16
MULTIPLE
INTEGRATION
16.1 Integration in Several Variables
(ET Section 15.1)
Preliminary Questions
1.
In the Riemann sum
S
8
,
4
for a double integral over
R
=[
1
,
5
]×[
2
,
10
]
, what is the area of each subrectangle and
how many subrectangles are there?
SOLUTION
Each subrectangle has sides of length
1
x
=
5
−
1
8
=
1
2
,1
y
=
10
−
2
4
=
2
Therefore the area of each subrectangle is
1
A
=
1
x
1
y
=
1
2
·
2
=
1, and the number of subrectangles is 8
·
4
=
32.
2.
Estimate the double integral of a continuous function
f
over the small rectangle
R
0
.
9
,
1
.
1
1
.
9
,
2
.
1
]
if
f
(
1
,
2
)
=
4.
Since we are given the value of
f
at one point in
R
only, we can only use the approximation
S
11
for the
integral of
f
over
R
.For
S
11
we have one rectangle with sides
1
x
=
1
.
1
−
0
.
9
=
0
.
2
y
=
2
.
1
−
1
.
9
=
0
.
2
Hence, the area of the rectangle is
1
A
=
1
x
1
y
=
0
.
2
·
0
.
2
=
0
.
04. We obtain the following approximation:
ZZ
R
fdA
≈
S
1
,
1
=
f
(
1
,
2
)1
A
=
4
·
0
.
04
=
0
.
16
3.
What is the integral of the constant function
f
(
x
,
y
)
=
5 over the rectangle
[−
2
,
3
2
,
4
]
?
The integral of
f
over the unit square
R
=[−
2
,
3
2
,
4
]
is the volume of the box of base
R
and height
5. That is,
R
5
dA
=
5
·
Area
(
R
)
=
5
·
5
·
2
=
50
4.
What is the interpretation of
R
f
(
x
,
y
)
if
f
(
x
,
y
)
takes on both positive and negative values on
R
?
The double integral
R
f
(
x
,
y
)
is the signed volume between the graph
z
=
f
(
x
,
y
)
for
(
x
,
y
)
∈
R
,
and the
xy
-plane. The region below the
-plane is treated as negative volume.
5.
Which of (a) or (b) is equal to
Z
2
1
Z
5
4
f
(
x
,
y
)
dydx
?
(a)
Z
2
1
Z
5
4
f
(
x
,
y
)
dx dy
(b)
Z
5
4
Z
2
1
f
(
x
,
y
)
The integral
R
2
1
R
5
4
f
(
x
,
y
)
is written with
dy
preceding
dx
, therefore the integration is ±rst with
respect to
y
over the interval 4
≤
y
≤
5, and then with respect to
x
over the interval 1
≤
x
≤
2. By Fubini’s Theorem,
we may replace the order of integration over the corresponding intervals. Therefore the given integral is equal to (b)
rather than to (a).