SECTION
16.5
Change of Variables
(ET Section 15.5)
1003
25.
With
8
as in Example 3, use the Change of Variables Formula to compute the area of the image of
[
1
,
4
]×[
1
,
4
]
.
SOLUTION
Let
R
represent the rectangle
[
1
,
4
1
,
4
]
. We proceed as follows. Jac
(8)
is easily calculated as
Jac
(
T
)
=
∂(
x
,
y
)
u
,v)
=
¯
¯
¯
¯
1
/v
−
u
/v
2
v
u
¯
¯
¯
¯
=
2
u
/v
Now, the area is given by the Change of Variables Formula as
ZZ
8(
R
)
1
dA
=
R
1

Jac
(8)

dud
v
=
R
1

2
u
/v

v
=
Z
4
1
Z
4
1
2
u
/v
v
=
Z
4
1
2
udu
·
Z
4
1
1
v
d
v
=
(
16
−
1
)(
ln 4
−
ln 1
)
=
15 ln 4
In Exercises 26–28, let
R
0
=[
0
,
1
0
,
1
]
be the unit square. The translate of a map
8
0
(
u
=
(
φ
(
u
,v),
ψ
(
u
,v))
is
amap
8(
u
=
(
a
+
(
u
b
+
(
u
where a
,
b are constants. Observe that the map
8
0
in Figure 16 maps
R
0
to the parallelogram
P
0
and the translate
8
1
(
u
=
(
2
+
4
u
+
2
v,
1
+
u
+
3
v)
maps
R
0
to
P
1
.
R
0
u
1
1
(4, 1)
(6, 4)
(2, 3)
P
0
(6, 2)
(2, 1)
(8, 5)
(4, 4)
P
1
x
y
Φ
0
(
u
,
)
=
(4
u
+
2
,
u
+
3
)
R
0
u
1
1
x
y
Φ
1
(
u
,
)
=
(2
+
4
u
+
2
, 1
+
u
+
3
)
(3, 2)
(
−
1, 1)
(1, 4)
P
3
x
y
(6, 3)
(2, 2)
(4, 5)
P
2
x
y
FIGURE 16
26.
Find translates
8
2
and
8
3
of the mapping
8
0
in Figure 16 that map the unit square
R
0
to the parallelograms
P
2
and
P
3
.
The parallelogram
P
2
is obtained by translating
P
0
two units upward and two units to the left. Therefore