Lecture 16 : Integral from upper and lower sums [Section 16.1]
However, it is not as easy for general regions as is for the regions with straight side. But it is possible to find an approximation to area using concept of area of the rectangles. The idea is to fill up (cover up) the required area by rectangles with sides parallel to the axes. To do this, let us make some definitions.
16.1.1
Definition:
Let be a closed bounded interval.
(i)
Let
be points in the interval . Then,
is called a partition of the interval .
(ii)
A partition with points divides the interval into closed subintervals:
A typical subintervals is called theth-subinterval of the partition.
be a closed bounded interval. 
. Then, 
. 
with 
points divides the interval 
into 
closed subintervals: 
is called the
th-subinterval of the partition.
, the number 
