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Spring 2004 Math 253/501–503
13 Multiple Integrals
13.1 Double Integrals over Rectangles
Thu, 19/Feb
c
±
2004, Art Belmonte
Summary
Single Riemann Integral
First, a little review from Calc 1. Let
f
be a function deﬁned on
I
=
[
a
,
b
]. Split [
a
,
b
]into
m
subintervals (typically of equal
length) whose endpoints constitute a partition
P
:
a
=
x
0
<
x
1
<
x
2
<
···
<
x
m

1
<
x
m
=
b
.
Let
x
*
i
be a point in the
i
th
subinterval,
±
x
i

1
,
x
i
²
.Nowle
tthe
number of subintervals increase indeﬁnitely, while the maximum
size of the subintervals (the
norm
of
P
) shrinks to 0. If the limit
lim
k
P
k→
0
m
X
i
=
1
f
(
x
*
i
)
1
x
i
exists, we call it the
deﬁnite integral
of
f
from
a
to
b
, written
R
b
a
f
(
x
)
dx
, and say that
f
is
integrable
.
Z
b
a
f
(
x
)
=
lim
k
P
k→
0
m
X
i
=
1
f
(
x
*
i
)
1
x
i
Double Riemann Integral
In Calc 3, we similarly deﬁne the double integral of a function
f
(
x
,
y
)
over a rectangular region
R
= {
(
x
,
y
)
:
a
≤
x
≤
b
,
c
≤
y
≤
d
}
by “slicing and dicing,” as it were. (Think of mincing that onion
with your Ginsu knife.
..) That is, split [
a
,
b
m
subintervals
and [
c
,
d
n
subintervals. (Typically, the
x
intervals are
equallength, as are the
y
subintervals.) The
norm
k
P
k
of the
resulting partition
P
is the length of the longest diagonal among
the subrectangles of the partition. We then form a double Riemann
sum and take the limit as
k
P
k
shrinks to 0. If this limit exists, we
obtain the
double integral
of
f
over
R
.
ZZ
R
f
(
x
,
y
)
dA
=
Z
b
a
Z
d
c
f
(
x
,
y
)
dydx
=
Z
d
c
Z
b
a
f
(
x
,
y
)
dxdy
=
lim
k
P
k→
0
m
X
i
=
1
n
X
j
=
1
f
³
x
*
i
,
y
*
j
´
1
x
i
1
y
j
Remarks
We’ll actually compute a double integral exactly in this handout
by having MATLAB take the limit of a double Riemann sum.
Since this is rather difﬁcult to do by hand, we’ll often merely use a
particular double Riemann sum as an approximation to a double
integral.
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