MAT136H1 Lecture Notes - Integral

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4 Feb 2013
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5.2 Integration & Anti-derivatives
Definite Integrals
Question #2 (Easy): Expressing the Limit as the Definite Integral
Strategy
By the definition of the definite integral, the limit expression is set equal to the definite integral  
over the interval   

  , where  
and    and
is integrable on  .
Therefore, direct correlation between the limit and definite integral is established:
1) Definite integral means the interval  is fixed. Definite integral describes this as 
.
2)  from the limit becomes in the integral since   .
3)  
 
in the limit becomes general function notation  in the integral.
Sample Question
Convert the limit into a definite integral over the given interval.
 
 , 
Solution
1) The interval is set as , thus     .
And the definite integral will start with 

.
2) The above step already took care of converting  into .
3) Then take  from the limit and let  .
a. This means  
b. By definition the function notation means:     
Therefore, converting the limit into definite integral over given interval is expressed as follows:
 
  

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