| Suppose that f
(x)
is defined on [a,b]. |
| Let P
= {x0, x1, ......, xn}
be
a set of points in [a,b] |
Let  i  [xi-1,
xi]
for i = 1, 2, ......
, n |
and  =
where  i
= xi - xi-1 |
When  0
as n |
the summation a
= 
exists such that |
|
(1)
|
a
is independent of the choice of i |
|
(2)
|
a
is independent of the the partition P |
Then, we say that f
(x)
is Riemann-integrable on [a,b] |
We can represents the defintion
in " - "
statement |
| Suppose there exists a fixed real number
I. |
If for any given >0,  >0, |
| such that for any partition P,
whenever <,
we have |
|
| then we call I
to be the definite integral of f
(x) on [a,b] |
and denote I
= |
Remarks:
| (1) |
When a,b
are constants, then I
is also a constant. |
| (2) |
In the definition, a
< b, when a
> b or a
= b,
|
is not defined. |
|
|
So we define
|
|
= |
 |
when a
> b |
|
|
|
|
|
| and |
 |
= |
0 |
when a
= b |
|
|
|
|
|
|
|
|
