Integral and its application

Primitive indefinite integral. Definition Antiderivative The function F(x) is an antiderivative of the function f(x) on (a, b) if F(x) = f(x) ∀x ∈ (a, b). 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: F(x) = G(x) + C Indefinite integral The set of antiderivatives of f (x), recorded as ∫f(x)dx is called the indefinite integral of the function f(x). If F (x) is an antiderivative of f (x) then we have

Group 3

Group 3

Group 3MATHEMATICAL ANALYSIS I

Content 1

Content 2

Content 3

Content 4

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The overview about antiderivative

- integrals

About the introduction and development of the theory of integration

Application of definite

integrals

Conclusion and references

Integral and its application

Group 3MATHEMATICAL ANALYSIS I

Integral and its application

Content 1

Content 2

Content 3

Content 4

The overview about antiderivative - integral

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1.Antiderivative - indefinite integral

2 Definite integral

3 Integration method

Group 3MATHEMATICAL ANALYSIS I

Integral and its application

Content 1

Content 2

Content 3

Content 4

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Some details on the birth and development of integral

Group 3MATHEMATICAL ANALYSIS I

Integral and its application

Content 1

Content 2

Content 3

Content 4

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Group NameMATHEMATICAL ANALYSIS I

Integral and its application

Content 1

Content 2

Content 3

Content 4

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Conclusion

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Conclusion

CONTENT 1

The overview about antiderivative

- integrals

1

CONTENT 1

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).

- 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:

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

B. Properties

CONTENT 1

2 Definite integral.

A. Definition

.Consider the function f(x) on a closed interval [a,b] and divide [a,b] into n points:

a= x0<x1<x2<…<xn=b, then we take a number ɛi ∈ [xi–1, xi], for i = 1, 2,…, n

We have the Riemann sum:

Sn== (xi–xi-1).f(ɛi) = (x1–x0).f(ɛ1) + (x2–x1).f(ɛ2) +…+ (xn–xn-1).f(ɛn)

If max |xi − xi–1| → 0, then Sn → α, that means f(x) is an integrable function

1 ≤ i ≤ n

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:

iii. Common special functions

1)

2) 3) 4) 5)

6)

CONTENT 1

B Methods of integral calculation

- 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, combining other mathematical tricks, we can easily solve all the fundamental integrals Moreover, in some definite integral problems, based on the two limits of the integral we can find the way to solve the problem

 So

I=

C Sub-conclusion

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

be an important skill If I realise them, the problem solving becomes much simpler

Like some of the following common types:

1) (Acosx ∓ Bsinx)dx = d(Asinx ± Bcosx + C) 2) (A ∓ B)sin2xdx = d(Asin2x ± Bcos2x + C) 3) −sin4xdx = d(sin4x + cos4x + C)

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Content 2

When was integral

introduced? How does

it develop?

Some details on the introduction and development of integrals

1 History of integrals

2 A brief overview of the application of integrals.

What is the application of integrals in the development

of the history of science?

Newton and Leibnitz, together with their students, used the relationship between differential and integral to broaden methods of integration, but methods of integral calculation are known in the equation The present degree is largely presented in Euler's work The contributions of mathematicians Tchébicheff and Ostro-gradski ended 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

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

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

who laid the first brick to enable integral math

Until the 17th century, the systematic development of the above method of area and volume

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

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

tangent finding of the relevant curve Not long after, Issac

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

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

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

in the environment with resistance for the bullet to go the farthest Today, the infinitesimal

calculus is one of the important tools in physics, economics and probability theory In the

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

But it's interesting that the first person to fly into space is Yuri Gagarin

(1934 - 1968) – a Soviet pilot and cosmonaut His space trip lasted 108 minutes on April 12, 1964 and became an important event in human’s history

2 Integral application

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1. General diagram using integral

A. The first method – sum of integrals

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 Qk, 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:

(we will learn more in part 3.1.2)

- 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 reduce the error and calculate the exact value of the quantity we are looking for.

by:

x-axis,

y = f(x) = cosx + 2 and two lines x = 0; x = 9.5.

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Content 3

B Second method – differential diagram

- Assumption: Quantity Q depends on the segments: ≤ < < ≤ � �′ � �′ �

We have: Q[a', b'] = Q[a', c] + Q[c, b']

and Q = Q[a, b]

To calculate Q we set Q(x) = Q[a, x] Considering increments:

∆Q = Q(x + ∆x) − Q(x) = Q[a, x + ∆x] − Q[a, x] = Q[x, x + ∆x]

And try to represent it as linear in terms of ∆x :

1 General diagram using integral

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+∆x

Content 3

2 Area of flat figure

Apply the sum of integrals method to calculate the area of

the plane figure S formed by two any functions f1(x) and f2(x) that

can be divided into an infinite number of curved trapeziums MPFE

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

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

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Content 3

If two functions y = f(x) and y = g(x) are continuous

on [a, b] then the area S under the curves:

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Content 3

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:

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Content 3

Example 3: Calculate the area under the curves:

We consider the difference of two functions:

Because at 3 points whose coordinates are -2;0;1 So the area of the figure is:

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

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.

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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:

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

In the formula to calculate the area I replace

+ 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 =

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

Then: S = ��

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

We have the parametric equation of (E) on the first quadrant on the plane (Oxy).

Converting the integral we get:

So

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3 Arc length

A Theoretical basis

 We can subdivide this curve into an infinite number of “nearly-straight” lines

Δl and sum them together Consider and Δx > 0 such that Δx

 With Δx small enough, we consider the length of the graph curve f(x) limiting

between the two lines x= and x= is the length of the line connecting 2 points A

and B as shown in figure 3 And also because Δx is small, so AB = (d) should be

considered as the tangent at x of f(x)

 So the arc length Δl AB = , with a is the angle formed by the tangent AB

Therefore: Δl=Δx (5)

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3 Acr length

B Curve length in Cartesian coordinates

From (5) we take the sum of the lengths of the small “straight” segments together, we get the formula to calculate the curve length limited by is L

Then: L=

Example 7: Calculate the length of the arc OA lying on the parapol y= with a≠0,

which O is the origin, A is a point on the parabol whose latitude is t

Therefore L = = = = [x+ ] = +

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3 Acr length

C Curve length with parametric equation

We demonstrate the complete analogy from how to convert the area of a plane figure in Cartesian coordinates to the area of a plane bounded by a line with parametric equations (in section 3.1.2)

From then, instead of (5) we get: l = dx in

=dx in

Example 8: Calculate the length of the arc of Archimedes’spiral

We have: x’(t) = -asint, y’(t) = acost, z’(t) = a

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Content 3

II ALGEBRAIC APPLICATIONS OF DEFINITE INTEGRAL

 We evalute the inequality in terms of functions and integrals

Example 11: Prove that: < <

Consider f(x) = is continuous on [0;]

We have 0 ≤ cosx ≤ 1, for all x ϵ [0;]

Then: 0 ≤ ≤ 1, for all x ϵ [0;]

=> , for all x ϵ [0;]



Example 12: Prove that: ln(1+a) > , for all a > 1

+) In the coordinate system Oxy, consider the graph of the function (C): y = and set ϵ [1;a] Let the points A, B, C, D be A(1; 0), B(1+a; 0), C(; 0), D()

We have: y’ = => y’’ = = > 0, for all x > 0

Then: y = is a concave function, for all x > 0

Two lines x = 1, x = 1+a intersect the tangent (d)

at D() at two points E, F and

intersect the graph (C): y = at two points H, G.

Content 4 Conclusion

Content 4

+ Is aimed at raising the reader's awareness of the integral and its application in mathematics

Through the above analysis, we will have an overview and a more positive view of integrals, about applications that are very close to reality But in any field, two methods

total integral and differential diagram are always two directions for easy access and use in practical problems Today, calculus - integration requires learners and researchers

to have a deep understanding of the nature so that they can come up with new and more practical applications, not just stopping at things that the group has already learned tell

In the process of implementation, the team has tried to limit errors but still cannot avoid mistakes, so the essay is still incomplete and profound The student group hopes to receive more comments from readers

Finally, the group would like to express their sincere thanks to Mr Nguyen Van Thin's support during the process of writing the essay.

Thanks For Watching!

Group 3