Thomas Calculus 13th [Solutions]

1136 Chapter 15 Multiple Integrals
7.
R
2 2
3x 14xy 8y dx dy
(3x 2 y)( x 4 y)
dx dy
R
( x, y) uv du dv 1 uv du dv;
( u, v) 10
G
G
We find the boundaries of G from the boundaries
of R, shown in the accompanying figure:
xy -equations for
the boundary of R
Corresponding
uv-equations
for the boundary of G
y 3 x 1
1 (3 v u) 3 (2 u v ) 1 u 2
2
10 10
Simplified
uv-equations
y 3 x 3
1 (3 v u) 3 (2 u v ) 3 u 6
2
10 10
y 1 x 1 (3 v u) 1 (2 u v ) v 0
4
10 20
y 1 x 1
1 (3 v u) 1 (2 u v ) 1 v 4
4
10 20
6 4 6 2
4 6
2 6
1 uv du dv 1 uv dv du 1 u v du 4 u du 4 u 4 (18 2) 64
10 10 2 0 10 2 2
0
5 2 5 2
2
5 5
G
8.
2( ) 2 ( x, y)
x y dx dy v du dv
( u, v)
2 v du dv ; the region G is sketched in Exercise 4
R G G
2 1 3 3 1 1 2
1
v du dv v
0 3 2 v du dv
0 2 v
v
(3 3 v 3 v ) dv
0 6 v dv 3 v 3 0
G
1 2
9. x u
y 2
2 ( x, y)
v uv 1 1
and y uv v and xy u ; J ( u, v) v u v u 2u
;
v
x
( u, v)
v
v u
y x uv u v 1, and y 4x v 2; xy 1 u 1, and xy 9 u 3; thus
v
y 3 2 3 2 2 3 2
2
xy dx dy ( v u) 2u dv du 2u 2u
dv du 2uv 2u ln v du
x 1 1 v 1 1 v
1 1
R
3 2 2 2
3
2 2
52
1 2 u 2 u ln 2 du u u
3 ln 2 8 1 3 (26)(ln 2) 8 3
(ln 2)
10. (a)
( x, y)
1 0
J ( u, v) u,
( u, v)
v u
is sketched at the right
and the region G
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