Graphics Reference
In-Depth Information
Figure 4.21.
Subrectangles of a rectangle
partition.
subrectangle [s
1
, s
2
,]x[t
2
, t
3
]
b
2
= t
4
t
3
t
2
t
1
a
2
= t
0
a
1
= s
0
s
1
s
2
b
1
= s
3
P
1
= {s
0
, s
1
, s
2
, s
3
}
P
2
= {t
0
, t
1
, t
2
, t
3
, t
4
}
A
refinement
of the partition P is a partition P¢=(P
1
¢,P
2
¢,...,P
n
¢) of
A
where P
i
¢ is a
refinement of P
i
. See Fig 4.21.
Definition.
Given a bounded function f :
X
Æ
R
, define min
X
(f) and max
X
(f) by
()
=
{
()
}
min
f
inf
f
xxX
xxX
Œ
,
X
()
=
{
()
}
max
f
sup
f
Œ
.
X
Let
A
be a rectangle in
R
n
and consider a function f :
A
Æ
R
.
Definition.
If P is a partition of
A
, then
Â
()
=
()
( )
L f P
,
min
f vol
B
B
subrectangle
B
of P
and
Â
()
=
()
( )
U f P
,
max
f vol
B
B
subrectangle
B
of P
are called the
lower
and
upper sum
for f over
A
with respect to P, respectively.
One can show that refining partitions increases the lower sums and decreases the
upper sums and that upper sums are always larger than or equal to lower sums. This
enables one to make the following definition.
Definition.
The function f is said to be
integrable
on
A
if f is bounded on
A
and
{
()
}
=
{
()
}
sup
LfP
,
P is a partition of
A
inf
UfP
,
P is a partition of
A
.
The common value is called the
integral
of f over
A
and is denoted by Ú
A
f.
