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4.1.2.+Definite+integral(4)-S

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ID NO:
4.1.2. Definite Integral(4)
NAME:
calculus1 journal
[Today’s Goal]
1. To Understand ‘Relationship between Definite Integral and Differentiation’
2. To Be Able To explain ‘Fundamental Theorem of Calculus’
[Key Words]
Fundamental Theorem of Calculus(미적분의 기본정리)
[Main contents]
[Warm up!] Definite integral+
➀ Symbol and Definition of Definite integral:
➁

  

➂ For a constant function    . find

  .

➃ Suppose that a function  is continuous over   as shown on the right. Both
  and   are the areas bounded by   , the vertical lines    and   , and

the -axis, where     and    . Evaluate
 .

***Relationship between Definite Integral and Differentiation***
[Q 1] Assume the relationship between Definite Integral and Differentiation
➀ By using geometric meaning of definite integral, find


 .

➁ Describe the relationship between the result of ➀ and the function  .
⇒ Assume:
[Q 2] Relationship between Definite Integral and Differentiation
Let    be a function which is continuous and non-negative over the closed interval
 . As shown in the diagram on the right, the area   of the region bounded by the
curve   , the lines    and    ( ≤  ≤ ), and the -axis is given by
  

 .

Note that   is a function of .
Our final step is to know how the definite integral and indefinite integral are related.
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4.1.2. Definite Integral(4)
calculus1 journal
ID NO:
NAME:
To analyze the function   , we will observe the derivative  ′  .
❶ However, we don't know whether  ′  exists or not (we don't know the equation for   at the moment).
❷ So, we need to approach the derivative ′  by definition.
❸ By definition, ′  
∆
, where ∆ is small, and the increase in  is ∆ : ∆     ∆    .
lim 
∆
∆→
Q1. Find the area that represents ∆ on the right.
∆
Q2. What value does  approach as ∆→?
∆
[Hint: We can approximate ∆ by applying quadrature by parts. That is,
we can approximate ∆ with a rectangle that has width ∆.]
  ∆
❹ Prove Q2.
Since it is difficult to evaluate ∆ (actually, we can evaluate ∆ by applying quadrature by parts), we will use the
Squeeze Theorem. Since    is continuous over    ∆, it attains both a maximum and a minimum value there.
∆
is a limit, so we should consider both the left-hand limit and the right-hand limit, that is, the
lim 
∆
(Note that
∆→
limits for ∆   and ∆  .)
When ∆  ,    attains a maximum value  and a minimum value  over the
closed interval    ∆.
Comparing areas, we have
 · ∆ ≤ ∆ ≤  · ∆ -----➀
When ∆  ,    attains a maximum value  and a minimum value  over the
closed interval    ∆.
Comparing areas, we have
 · ∆ ≥ ∆ ≥  · ∆ -----➁
∆
Using ➀ and ➁, we can get the inequality for  :
∆
∆
 ≤  ≤ .
∆
Letting ∆→:
lim 
≤
∆→
∆
≤ lim  .
lim 
∆
∆→
∆→
Since    is continuous over the closed interval    ∆, as ∆→, both → and →. Thus,


 ≤    ≤ . That is,     .



∴


   

The Relationship Between Integration and Differentiation
Given a function  which is continuous over a closed interval   ,




    ( ≤  ≤ ).


[Note:
  is an antiderivative of  .]

[Example 1] Evaluate the following:

(1) 

- 2 -

  



(2) 



  

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4.1.2. Definite Integral(4)
calculus1 journal
ID NO:
NAME:
[Exercise 1] Find the derivatives of the following functions:

(1)

  

(2)


(3)
     

 


       

(4)

       


***The Fundamental Theorem of Calculus***
Using the relationship between integration and differentiation, we can obtain a relationship between indefinite integrals and
definite integrals. The result is called the Fundamental Theorem of Calculus.
Given a function  which is continuous over a closed interval  , let
  

  ( ≤  ≤ ).

The relationship between integration and differentiation tells us that
 ′     ≤  ≤ .
That is,   is an antiderivative of ; thus, if   is any antiderivative of , then

  
      , for some real number  .

[Note: Since   is a function,  is a constant defined by   .]
Since   ____, we have
         ⇒     , and
  

        .

Normally, we write this expression without  :

        ---➀


Q1. Using ➀, express
  in terms of  .


  .

The Fundamental Theorem of Calculus
If a function  is continuous over a closed interval   and has an antiderivative   , then

        .

[Note: We can denote      as
  .]
[Example 2] Evaluate the following definite integrals:

(1)
 

- 3 -

(2)
  


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ID NO:
4.1.2. Definite Integral(4)
NAME:
calculus1 journal

[Q 3]

  and  



By using FTC, describe the relationship between

  and  .


Properties of Definite Integrals(1)
If a function  is continuous on a closed interval   , then the following properties hold:

❶
  


❷

   


[Exercise 2] Evaluate the following definite integrals:

(1)
 


(2)
 


(3)
   


[Exercise 3] p.148 [문제 02], [문제 03], [문제 04] [Exercise 4] p.153 [문제 08]
summarize
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