Let be a continuous function, not necessarily nonnegative. Partition
into
consecutive sub-intervals
(
) each of length
, where we set
,
and
to be successive points between
and
with
. Let
be any intermediate point in the sub-interval
. Then the sum
is called a Riemann sum for on
.
Suppose we let the number of partition in tends to infinity.
We call the Riemann integral (or definite integral) of
over
and we write
In other words,
if the limit on the right side exists.
If we put we the obtain
Example 1. Find
Solution. Clearly
Then if we choose ,
we then get
With this it is easy to see that
since
If we put we the obtain
Example 2. Find
Solution. Clearly,
which yields
Remark. It is worth mentioning that in general it is not true that
For example, we all know that for each fixed
but
The point is
holds true only for finite summation.