Thomas Calculus 13th [Solutions]

Section 15.8 Substitutions in Multiple Integrals 1137
(b) x 1 u 1, and x 2 u 2; y 1 uv 1 v 1 ,
u
and y 2 uv 2 v 2 ; thus,
u
2 2 2 2/ 2 2/ 2 2 2/ u
y u
uv
u
v
2
2 1
1 1 x
dy dx 1 1/ u u
u dv du 1 1/ u uv dv du 1 u 2
du 1 u du
2 2
1/ u
u 2u
3
2
3 2
u 1 du ln u 3 ln 2;
1 u
1
2 2 2 2
2 2 y 2
1 y 3
2
3 2
dy dx dx dx ln x 3 ln 2
1 1 x 1 x 2 2 1 x 2 1 2
1
( x, y) a cos ar sin
2 2
11. x ar cos and y ar sin J ( r, ) abr cos abr sin abr;
( r, )
b sin br cos
2 2 2 1 2 2 2 2 2
I0
x y dA r a cos b sin J ( r, ) dr d
0 0
R
2 1 3 2 2 2 2 2 2 2 2 2
abr a cos b sin drd ab a cos b sin d
0 0 4 0
2 2 2 2 2
2 2
ab a b
ab a a sin 2 b b sin 2
4 2 4 2 4
0
4
2 2
12.
( x, y)
a 0
J ( u, v) ab;
1 1 2
u
A dy dx ab du dv 1 2
( u, v)
2 ab dv du 2ab 1 u du
0 b
1 1 u
1
R G
1
2 1 1 1 1
2ab u 1 u sin u ab sin 1 sin ( 1) ab ab
2 2 2 2
1
13. The region of integration R in the xy -plane is
sketched in the figure at the right. The boundaries of
the image G are obtained as follows, with G
sketched at the right:
xy -equations for
Corresponding
uv-equations
Simplified
the boundary of R for the boundary of G uv-equations
x y 1 ( u 2 v) 1 ( u v )
v 0
3 3
x 2 2y 1 ( u 2 v) 2 2 ( u v )
u 2
3 3
y 0
0 1 ( u v)
v u
3
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