals x plus 3

How to find the corresponding area

Taking the following point

taking the original point x0 equals

lower limit means minus 2

The xn equals our upper limit means 3

How to write it

x1 equals beginning point plus the

interval distance

xn equals the left end point plus

5 over n time 2

and go on

How to do

Blah blah blah

xi equals

the left end point minus 2 plus

5 over n time i

We can take the different points

Using the idea

pay attention to xi corresponding values

So now

we have f(ξi)

Do you remember

f(x) equals x plus 3

So f(ξi) equals (f(ξi)

f(xi) f(xi) plus 3

Together

xi

do you remember

minus 2 plus i time 5 over n

minus 2 plus 3 equals positive 1

So equals 1 plus i time 5 over n

Try to understand f(ξi)

It is our height

Do the sum

integral sum

The sum equals f(ξi) time Δxi

What is our f(ξi)

Yes

f(xi)

what is our Δxi

Do you remember

Yes

5 over n

Taking the corresponding values

f(ξi) equals this number ok

Δxi

Try to do the summation

Have the similar idea in Example 1

Pay attention to the sum

What is changeable

Pay attention to i

Yes

i is changeable

How about n

From 1 to n

This means n is fixed

So i is changeable

So now we do the summation

equals 5 plus 25 over 2

plus 25 over double n

This is our sum

integral sum

Finally the sum is a function of n

not i

Why

Because i we have computed

So corresponding

Definite integral means our area ok

equals the limit of integral sum

Taking in Corresponding

if λ tends to 0

equals what

Equivalent to

n tends to positive infinity

Pay attention to this formula

In this formula

if n tends to positive infinity

at this time

constant is still constant equals itself

n tends to positive infinity

This term equals 0

So the final result

the limit equals 5 plus 25 over 2

together equals 35 over 2

It is our final result

means the definite integral value

Try to understand the idea

how to take the integral sum

how to compute the limit

According to the two examples

we try to get ome summary of

the definite integral

Look at property 1

if a equals b

You can think the figure in your mind

If a equals b

at this time

a equals b

this means 0

Why

if a equals b

at this time

the corresponding figure

the region become one segment

The area of segment equals 0

So from 1 to 1

3 to 3

4 to 4

equals 0

Why

At this time

the geometrical meaning means

the area of the segment

so equals 0

Now look at

if a larger than b

This means from a to b

f(x) dx equals corresponding b, a, minus

How to understand

Try to understand

the formula using the limit property

From a to b

from b to a

They are corresponding

if you equals 9

I will equals minus 9

Change the order of

upper limit and lower limit

change the order

So a b become b a minus

Try to remember

if a function has definite integral

What means has definite integral

This means the limit exists

And then

if we don't consider the values of a and b

We don't care the values

Maybe a is larger than b

Maybe a is less than b

Maybe a equals b

We don't care which one is bigger

And then

we will have the following properties

Five properties

so now look at Property 1

If a, b integral function

f(x) plus or minus g(x) equals what

Yes

equals the corresponding summation

or minus of the two definite integrals

How to prove

You can prove by yourself

using the limit idea

Now look at Property 2

a b

the integral function is k time f(x)

k is a constant

k is a constant means

k is taken out

from the definite integral

equals k time the definite integral

where k is a constant number

The property 1 and property 2 together mean

the linear characteristics

Try to understand

in the future computation

we always use the linear property

So now look at Property 3

If c between a and b

means c is a number between a and b

How to write it

Yes

the definite integral a, b

f(x) dx

equals from a to b

How to write it c

Any c

Yes c

from a c plus c b

The property means interval additivity

c is a number between a and b

We can prove in the future

c is any real number for the formula

is always true

This means interval additivity

Now look at Property 4

If the integral function

is a special function equals 1

You can think it

In your mind at this time

the curve trapezoid becomes a

regular rectangle

Yes

at this time equals base

The base means b minus a

How about the height

Yes

the height is constant

always equals 1

So the final value is

b minus a time 1

so equals b minus a

Now we study the last property 5

Property 5 means

if f(x) is non-negative

Negative means less than 0

Non-negative means large than or equals 0

At this time

a b f(x) dx

it is our definite integral

still non-negative

Still non-negative

this means "sign preserving" property

Try to understand the 5 properties

So now

we will use the 5 properties

and the definition of definite integral

Try to answer the questions

Look at the question 1

Question 1 means try to find the limit

The limit

pay attention to this

n tends to positive infinity

Limit the first term

1 over n plus 1

The second term

1 over n plus 2

plus blah blah blah

plus the last term 1 over n plus n

Maybe you can write it like this

n tends to positive infinity

So the first limit equals 0

ok

The second equals 0

blah blah blah

Together

the last equals 0

0 0 plus 0

infinite 0

together equals 0

Some students write it

If in your test

you write it like this

It is wrong

Why

Do you remember

if the limit of summation equals

the summation of the limit

The given condition

what

Yes

the number of expression

means the number of limit is finite

Now we can count

1 2 3 4 … n

How many

n

n tends to positive infinity

so using the limit property is wrong

Now how to do

We have the following idea

We can write it the limit

in this expression

Pay attention to n plus 1

n plus 2 until n plus n

Thinking yourself

the integral summation

yes

f(ξi) time Δxi

So we can write it Δxi

Write it 1 over n

How to write it 1 over n

means doesn't change ok

Equivalent to taking n

from every term

The first term

taking n out

become 1 plus 1 over n

the second term

taking n out

n taken out become 1

This term become 2 over n

until the last term

Yes how to write it

f(ξi) means f(xi)

What is our i

How to find it

We can write it

Copy the limit

n tends to positive infinity

summation 1 plus 1 over n

1 plus 2 over n

Blah blah blah

Try to find the regulation

Try to find the rule

The rule means 1 plus i

Yes

i here

1 2 3 4 until n

So this is our regulation i over n

time 1 over n

Yes

so now we find it the ξi

ξi means i over n

So we can write it

the limit summation

become the definite signs

How about f(ξi)

f(ξi) means our f(x)

1 over 1 plus ξi

This means

1 over 1 plus x

1 over n means our interval distance

We can write it

d integral variable

So write it dx

So pay attention to limit summation

corresponding

the definite signs

f(ξi) corresponding

our integral function

1 over n doesn't change

1 over n is our interval distance

We can write it corresponding

x is our interval variable

So write it this form

We have known the corresponding

the definite integral anti-derivative function is

ln bracket 1 plus x

So equals

ln 1 plus 1

minus ln 1 plus 0

So final result equals

ln2 minus 0 equals ln2

It's our final limit

Try to understand the question

means using the idea of definite integral

Try to find our limit

Now look at Question 2

Question 2 means

try to find the limit

Limit n tends to positive infinity

The first term

look at the characteristic

Try to find the rule

Square root of 4 n square minus 1

The second term

4 n square minus 4

blah blah blah

The last term

4 n square minus n square

Compare the question 1

How to find it

Yes

taking out the interval distance 1 over n

And then find i

What is changeable

4 n square

doesn't change

Changeable here

1 4

next

Yes 9

You are clever

Next

16

until n square

1 4 9 16 means i square

So we have the idea

We can write it in this step

Try to find

Try to find what

Yes

interval distance

So taking out this term 1 over n

1 over n is our interval distance

How to write it

Write it how many terms

1 2 3 4 … n

so there are n terms

In the similar idea

if you write it

0 plus 0 plus 0

It’s wrong

Using the idea of

definite integral means the limit idea

So we can write it

Taking n out

Square root means taking n square out

Square root n square means

1 over n

Square and square root

together become n

So now look at the first term

4 n square minus 1

taking out n square

The first becomes 4

The second becomes 1 over n

bracket square

Similar meaning

the second term

becomes 2 over n

bracket square

until the last term

You can find it

Yeah

this is our interval distance

How about ξi

1 over n

2 over n

n over n

yes

1 2 3 until n

So the 1 over n

2 over n

these terms are changeable

Changeable means our ξi

So we can write it

Yes

4 minus 1 over n

bracket square.

Pay attention to i

At this time

we find it ξi

ξi means 4 minus ξi square

ξi means our xi

So the summation

corresponding function

we can find it interval

corresponding function becomes

1 over square root of 4 minus x square

So now how to write it

The lower limit and the upper limit

Do you remember

Yes

1 over n

It is our interval distance

In the first interval

the left end point means 0 over n

The left point ok 0 over n

Another point

1 over n until ok

so i minus 1 over n

i over n

continue

We can find it

1 over n is our interval distance

So how to write it

Look at the lower limit

0 over n is 0

So it is our a equals 0

How about the upper limit

You can think it by yourself

Yes

n over n means 1

So it is our upper limit

So from 0 to 1

we can finish this

Together

write it like this ok

Limit summation means definite integral

from 0 to 1

The corresponding

the summation function

we can write it the integral function

So we do

this equals π over 6

It is our final limit

equals π over 6

So the definition of definite integral

gives us some idea

Try to find the limit

The class is over

Try to understand

See you next time