Riemann Integral:
Let
f
be a bounded function on [,]ab.
Let P: 01
... n
ax x x b=<<<= be a partition of [,]ab. The norm of the partition,
denoted P, is equal to 1
max{ : 1,2,... }
ii
x
xi n
−
−= .
1
For 1, 2, ... , let [ , ]
inn
incxx
−
=∈.
1
1
()( )
n
iii
i
f
cxx
−
=
−
∑ is called a Riemann Sum for f over [a, b] .
The function f is said to be Riemann integrable on the interval [a, b] provided
1
01
lim ( ) ( )
n
iii
Pi
f
cxx L
−
→=
−=
∑ .
That is, for each 0
ε
>there exists 0
δ
> such that given any partition
P: 01
... n
ax x x b=<<<= with P
δ
<
, and given any choice
of 1
[ , ], 1, 2, ...
inn
cxxi n
−
∈=, it follows that
1
1
()( )
n
iii
i
fc x x L
ε
−
=
−−<
∑ .
Notation: The limit is usually denoted by ()
b
a
f
xdx
∫.
