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
x
=
5
−
1
8
=
1
2
,
y
=
10
−
2
4
=
2
Therefore the area of each subrectangle is
A
=
x
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.
SOLUTION
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
x
=
1
.
1
−
0
.
9
=
0
.
2
,
y
=
2
.
1
−
1
.
9
=
0
.
2
Hence, the area of the rectangle is
A
=
x
y
=
0
.
2
·
0
.
2
=
0
.
04. We obtain the following approximation:
R
f d A
≈
S
1
,
1
=
f
(
1
,
2
)
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
]
?
SOLUTION
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
d A
=
5
·
Area
(
R
)
=
5
·
5
·
2
=
50
4.
What is the interpretation of
R
f
(
x
,
y
)
d A
if
f
(
x
,
y
)
takes on both positive and negative values on
R
?
SOLUTION
The double integral
R
f
(
x
,
y
)
d A
is the signed volume between the graph
z
=
f
(
x
,
y
)
for
(
x
,
y
)
∈
R
,
and the
xy
plane. The region below the
xy
plane is treated as negative volume.
5.
Which of (a) or (b) is equal to
2
1
5
4
f
(
x
,
y
)
dy dx
?
(a)
2
1
5
4
f
(
x
,
y
)
dx dy
(b)
5
4
2
1
f
(
x
,
y
)
dx dy
SOLUTION
The integral
2
1
5
4
f
(
x
,
y
)
dy dx
is written with
dy
preceding
dx
, therefore the integration is first 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
)
=
xy
2
(c)
f
(
x
,
y
)
=
sin
x
(d)
f
(
x
,
y
)
=
e
x
x
y
−
1
1
1
FIGURE 16
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S E C T I O N
16.1
Integration in Several Variables
(ET Section 15.1)
451
SOLUTION
The double integral is the signed volume of the region between the graph of
f
(
x
,
y
)
and the
xy
plane over
R
. In (c) and (d) the function satisfies
f
(
−
x
,
y
)
= −
f
(
x
,
y
)
, hence the region below the
xy
plane, where
−
1
≤
x
≤
0
cancels with the region above the
xy
plane, where 0
≤
x
≤
1. Therefore, the double integral is zero. In (a) and (b), the
function
f
(
x
,
y
)
is always positive on the rectangle, so the double integral is greater than zero.
Exercises
1.
Compute the Riemann sum
S
4
,
3
to estimate the double integral of
f
(
x
,
y
)
=
xy
over
R
= [
1
,
3
] × [
1
,
2
.
5
]
. Use the
regular partition and upperright vertices of the subrectangles as sample points.
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 Spring '08
 Hubscher
 Angles, dx, Riemann

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