7.3
1. The region of integration is a circle, ellipse, or rhombus. Then use the change of variables.
(a)
Then we use the polarcoordinates,
. Then
.Thus,
is maps to
(b)
Then we use the polarcoordinates.
implies
.Then
is maps to
,
(c)
Then we use the change of variables
.
Since
, a line
maps to
,Also since
,
maps to
. Aline
maps to
. Then
maps to
Thus,,
(d)
Then we use the polarcoordinates.
implies
.Thus
maps to
,
(e)
Then we use the polarcoordinates.
implies
.Then
maps to
,
(f)
Then we use the polarcoordinates.
implies
.Also,
implies
.Thus
maps to
2.
Then we use the change of variables
.
Aline
maps to
by
,A line
mpas to
,A line
maps to
by
,
A line
maps to
. Then
maps to
Also,
implies
Thus,
ここで,
したがって,