Thomas Calculus 13th [Solutions]

Section 15.8 Substitutions in Multiple Integrals 1141
27. The region R is shaded in the graph below.
y
y 2x
2
1
1,
2
1 1
2 ,
1,
1 2
2,
1
0 1 2
y
y
y
x
x
2
2
x
1
2x
Solving explicitly for the transformation that gives x and y in terms of u and v yields a complicated
expression for
( x, y) .
( u, v)
However, its reciprocal,
( u, v)
is relatively easy to compute.
( x, y)
y x
y
Since u( x, y)
xy and v( x, y) y/ x,
J ( x, y) y 1 2 2 v.
Thus J ( u, v) 1/2 v . In the uv-plane
x
2
x x
1 1
the region corresponding to R is G : u 2, v 2. Thus v is positive and J ( u, v) 1/2 v.
2 2
2 2
2 2
1 ln u
2 3
dA du dv dv ln 2 dv ln 2
1/2
1/2
1/2 2v
2
1/2
1/2
2
R
28. Under the given transformation,
y
2
uv , so
R
2
2 2
2
2 2
2
uv
u
15 45
y dA du dv dv dv
1/2 1/2 2v
4 1/2 16 32
1/2
1/2
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