Chapter 6
Integration
6.1
Integrals of continuous functions
1
6.1.1
Existence of the integral
Let
f
(
x
) be a function defined on [
a, b
]. We want to define its integral
R
b
a
f
(
x
)
dx
.
Definition 6.1.1.
A
partition
P
of the interval [
a, b
] is a finite set of points
{
a
=
x
0
, x
1
, x
2
, . . . , x
n
=
b
}
,
x
i
< x
i
+1
.
Definition 6.1.2.
On each subinterval [
x
i
, x
i
+1
], of the partition
P
, define
M
i
= sup
f
(
x
)
x
i

1
≤
x
≤
x
i
m
i
= inf
f
(
x
)
x
i

1
≤
x
≤
x
i
U
(
f, P
) =
n
X
i
=1
M
i
(
x
i

x
i

1
)
L
(
f, P
) =
n
X
i
=1
m
i
(
x
i

x
i

1
)
U
(
f, P
) is the
upper sum
of
f
on the partition
P
, and
L
(
f, P
) is the
lower sum
of
f
on
the partition
P
.
1
May 2, 2007
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82
Math 413
Integration
Definition 6.1.3.
If sup
P
L
(
f, P
) = inf
P
U
(
f, P
), then the integral of
f
on [
a, b
] is defined
to be the common value, denoted
R
b
a
f
(
x
)
dx
. We say
f
is (
Riemann
)
integrable
on [
a, b
]
and write
f
∈ R
[
a, b
].
Definition 6.1.4.
The partition
P
0
is a
refinement
of
P
iff
P
⊆
P
0
.
Note that adding points to the partition has two effects:
1. the lengths of the subintervals decreases, and
2.
L
(
f, P
) increases and
U
(
f, P
) decreases.
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 Fall '11
 Wong
 Integrals, partitions P1

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