- Trang chủ >>
- Sư phạm >>
- Sư phạm toán
THUYẾT TRÌNH TÍCH PHÂN BẰNG TIẾNG ANH INTEGRALS
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (5.38 MB, 30 trang )
INTEGRALS
• Instructor: …
• Presentation group 4: …
ISAAC NEWTON
was scientist who invented derivatives,
integrals and differentials
INTEGRALS
DEFINITION OF
INTEGRALS
PROPERTIES OF
AN INTEGRAL
METHODS OF
COMPUTING INTEGRALS
I. Definition of integrals
Suppose that is a continuous function on the closed interval
and are two antiderivatives of Prove that , (that means the
difference does not depend on the choice of antiderivative).
Let be a continuous function on the closed interval . Suppose that is
an antiderivative of on the closed interval .
The difference is called the integral from a to b (or the definite
integral on the closed interval of the function denoted by
Closed interval: khoảng đóng /đoạn
Antiderivative: nguyên hàm
Depend on: phụ thuộc vào
Definite Integral: tích phân xác định
I. Definition of integrals
We also use the notation to demonstrate the
difference
Therefore
We call the integral sign, a is the lower limit, b is
the upper limit, is the expression under integral sign
and is the integrand.
Notation: ký hiệu
Demonstrate: chỉ/ biểu thị
Difference: hiệu số
Integral sign: dấu tích phân
Lower limit: giới hạn dưới
Upper limit: giới hạn trên
Expression: biểu thức
Integrand: hàm số dưới dấu tích phân
Note
In cases hoặc , we make the
convention that
In cases : trong trường hợp
Convention: quy ước
R
E
M
A
X
a) The integral of the function from
to can be denoted by or
That integral only depends on and
limits does not depends on variables
.
Variable: biến số
R
E
M
A
X
b) The geometrical meaning of integrals.
If the function is continuous and
non - negative on the closed interval ,
then integral is the area S of the curved trapezoid
enclosed by the graph of the , the axits and two
straight lines Therefore
Geometrical: hình học
Non – negative: khơng âm
Curved trapezoid: hình thang cong
Area: diện tích
Enclose: giới hạn/ bao quanh
Straight line: đường thẳng
Example:
1)
2)
II. PROPERTIES OF AN INTEGRAL
(a
Example 1: Compute:
Solution: We have
Compute: Tính
Example
2: Compute
Solution: We have
Because
So:
III. METHODS OF COMPUTING INTEGRALS
1. METHOD OF
INTEGRATION BY PARTS
INTEGRATION: lấy tích phân
METHOD: phương pháp
COMPUTING: cách tính
2. METHOD OF
CHANGE OF
VARIABLES
1. Method of change of variables
Theorem
•Let be a function continuous on the closed interval
•Suppose that the function is continuously differentable on the
closed interval such that
and for every . Then
Example 1: Compute
Solution
Let We have
• When
• When
The assumptions of the above theorem are satisfied.
So
=
Note
In many cases we also use the method of change of variables
in the from below:
Let be a function continuous on the closed interval . To
compute , sometimes we use the function as a new
variable, where is continuously differentiable on the closed
interval and .
Suppose that we can write
,
Where is continuous on the closed interval . Then, we have
Example 1: Compute
Solution
Let We have
• When
• When
Therefore
Example 2: Compute
Solution
Let We have
• When
• When
Therefore
- 1)
2. Method of integration by parts
Theorem
If and are two function which are
continuously differentiable on the closed
interval . Then
or
Example 1: Compute
Solution
Letting and , we have :
.
So
Example 2: Compute
Solution
Letting and , we have:
. So
= 1-
THANKS
FOR
LISTENIN
G
Hello
!
GAME
START
Hi
!
PART
Groups have 5 minutes to review
vocabulary. Then we give each
group of Vietnamese word
boards. The task of 3 groups is to
go up to the board and fill in
English words that exactly match
the meaning of that Vietnamese
word. For each correct word, the
group gets 5 points.
