Lecture 16 : Integral from upper and lower sums [Section 16.1]
16.1.6
Examples:
(i)
For an interval , a natural sequence of refinement partitions is given by
(ii)
For an interval , let,
Then is not a sequence of refinement partition of .
16.1.7
Theorem:
Letbe a sequence of refinement partitions of . Then the following hold:
(i)
The sequence is monotonically decreasing and is bounded below by .
(ii)
The sequence is monotonically increasing and is bounded above by .
(iii)
Both and are convergent sequences. In fact, for is continuous,
and the limit is independent of the sequence of refinement partition of .
This motivates for the following definition:
16.1.8
Definition:
For a continuous function , the real number as given by theorem
16.1.7 (ii) is called the definite integral or just the integral of over and is denoted by
, a natural sequence 
of refinement partitions is given by 
, let, 
is not a sequence of refinement partition of 
. 
be a sequence of refinement partitions of 
. Then the following hold: 
is monotonically decreasing and is bounded below by 
. 
is monotonically increasing and is bounded above by 
. 
and 
are convergent sequences. In fact, for 
is continuous, 
of refinement partition of 
. 
, the real number 
as given by theorem 
over 
and is denoted by 
