Lecture 16 : Integral from upper and lower sums [Section 16.1]
(ii)
Let
The sum is called upper sum of and the sum is called lower sum of for the partition .
16.1.3
Note:
For , each is the area of the rectangle with base and height . The number is the sum of the areas of all such rectangles. These rectangles cover of the region . Similarly, each and is the area of the rectangle with base and height . The number is the sum of all these rectangles which try to fill up the region . The sum under estimates the ‘area' of and over estimates the area of , i.e.,
Geometrically, the required area ‘Area(S)' of the region is captured between and i.e., for every partition of .
is called 
and the sum 
is called 
for the partition 
. 
, each 
is the area of the rectangle with base 
and height 
. The number 
is the sum of the areas of all such rectangles. These rectangles cover of the region 
. Similarly, each 
and 
is the area of the rectangle with base 
and height 
. The number 
is the sum of all these rectangles which try to fill up the region 
. The sum 
under estimates the ‘area' of 
and 
over estimates the area of 
, i.e., 
is captured between 
and 
i.e., for every partition 
of 
. 
