The natural question that arises is the following:
Can we improve the approximations in (1) so that upper and lower sums come closer to the actual ‘area(S)'?
To answer this, let us observe the following:
Let be two partitions of such that
and
i.e., has all the points of and an extra point in the subinterval. Then
Let and be two partition of such that every point of is also a point of . Then we say is a
Can we improve the approximations in (1) so that upper and lower sums come closer to the actual ‘area(S)'? 
be two partitions of 
such that 
has all the points of 
and an extra point 
in the 
subinterval. Then 
and 
be two partition of 
such that every point of 
is also a point of 
. Then we say 
is a 
. 
of partitions of 
is called a sequence of refinement partitions if for all 
is a 
. 
