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).
6.
For which of the following functions is the double integral over the rectangle in Figure 16 equal to zero? Explain
your reasoning.
(a)
f
(
x
,
y
)
=
x
2
y
(b)
f
(
x
,
y
)
=
2
(c)
f
(
x
,
y
)
=
sin
x
(d)
f
(
x
,
y
)
=
e
x
x
y
−
11
1
FIGURE 16