Integral and its application

Group 3


Group 3


Group 3
Mathematical Analysis I

Quỳnh Anh

Dũng

Hồng Mai

Quỳnh 16
Thảo

Mạnh

Vũ

Ngọc


Group 3

MATHEMATICAL ANALYSIS I
Integral and its application

About the introduction
The overview about

and development of the

antiderivative

theory of integration

- integrals

Content 1
Ghi chú Nội dung 1

Content 2

Content 3

Content 4

Application of definite

Conclusion and

integrals

references


Group 3

MATHEMATICAL ANALYSIS I
Integral and its application

Content 1
The overview about antiderivative - integral

1.Antiderivative indefinite integral

Content 2
Ghi chú Nội dung 2

Content 3

Content 4

2. Definite integral
3. Integration method


Group 3

MATHEMATICAL ANALYSIS I
Integral and its application

Content 1
Ghi chú Nội dung 1

Content 2
Some details on the birth and development of integral

theory

1. History of integrals
2. A brief overview of the
application of integrals.

Content 3
Ghi chú Nội dung 3

Content 4


Group 3

MATHEMATICAL ANALYSIS I
Integral and its application

Content 1
Ghi chú Nội dung 1

definite integrals

Content 2
Ghi chú Nội dung 2

2.Algebraic application of definite
integrals

Content 3
Application of

definite integrals

Ghi chú Nội dung 4

Content 4

1.Geometrical applications of


Group Name

MATHEMATICAL ANALYSIS I
Integral and its application

Content 1
Ghi chú Nội dung 1

Content 2
Ghi chú Nội dung 2

Conclusion

Content 3
Ghi chú Nội dung 3

Content 4
Conclusion


1

CONTENT 1
The overview about antiderivative
- integrals


1.
-

CONTENT 1
Primitive - indefinite integral.

A.

Definition

i.

Antiderivative

The function F(x) is an antiderivative of the function f(x) on (a, b) if F'(x) = f(x) ∀x ∈ (a, b).

F(x) = G(x) + C

ii.

Indefinite integral

The set of antiderivatives of f (x), recorded as ∫f(x)dx is called the indefinite integral of the
function f(x).

-

Properties

1) [ f(x)dx]’ = f(x)
∫

Let F(x) and G(x) be two antiderivatives of the function f(x) on (a, b), then there exists a
constant C such that:

-

B.

If F (x) is an antiderivative of f (x) then we have

∫

�(�)�� = F(x) + C (C =

const)

2) [f(x) ± g(x)]dx = f(x)dx ± g(x)dx
∫
∫
∫
3) kf(x)dx = k f(x)dx (với k ∈ R)
∫
∫

CONTENT 1
 

2.

Definite integral.

B.
A.
 

Properties
Definition

. Consider the function f(x) on a closed interval [a,b] and divide [a,b] into n points:
1.a= x-02.We Ifhave
f(x) ≥the
0, Riemann
x [a,b], then
sum:
3.Sn==If f(x)
g(x),
(xi–x≥i-1
).f(ɛxi) =[a,b],
(x1–xthen
0).f(ɛ1) + (x2–x1).f(ɛ2) +…+ (xn–xn-1).f(ɛn)
4. |
If max |x − x | → 0, then Sn → α, that means f(x) is an integrable function

5. = + (ai b c)i–1
�
�

∫

1≤i≤n

�

�(�)�� = F(b) – F(a) ≡ [�(�)]|

and α is called integral of f(x) on [a,b], or = α

. If a function f(x) is continuous on [a,b], then f(x) is integrable on [a,b]
. Instead of the above definition, we can also define an integral by Newton – Leibnitz formula:

�


CONTENT 1

3.

Integration method

A.

The antiderivative of a basic function

ii.i.
iii.

   special
Elementary
function function
Inverse
trigonometric
Common
functions

1)

1)

2)

2)

2) ) ϵ (

3)

3) ) ϵ [

4)

4) ) ϵ [

3)

4)

1) ) ϵ (

5)

6)
7)

8)

5)

9)
6)


CONTENT 1

3.

Integration method

B.
C.

Methods
of integral calculation
Sub-conclusion

 Example 1:

 
  So
From this we see that in addition to applying basic trigonometric transformations, the recognition of “hidden” trigonometric differential formulas will also

Solution:
- As learned in high school, to determine an integral expression we usually use two methods: By substitution and by parts. Use the above two methods,
be an important skill. If I realise them, the problem solving becomes much simpler.
set
I=combining
other mathematical tricks, we can easily solve all the fundamental integrals. Moreover, in some definite integral problems, based on the two
: some of the following common types:
Like
limits
of the integral we can find the way to solve the problem.

1) (Acosx ∓ Bsinx)dx = d(Asinx ± Bcosx + C)
2
2
2) (A ∓ B)sin2xdx = d(Asin x ± Bcos x + C)
4
4
3) −sin4xdx = d(sin x + cos x + C)

 


NỘI DUNG 1

Content 2
Some details on the introduction and development of

When was integral

integrals

introduced? How does
it develop?

What is the application of
integrals in the development
of the history of science?

1. History of

2. A brief overview of

integrals

the application of
integrals.


NỘI DUNG 1

Content 2
1.
2.

History
intergrals
Integralofapplication

the force used on the ship; or it is still not possible to calculate the appropriate angle of fire

But it's interesting that the first person to fly into space is Yuri Gagarin
Newton and Gottfried Leibniz abstracted the above connection into
(1934 - 1968) – a Soviet pilot and cosmonaut. His space trip lasted 108
the connection between differential and integral, two important
minutes on April 12, 1964 and became an important event in human’s
shapes of calculus.
history.
Newton and Leibnitz, together with their students, used the

in the
with
resistance
the bullet to go the farthest. Today, the infinitesimal
who
laidenvironment
the first brick to
enable
integralfor
math.

relationship between differential and integral to broaden methods of

Until

the 17th
the systematic
of the above
of area
andIn volume
calculus
is onecentury,
of the important
toolsdevelopment
in physics, economics
and method
probability
theory.
the

integration, but methods of integral calculation are known in the

calculation was done by mathematicians such as Cavalieri, Torricelli, Fermat, Pascal and many other

equation. The present degree is largely presented in Euler's work. The

mathematicians. In 1659, Barrow established the relationship between the area calculation and the

contributions of mathematicians Tchébicheff and Ostro-gradski ended

tangent finding of the relevant curve. Not long after, Issac

the process of developing this calculation.

Calculating the area of a plane or calculating the volume of an object in space, whose shape could not

Before the advent of integrals, mathematicians were able to calculate the speed of a

apply the formulas of elementary geometry had long been devised. Since ancient times, the eminent

ship. But they still could not calculate the relation between the acceleration and the ratio of

mathematician and physicist Archimède has used elementary mathematical tools to calculate the area
of a number of planes limited by curves such as spheres and cones. He is considered to be the person

1960s, the first infinitesimal functions had helped aerospace engineers in project Apollo to
calculate the data that landed the first human – Neil Armstrong on the Moon.


Content 3
I.

DUNG 1
Application ofNỘIdefinite
integrals

Geometrical applications of definite integrals

1.

A.

General diagram using integral
The first method – sum of integrals

Integral value:18.92

Integral value:18.92

Total area: 10 trapezoids

Total area: 20 trapezoids

 We want to calculate any quantity Q such as volume, area, length, etc. We divide Q into n parts in an appropriate way:
Q = Q1 + Q2 +…+Qn

(1)

Then calculate the approximate value of Q k, plug in (1) and use the limit to find the sum Q.
- We used the above method to calculate the area under a curve and got:

- The larger n is (the greater the number of division), the higher the accuracy, so the calculation is very hard. The integral helps
us to reduceIntegral
the error
and calculate the exact value of the quantity we are looking for. Integral value:18.92
value:18.92
Total area: 50 trapezoids

Integral value:18.92
Total area: 100 trapezoids

by:
x-axis,
y = f(x) = cosx + 2 and
two lines x = 0; x = 9.5.

(we will learn more in part 3.1.2)

Total area: 30 trapezoids

Ex : Find the the area of the plane bounded
1


NỘI DUNG 1

Content 3
1.

General diagram using integral

B.

Secondmethod–differentialdiagram

-Assumpti
on:Quanti
<�<�′
≤�
∆Q=q(x)∆x
+0(∆x)tyQdependson
(2)thesegments: �≤�′
W=>
ehave:Q[a',b']=Q[a',c]+Q[c,dQ(x)
b'] =q(x)dx,so:

and
=Q[a,
QQ
=Q[a,
b]b]
= (3)
To-calculateQwesetQ(x)=Q[a,Sox].the
Consi
deringofinthe
crements:
content
differentialdiagrammethodistoestablishtherelation(2),whosefinalresultis
(3).=Q(x+∆x)−Q(x)=Q[a,x+∆x]−Q[a,x]=Q[x,x+∆x]
∆Q
Andtrytorepresentitaslinearintermsof∆x:


2.Area of flat figure

NỘI DUNG 1

Content 3

A. Area under the curves in Cartesian coordinates

Apply the sum of integrals method to calculate the area of
the plane figure S formed by two any functions f 1(x) and f2(x) that

y

can be divided into an infinite number of curved trapeziums MPFE y+∆y

P

y

(Figure 1). It is easy to prove the following cases:

M

F

E
a

O

If the function y = f(x) is continuous on [a, b] then the area S under:

x

x +∆x

b

x

y

 

(C): y = f(x)

+∆x
a

 
O

b
x


NỘI DUNG 1

Content 3
 

Example 2: Calculate the area S under the curves:

 

We have:
S=
S=
=
=.


Content 3

 

If two functions y = f(x) and y = g(x) are continuous
on [a, b] then the area S under the curves:

NỘI DUNG 1


Content 3

NỘI DUNG 1

Note: Four lines of the graph formed by two continuous functions f(x) and g(x) and two lines
�= �; = b always create a plane. However, just two graph of two continuous functions f(x) and
g(x) are enough to form a plane figure, for example as shown in the following figure:

 

Problem: Find the area of a plane figure S bounded by are two open curves and intersect to each other.
+ Step 1: Solve the equation
+ Step 2: Use the formula:


NỘI DUNG 1

Content 3
Example 3: Calculate the area under the curves:
We consider the difference of two functions:

Note: Equation (4) is only used to calculate the area of a limited simple plane (H) by the graph of the function f(x), g(x) and two lines x = a, x = b.In the case of
Because at 3 points whose coordinates are -2;0;1. So the area of the figure is:

a plane figure (H) is defined by more than two graphs of the function, then to calculate the area (H), we must draw figure (H) – ie draw defined lines (H).
Based on that, we divide the plane figure (H) into simple shapes simple that we know how to calculate.


Content 3
 

Example 4: Calculate the area S of the plane figure (H) under by the graph of the function
and straight line
Equation of the intersection point:
1,
2,
3,
We can easily see the area of the plane figure (H):
(H) =
=

NỘI DUNG 1


Content 3
 

2.

NỘI DUNG 1

Area of a plane bounded by a line with parametric equations

 If the curve (C): y = f(x) has parametric equation

Example 5 : Calculate the area of an ellipse formed by (E)

In the formula to calculate the area I replace

We
have the parametric equation of (E) on the first quadrant on the plane (Oxy).
+ y = f(x) by y = ω(t)
′
+ dx by φ (t)dt
+ Two limits a and b are replaced by α and β which are solutions of a = φ(α) and b = φ(β) respectively.
Then :

S=

Converting the integral we get:
If the curve (C) is closed, counter-clockwise and the area S is limited to the left and (C) has a

parametric equation with 0 ≤ t ≤ T where T
is the period of it.

So