~ ?yvD ?Dy
? ??
US9
=×s
~ ?yvD ?Dy
A
o
D
i
σ?
i
rr =
ii
rrr?+=
ii
θθθ?+=
i
θθ =
iiiiii
rrr θθσ+=?
2
2
2
1
)(
2
1
iiii
rrr θ+= )2(
2
1
ii
iii
r
rrr
θ
++
=
2
)(
,
iii
rr θ=
.)sin,cos(),(
∫∫∫∫
=
DD
rdrdrrfdxdyyxf θθθ
B a ?¨ US"9
=×s
~ ?yvD ?Dy
.)sin,cos(
)(
)(
2
1
∫∫
=
θ?
θ?
β
α
θθθ rdrrrfd
α
β
A
D
o
)(
1
θ?=r )(
2
θ?=r
∫∫
D
rdrdrrf θθθ )sin,cos(
=×s1=Qs￥
T
u×+? ?m
,βθα ≤≤
).()(
21
θ?θ? ≤≤ r
~ ?yvD ?Dy
u×+? ?m
,βθα ≤≤
).()(
21
θ?θ? ≤≤ r
.)sin,cos(
)(
)(
2
1
∫∫
=
θ?
θ?
β
α
θθθ rdrrrfd
∫∫
D
rdrdrrf θθθ )sin,cos(
α
β
A
o
D
)(
2
θ?=r
)(
1
θ?=r
~ ?yvD ?Dy
A
o
D
)(θ?=r
.)sin,cos(
)(
0
∫∫
=
θ?β
α
θθθ rdrrrfd
=×s1=Qs￥
T
u×+? ?m
,βθα ≤≤
).(0 θ?≤≤ r
∫∫
D
rdrdrrf θθθ )sin,cos(
α
β
~ ?yvD ?Dy
∫∫
D
rdrdrrf θθθ )sin,cos(
.)sin,cos(
)(
0
2
0
∫∫
=
θ?π
θθθ rdrrrfd
US"/ u×￥
,
∫∫
=
D
rdrdθσ
=×s1=Qs￥
T
u×+? ?m
).(0 θ?≤≤ r
D
o
A
)(θ?=r
,2π
≤≤θ0
~ ?yvD ?Dy
è s
∫∫
D
dxdyyxf ),( ￥ US=Qs?
T ?s u×
,11|),{(
2
xyxyxD?≤≤?= }10 ≤≤ x,
1=+yx
1
22
=+yx
3 US"/
=
=
θ
θ
sin
cos
ry
rx
[?Z?1 1=r,
°LZ?1
θθ cossin
1
+
=r,
∫∫
D
dxdyyxf ),(,)sin,cos(
2
0
1
cossin
1
∫∫
+
=
π
θθ
θθθ rdrrrfd
~ ?yvD ?Dy
è 9
dxdye
D
yx
∫∫
22
? D
^???
e???1 a￥??
??￥> u×,
3
US"/
D
ar ≤≤0 π≤θ≤ 20
.
dxdye
D
yx
∫∫
22
∫∫
π
θ=
a
r
rdred
0
2
0
2
).1(
2
a
e
π=
~ ?yvD ?Dy
è p<ls
∫
∞+
0
2
dxe
x
,
3 }|),{(
222
1
RyxyxD ≤+=
}2|),{(
222
2
RyxyxD ≤+=
}0,0{ ≥≥ yx
}0,0|),{( RyRxyxS ≤≤≤≤=
A ?μ
21
DSD
,0
22
>
yx
eQ
∴
∫∫
1
22
D
yx
dxdye
∫∫
≤
S
yx
dxdye
22
.
2
22
∫∫
≤
D
yx
dxdye
1
D
2
D
S
S
2
D
R
R2
~ ?yvD ?Dy
?
∫∫
=
S
yx
dxdyeI
22
Q
∫∫
=
R
y
R
x
dyedxe
00
22;)(
2
0
2
∫
=
R
x
dxe
=
1
I
∫∫
1
22
D
yx
dxdye
∫∫
π
θ=
R
r
rdred
00
2
2
);1(
4
2
R
e
π
=
] ? =
2
I
∫∫
2
22
D
yx
dxdye );1(
4
2
2R
e
π
=
~ ?yvD ?Dy
? ∞→R
H,,
4
1
π
→I
,
4
2
π
→I
#? ∞→R
H,,
4
π
→I
' =
∫
∞
2
0
)(
2
dxe
x
4
π
,
p<ls
=
∫
∞
0
2
dxe
x
2
π
.
,
21
III <<Q
);1(
4
)()1(
4
222
22
0
R
R
xR
edxee
π
<<?
π
∴
∫
~ ?yvD ?Dy
è 9
dxdyyx
D
)(
22
∫∫
+ D1??
yyx 2
22
=+ yyx 4
22
=+ #°L yx 3? 0=
03 =? xy
??￥
ü
> u×,
3
3
2
π
θ =?
6
1
π
θ =?
θsin4=? r
θsin2=? r
dxdyyx
D
)(
22
∫∫
+
∫∫
π
π
θ
θ
θ=
3
6
sin4
sin2
2
rdrrd
).3
2
(15?
π
=
yyx 4
22
=+
yyx 2
22
=+
03 =? yx
03 =? xy
~ ?yvD ?Dy
è 9
=×s
∫∫
+
+π
D
dxdy
yx
yx
22
22
)sin(
,
?s u×1 }41|),{(
22
≤+≤= yxyxD,
3 ??? Vo I
n?B`K?s
?i $f
91μ??,
∫∫
+
+π
D
dxdy
yx
yx
22
22
)sin(
4=
∫∫
+
+π
1
22
22
)sin(
D
dxdy
yx
yx
∫∫
π
θ=
π
2
10
sin
4
2
rdr
r
r
d,4?=
1
4DD =
1
D
~ ?yvD ?Dy
è p wL )(2)(
222222
yxayx?=+
222
ayx ≥+
??￥m?￥
,
3
? ??μ
1
4DD=
US"/
)(2)(
222222
yxayx?=+,2cos2 θar =?
,
222
arayx =?=+
1
D
~ ?yvD ?Dy
?
=
=
ar
ar θ2cos2
,
¤?? )
6
,(
π
= aA
,
p
∫∫
=
D
dxdyσ
∫∫
=
1
4
D
dxdy
∫∫
θ
π
θ=
2cos2
0
6
4
a
a
rdrd
).
3
3(
2
π
= a
~ ?yvD ?Dy
= a=×s￥DíE
θ=
θ=
.sin
,cos
ry
rx
W￥1"1
USD US-
ü
]B??°?
￥B?MDUS
ü
?°?
ü
T V A?
^V US
xoy
roθ
D
^BB￥
O??M
ü
￥B??
YV
TMDM
￥B?
ü'?
),(
),(
yxMxoy
rM
ro
θ
′
θ
~ ?yvD ?Dy
.),()],(),,([),(
:)3(;0
),(
),(
),()2(
),(),,()1(
),(),,(:
),(
∫∫∫∫
′
=
→
′
≠
=
′
′
′
==
DD
dudvvuJvuyvuxfdxdyyxf
DDT
vu
yx
vuJD
Dvuyvux
DxoyDuov
vuyyvuxxT
Dxoyyxf
^BB￥5μMD
? V1
T
μB¨ ??
ê?
O
@
ü
￥M1
ü
￥> u×|
??MD
ü
￥> u×
!? ?
~ ?yvD ?Dy
è
3
??￥> u×L
à°à a? ?9
2
,
=+
∫∫
+
yx
yxDdxdye
D
xy
xy
,,xyvxyu +=?=
7
.
2
,
2
uv
y
uv
x
+
=
=5
,DD
′
→
D
x
y
o
2=+ yx
D
′
u
v
o
vu=vu?=
2=v
.22;0;0
=→=+
=→=
=→=
vyx
vuy
vux'
~ ?yvD ?Dy
),(
),(
vu
yx
J
=
,
2
1
2
1
2
1
2
1
2
1
=
=
∫∫∫∫
′
+
=
D
v
u
D
xy
xy
dudvedxdye
2
1
#
∫∫
=
v
v
v
u
duedv
2
0
2
1
∫
=
2
0
1
)(
2
1
vdvee
.
1?
= ee
~ ?yvD ?Dy
è
3
??￥> u×??
1 ?9
1
,1
2
2
2
2
2
2
2
2
=+
∫∫
b
y
a
x
Ddxdy
b
y
a
x
D
.20,0,0,0 π≤θ≤≥>> rba ?
θ=
θ=
,sin
,cos
bry
arx
T<l USMD
},20,10),{( π≤θ≤≤≤θ=
′
→ rrDD ?MD/
~ ?yvD ?Dy
.
),(
),(
abr
r
yx
J =
=
θ
#Dí
T ˉ? ?
)1
,
=?? 0=
′
rDJ
θdrdabrrdxdy
b
y
a
x
DD
∫∫∫∫
′
=∴
2
2
2
2
2
11,
3
2
abπ=
~ ?yvD ?Dy
è
3
.
00
)(
22
dxdye
yx
∫ ∫
∞+ ∞+
+?
9
.20,0 πθ ≤≤≥r ?
=
=
,sin
,cos
θ
θ
ry
rx
T USMD
},20,0),{( πθθ ≤≤≤=
′
→ rrDD ?MD/
.
),(
),(
r
r
yx
J =
=
θ
θdrdredxdye
D
r
R
yx
∫∫∫∫
′
+∞→
∞+ ∞+
+?
=∴
222
lim
00
)(
.π=
drred
R
r
R
∫∫
+∞→
=
0
2
0
2
lim
π
θ
)1(lim
2
R
R
e
+∞→
= π
~ ?yvD ?Dy
1 a=×s US/￥9
T
s??i
P¨ ??
? al2
∫∫
D
rdrdrrf θθθ )sin,cos(
.)sin,cos(
)(
)(
2
1
∫∫
=
θ?
θ?
β
α
θθθ rdrrrfd
.)sin,cos(
)(
0
∫∫
=
θ?β
α
θθθ rdrrrfd
.)sin,cos(
)(
0
2
0
∫∫
=
θ?π
θθθ rdrrrfd
a=×s￥DíE2
'1 p MDa?KeL p ?^
~ ?yvD ?Dy
1 a?DsQ?,
).0(),(
cos
0
2
2
≥θθ=
∫∫
θ
π
π
adrrfdI
a
± I5
2 a9
σde
yx
y
yx
D
2
)( +
∫∫
+
? D 1=+ yx
0=x 0=y
??,
~ ?yvD ?Dy
,
cos0
22
:
θ≤≤
π
≤θ≤
π
ar
D
o
x
y
± I5 13s
θcosar =
D
a
a
r
arccos?=θ
a
r
arccos=θ
.),(
arccos
arccos0
∫∫
=
a
r
a
r
a
drfdrI θθ
~ ?yvD ?Dy
7
=
+=
yv
yxu
,
=
=
vy
vux
? V1?
T 1
),(
),(
=
=
vu
yx
J,
MDa u×1
± I5 23s
o
x
y
1=+ yx
D
o
u
v
vu =
D
′
~ ?yvD ?Dy
σde
yx
y
yx
D
2
)( +
∫∫
+
∫∫
′
=
D
dudvJvuf ||),(
dve
u
v
du
u
u
2
0
1
0
∫∫
= due
u
u
2
1
0
2
∫
= ).1(
4
1
= e
D
′
1=+ yx
1=? u
0=x 0= vu
0=y 0=? v
~ ?yvD ?Dy
°z A b5
a |
∫∫
D
dxdyyxf ),(
D1 xyx 2
22
≤+
V
U1 U
S?
T￥=Qs
1 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
a |
∫∫
D
dxdyyxf ),(
D1 xy?≤≤ 10
10 ≤≤ x
V
U1 US?
T￥=Qs1 @ @ @ @ @ @ @ @ @ @ @ @ @ @
a |
∫∫
+
x
x
dyyxfdx
3
22
2
0
)(1 US?
T￥=
Qs1 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
a |
∫∫
2
0
1
0
),(
x
dyyxfdx1 US?
T￥=Qs
1 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
5B
~ ?yvD ?Dy
a |
∫∫
+
x
x
dyyxdx
2
2
1
)(
22
1
0
1 US?
T￥=Q
s 1 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
′ 1
@ @ @ @ @ @ @ @ @ @ @ @ @ @ @
?z 9
/
=×s
a
∫∫
++
D
dyx σ)1ln(
22
? D
^??? 1
22
=+ yx
#USà
??￥ ?B`K
=￥ u×
a
∫∫
+
D
dyx σ)(
22
? D
^?°L xy =
)0(3,,>==+= aayayaxy
??￥ u×
a
∫∫
D
dyxR σ
222
? D
^???
Rxyx =+
22
??￥ u×
a
∫∫
+
D
dyx σ2
22
? D 3
22
≤+ yx
~ ?yvD ?Dy
? a
k| US￥=Qs
∫∫
θ
π
π
θθθ=
cos2
0
4
4
)sin,cos(
a
rdrrrfdI ?DsQ?
1 a
!
ü
]￥> u× D
^?
L θ2=r
B
2
0
π
≤θ≤
D°L
2
π
=θ
??
￥
á1
22
),( yxyx +=ρ
p?
￥é
z 9
[ xoy
￥?? axyx =+
22
??￥> u×1
?7[ w
22
yxz += 1?￥ w??8￥8
~ ?yvD ?Dy
B a a rdrrrfd
∫∫
θ
π
π
θθθ
cos2
0
2
2
)sin,cos(
a
∫∫
θ+θ
π
θθθ
1
)sin(cos
0
2
0
)sin,cos( rdrrrfd
a
∫∫
θ
π
π
θ
sec2
0
3
4
)( rdrrfd
a
∫∫
θ
θθ
π
θθθ
sec
tansec
4
0
)sin,cos( rdrrrfd
a
∫∫
θ
θπ
θ
2
cos
sin
0
4
0
1
rdr
r
d
12?
= a a )12ln2(
4
π
a
4
14a
5Bs?
~ ?yvD ?Dy
a )
3
4
(
3
3
π
R
a π
2
5
? a
∫∫
π
π
θθθ=
4
4
2
0
)sin,cos( drrfrdrI
a
∫∫
θθθ+
a
r
a
r
a
a
drrfrdr
2
arccos
2
arccos
2
2
)sin,cos(
1 a
40
5
π
? a
4
32
3
a
π
~ ?yvD ?Dy
°z T
a?￥MD
9
/
=×s
a
∫∫
D
dxdyyx
22
? D
^?
H
wL 1=xy
2=xy
°L xy = xy 4=
??￥ ? ú`K
￥> u×
a
∫∫
+
D
dxdyyx )(
22
? D
^?? u×
14
22
≤+ yx
?z
!D
^? wL
333
,4,yxxyxy ===
3
4yx =
??￥? ú`K?s￥> u×
p
? a
k￡
∫∫
++
D
dxdycbyaxf )(
∫
++?=
1
1
222
)(12 ducbaufu
? D1
0,1
2222
≠+≤+ bayx O
5=
~ ?yvD ?Dy
B a a 2ln
3
7
a π
32
5
= a
8
1
5=s?
? ??
US9
=×s
~ ?yvD ?Dy
A
o
D
i
σ?
i
rr =
ii
rrr?+=
ii
θθθ?+=
i
θθ =
iiiiii
rrr θθσ+=?
2
2
2
1
)(
2
1
iiii
rrr θ+= )2(
2
1
ii
iii
r
rrr
θ
++
=
2
)(
,
iii
rr θ=
.)sin,cos(),(
∫∫∫∫
=
DD
rdrdrrfdxdyyxf θθθ
B a ?¨ US"9
=×s
~ ?yvD ?Dy
.)sin,cos(
)(
)(
2
1
∫∫
=
θ?
θ?
β
α
θθθ rdrrrfd
α
β
A
D
o
)(
1
θ?=r )(
2
θ?=r
∫∫
D
rdrdrrf θθθ )sin,cos(
=×s1=Qs￥
T
u×+? ?m
,βθα ≤≤
).()(
21
θ?θ? ≤≤ r
~ ?yvD ?Dy
u×+? ?m
,βθα ≤≤
).()(
21
θ?θ? ≤≤ r
.)sin,cos(
)(
)(
2
1
∫∫
=
θ?
θ?
β
α
θθθ rdrrrfd
∫∫
D
rdrdrrf θθθ )sin,cos(
α
β
A
o
D
)(
2
θ?=r
)(
1
θ?=r
~ ?yvD ?Dy
A
o
D
)(θ?=r
.)sin,cos(
)(
0
∫∫
=
θ?β
α
θθθ rdrrrfd
=×s1=Qs￥
T
u×+? ?m
,βθα ≤≤
).(0 θ?≤≤ r
∫∫
D
rdrdrrf θθθ )sin,cos(
α
β
~ ?yvD ?Dy
∫∫
D
rdrdrrf θθθ )sin,cos(
.)sin,cos(
)(
0
2
0
∫∫
=
θ?π
θθθ rdrrrfd
US"/ u×￥
,
∫∫
=
D
rdrdθσ
=×s1=Qs￥
T
u×+? ?m
).(0 θ?≤≤ r
D
o
A
)(θ?=r
,2π
≤≤θ0
~ ?yvD ?Dy
è s
∫∫
D
dxdyyxf ),( ￥ US=Qs?
T ?s u×
,11|),{(
2
xyxyxD?≤≤?= }10 ≤≤ x,
1=+yx
1
22
=+yx
3 US"/
=
=
θ
θ
sin
cos
ry
rx
[?Z?1 1=r,
°LZ?1
θθ cossin
1
+
=r,
∫∫
D
dxdyyxf ),(,)sin,cos(
2
0
1
cossin
1
∫∫
+
=
π
θθ
θθθ rdrrrfd
~ ?yvD ?Dy
è 9
dxdye
D
yx
∫∫
22
? D
^???
e???1 a￥??
??￥> u×,
3
US"/
D
ar ≤≤0 π≤θ≤ 20
.
dxdye
D
yx
∫∫
22
∫∫
π
θ=
a
r
rdred
0
2
0
2
).1(
2
a
e
π=
~ ?yvD ?Dy
è p<ls
∫
∞+
0
2
dxe
x
,
3 }|),{(
222
1
RyxyxD ≤+=
}2|),{(
222
2
RyxyxD ≤+=
}0,0{ ≥≥ yx
}0,0|),{( RyRxyxS ≤≤≤≤=
A ?μ
21
DSD
,0
22
>
yx
eQ
∴
∫∫
1
22
D
yx
dxdye
∫∫
≤
S
yx
dxdye
22
.
2
22
∫∫
≤
D
yx
dxdye
1
D
2
D
S
S
2
D
R
R2
~ ?yvD ?Dy
?
∫∫
=
S
yx
dxdyeI
22
Q
∫∫
=
R
y
R
x
dyedxe
00
22;)(
2
0
2
∫
=
R
x
dxe
=
1
I
∫∫
1
22
D
yx
dxdye
∫∫
π
θ=
R
r
rdred
00
2
2
);1(
4
2
R
e
π
=
] ? =
2
I
∫∫
2
22
D
yx
dxdye );1(
4
2
2R
e
π
=
~ ?yvD ?Dy
? ∞→R
H,,
4
1
π
→I
,
4
2
π
→I
#? ∞→R
H,,
4
π
→I
' =
∫
∞
2
0
)(
2
dxe
x
4
π
,
p<ls
=
∫
∞
0
2
dxe
x
2
π
.
,
21
III <<Q
);1(
4
)()1(
4
222
22
0
R
R
xR
edxee
π
<<?
π
∴
∫
~ ?yvD ?Dy
è 9
dxdyyx
D
)(
22
∫∫
+ D1??
yyx 2
22
=+ yyx 4
22
=+ #°L yx 3? 0=
03 =? xy
??￥
ü
> u×,
3
3
2
π
θ =?
6
1
π
θ =?
θsin4=? r
θsin2=? r
dxdyyx
D
)(
22
∫∫
+
∫∫
π
π
θ
θ
θ=
3
6
sin4
sin2
2
rdrrd
).3
2
(15?
π
=
yyx 4
22
=+
yyx 2
22
=+
03 =? yx
03 =? xy
~ ?yvD ?Dy
è 9
=×s
∫∫
+
+π
D
dxdy
yx
yx
22
22
)sin(
,
?s u×1 }41|),{(
22
≤+≤= yxyxD,
3 ??? Vo I
n?B`K?s
?i $f
91μ??,
∫∫
+
+π
D
dxdy
yx
yx
22
22
)sin(
4=
∫∫
+
+π
1
22
22
)sin(
D
dxdy
yx
yx
∫∫
π
θ=
π
2
10
sin
4
2
rdr
r
r
d,4?=
1
4DD =
1
D
~ ?yvD ?Dy
è p wL )(2)(
222222
yxayx?=+
222
ayx ≥+
??￥m?￥
,
3
? ??μ
1
4DD=
US"/
)(2)(
222222
yxayx?=+,2cos2 θar =?
,
222
arayx =?=+
1
D
~ ?yvD ?Dy
?
=
=
ar
ar θ2cos2
,
¤?? )
6
,(
π
= aA
,
p
∫∫
=
D
dxdyσ
∫∫
=
1
4
D
dxdy
∫∫
θ
π
θ=
2cos2
0
6
4
a
a
rdrd
).
3
3(
2
π
= a
~ ?yvD ?Dy
= a=×s￥DíE
θ=
θ=
.sin
,cos
ry
rx
W￥1"1
USD US-
ü
]B??°?
￥B?MDUS
ü
?°?
ü
T V A?
^V US
xoy
roθ
D
^BB￥
O??M
ü
￥B??
YV
TMDM
￥B?
ü'?
),(
),(
yxMxoy
rM
ro
θ
′
θ
~ ?yvD ?Dy
.),()],(),,([),(
:)3(;0
),(
),(
),()2(
),(),,()1(
),(),,(:
),(
∫∫∫∫
′
=
→
′
≠
=
′
′
′
==
DD
dudvvuJvuyvuxfdxdyyxf
DDT
vu
yx
vuJD
Dvuyvux
DxoyDuov
vuyyvuxxT
Dxoyyxf
^BB￥5μMD
? V1
T
μB¨ ??
ê?
O
@
ü
￥M1
ü
￥> u×|
??MD
ü
￥> u×
!? ?
~ ?yvD ?Dy
è
3
??￥> u×L
à°à a? ?9
2
,
=+
∫∫
+
yx
yxDdxdye
D
xy
xy
,,xyvxyu +=?=
7
.
2
,
2
uv
y
uv
x
+
=
=5
,DD
′
→
D
x
y
o
2=+ yx
D
′
u
v
o
vu=vu?=
2=v
.22;0;0
=→=+
=→=
=→=
vyx
vuy
vux'
~ ?yvD ?Dy
),(
),(
vu
yx
J
=
,
2
1
2
1
2
1
2
1
2
1
=
=
∫∫∫∫
′
+
=
D
v
u
D
xy
xy
dudvedxdye
2
1
#
∫∫
=
v
v
v
u
duedv
2
0
2
1
∫
=
2
0
1
)(
2
1
vdvee
.
1?
= ee
~ ?yvD ?Dy
è
3
??￥> u×??
1 ?9
1
,1
2
2
2
2
2
2
2
2
=+
∫∫
b
y
a
x
Ddxdy
b
y
a
x
D
.20,0,0,0 π≤θ≤≥>> rba ?
θ=
θ=
,sin
,cos
bry
arx
T<l USMD
},20,10),{( π≤θ≤≤≤θ=
′
→ rrDD ?MD/
~ ?yvD ?Dy
.
),(
),(
abr
r
yx
J =
=
θ
#Dí
T ˉ? ?
)1
,
=?? 0=
′
rDJ
θdrdabrrdxdy
b
y
a
x
DD
∫∫∫∫
′
=∴
2
2
2
2
2
11,
3
2
abπ=
~ ?yvD ?Dy
è
3
.
00
)(
22
dxdye
yx
∫ ∫
∞+ ∞+
+?
9
.20,0 πθ ≤≤≥r ?
=
=
,sin
,cos
θ
θ
ry
rx
T USMD
},20,0),{( πθθ ≤≤≤=
′
→ rrDD ?MD/
.
),(
),(
r
r
yx
J =
=
θ
θdrdredxdye
D
r
R
yx
∫∫∫∫
′
+∞→
∞+ ∞+
+?
=∴
222
lim
00
)(
.π=
drred
R
r
R
∫∫
+∞→
=
0
2
0
2
lim
π
θ
)1(lim
2
R
R
e
+∞→
= π
~ ?yvD ?Dy
1 a=×s US/￥9
T
s??i
P¨ ??
? al2
∫∫
D
rdrdrrf θθθ )sin,cos(
.)sin,cos(
)(
)(
2
1
∫∫
=
θ?
θ?
β
α
θθθ rdrrrfd
.)sin,cos(
)(
0
∫∫
=
θ?β
α
θθθ rdrrrfd
.)sin,cos(
)(
0
2
0
∫∫
=
θ?π
θθθ rdrrrfd
a=×s￥DíE2
'1 p MDa?KeL p ?^
~ ?yvD ?Dy
1 a?DsQ?,
).0(),(
cos
0
2
2
≥θθ=
∫∫
θ
π
π
adrrfdI
a
± I5
2 a9
σde
yx
y
yx
D
2
)( +
∫∫
+
? D 1=+ yx
0=x 0=y
??,
~ ?yvD ?Dy
,
cos0
22
:
θ≤≤
π
≤θ≤
π
ar
D
o
x
y
± I5 13s
θcosar =
D
a
a
r
arccos?=θ
a
r
arccos=θ
.),(
arccos
arccos0
∫∫
=
a
r
a
r
a
drfdrI θθ
~ ?yvD ?Dy
7
=
+=
yv
yxu
,
=
=
vy
vux
? V1?
T 1
),(
),(
=
=
vu
yx
J,
MDa u×1
± I5 23s
o
x
y
1=+ yx
D
o
u
v
vu =
D
′
~ ?yvD ?Dy
σde
yx
y
yx
D
2
)( +
∫∫
+
∫∫
′
=
D
dudvJvuf ||),(
dve
u
v
du
u
u
2
0
1
0
∫∫
= due
u
u
2
1
0
2
∫
= ).1(
4
1
= e
D
′
1=+ yx
1=? u
0=x 0= vu
0=y 0=? v
~ ?yvD ?Dy
°z A b5
a |
∫∫
D
dxdyyxf ),(
D1 xyx 2
22
≤+
V
U1 U
S?
T￥=Qs
1 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
a |
∫∫
D
dxdyyxf ),(
D1 xy?≤≤ 10
10 ≤≤ x
V
U1 US?
T￥=Qs1 @ @ @ @ @ @ @ @ @ @ @ @ @ @
a |
∫∫
+
x
x
dyyxfdx
3
22
2
0
)(1 US?
T￥=
Qs1 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
a |
∫∫
2
0
1
0
),(
x
dyyxfdx1 US?
T￥=Qs
1 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
5B
~ ?yvD ?Dy
a |
∫∫
+
x
x
dyyxdx
2
2
1
)(
22
1
0
1 US?
T￥=Q
s 1 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
′ 1
@ @ @ @ @ @ @ @ @ @ @ @ @ @ @
?z 9
/
=×s
a
∫∫
++
D
dyx σ)1ln(
22
? D
^??? 1
22
=+ yx
#USà
??￥ ?B`K
=￥ u×
a
∫∫
+
D
dyx σ)(
22
? D
^?°L xy =
)0(3,,>==+= aayayaxy
??￥ u×
a
∫∫
D
dyxR σ
222
? D
^???
Rxyx =+
22
??￥ u×
a
∫∫
+
D
dyx σ2
22
? D 3
22
≤+ yx
~ ?yvD ?Dy
? a
k| US￥=Qs
∫∫
θ
π
π
θθθ=
cos2
0
4
4
)sin,cos(
a
rdrrrfdI ?DsQ?
1 a
!
ü
]￥> u× D
^?
L θ2=r
B
2
0
π
≤θ≤
D°L
2
π
=θ
??
￥
á1
22
),( yxyx +=ρ
p?
￥é
z 9
[ xoy
￥?? axyx =+
22
??￥> u×1
?7[ w
22
yxz += 1?￥ w??8￥8
~ ?yvD ?Dy
B a a rdrrrfd
∫∫
θ
π
π
θθθ
cos2
0
2
2
)sin,cos(
a
∫∫
θ+θ
π
θθθ
1
)sin(cos
0
2
0
)sin,cos( rdrrrfd
a
∫∫
θ
π
π
θ
sec2
0
3
4
)( rdrrfd
a
∫∫
θ
θθ
π
θθθ
sec
tansec
4
0
)sin,cos( rdrrrfd
a
∫∫
θ
θπ
θ
2
cos
sin
0
4
0
1
rdr
r
d
12?
= a a )12ln2(
4
π
a
4
14a
5Bs?
~ ?yvD ?Dy
a )
3
4
(
3
3
π
R
a π
2
5
? a
∫∫
π
π
θθθ=
4
4
2
0
)sin,cos( drrfrdrI
a
∫∫
θθθ+
a
r
a
r
a
a
drrfrdr
2
arccos
2
arccos
2
2
)sin,cos(
1 a
40
5
π
? a
4
32
3
a
π
~ ?yvD ?Dy
°z T
a?￥MD
9
/
=×s
a
∫∫
D
dxdyyx
22
? D
^?
H
wL 1=xy
2=xy
°L xy = xy 4=
??￥ ? ú`K
￥> u×
a
∫∫
+
D
dxdyyx )(
22
? D
^?? u×
14
22
≤+ yx
?z
!D
^? wL
333
,4,yxxyxy ===
3
4yx =
??￥? ú`K?s￥> u×
p
? a
k￡
∫∫
++
D
dxdycbyaxf )(
∫
++?=
1
1
222
)(12 ducbaufu
? D1
0,1
2222
≠+≤+ bayx O
5=
~ ?yvD ?Dy
B a a 2ln
3
7
a π
32
5
= a
8
1
5=s?
