Lecture 16 : Integral as a limit of Riemann sums [Section 16.2]
Note that for every partition , the sum depends not only on the partition , but also on the choice of points . However, for every partition , the following holds:
.
We hope that as we make smaller and smaller, the Riemann sum will approximate the required area better and better. This actually does happen for integrable functions. In fact, we have the following:
16.2.2
Theorem (Riemann):
A function is integrable if and only if there exists such that for every sequence of partitions of with , and every sequence of Riemann sums .
Further, in that case,
16.2.3
Note:
(i)
In view of the above theorem, for a function , which we know is integrable, to compute we can use
any convenient sequence of partition of with and compute
, the sum 
depends not only on the partition 
, but also on the choice of points 
. However, for every partition 
, the following holds: 
. 
smaller and smaller, the Riemann sum 
will approximate the required area better and better. This actually does happen for integrable functions. In fact, we have the following: 
is integrable if and only if there exists 
such that for every sequence 
of partitions of 
with 
, and every sequence 
of Riemann sums 
. 
, which we know is integrable, to compute 
we can use
of partition of 
with 
and compute 
