Page 50 - DMTH202_BASIC_MATHEMATICS_II

Unit 4: Definite Integral

Notes
b
x
dx
We take this expression as the definition of a definite integral and denote it by  f ( ) .
a
Hence,
b
x
f
a
h
h
h
h



 f ( )dx  lim [ ( ) ( f a h ) f (a  2 ) f (a  3 ) f  (a n  1 ]

h  0
a
where nh = b – a.
The process of evaluating a definite integral by using the above definition is called integration
from first principles or integration by ab-initio method or integration as the limit of a sum.
The following results are useful in evaluating definite integrals as limit of sums:
( n n  1)


1. 1 2 3   (n  1)
2
2 n
2
2
2
2. 1  2  3    (n 1)  (n  1)(2n  1)
6
 2
( n n 
1)
3 
3
3
3
3. 1  2  3    (n 1)   
 2 
n
( a r  1)
1
n
2

4. a ar  ar   ar 
r  1



5. sin a  sin(a h) sin(a 2h)   sin[a n–1]

  n  1    nh 
h
sin a     sin  

   2    2 
 h 
sin  
 2 

h
h

6. cosa  cos(a h ) cos(a  2 )   cis [a (n 1) ]
  n  1    nh 
h
cos a     sin  

   2    2 
 h 
sin  
 2 
!
Caution The value of the definite integral of a function over any particular interval depends
on the function and the interval, but not on the variable of integration that we choose to
represent the independent variable.
The variable of integration is known as a dummy variable.
Example: Evaluate the following definite integrals as limit of sums:
2 1
1.  (2x  3)dx 2.  (x  3)dx
1  1
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