Bài tập toán cao cấp tập 3 part 5 ppt

132 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
51.

D
ln(x
2
+ y
2
)
x
2
+ y
2
dxdy; D :1 x
2
+ y
2
 e.(DS. 2π)
52.

D
(x
2

+ y
2
)dxdy; D gi´o
.
iha
.
nbo
.
’
ic´acd
u
.
`o
.
ng tr`on
x
2
+ y
2
+2x − 1=0,x
2
+ y
2
+2x =0. (DS.
5π
2
)
Chı
’
dˆa

˜
n. D
˘a
.
t x − 1=r cos ϕ, y = r sin ϕ.
T´ınh thˆe
’
t´ıch cu
’
avˆa
.
tthˆe
’
gi´o
.
iha
.
nbo
.
’
i c´ac m˘a
.
td
˜achı
’
ra.
53. x =0,y =0,z =0,x + y + z = 1. (D
S.
1
6

)
54. x =0,y =0,z =0,x + y =1,z = x
2
+ y
2
.(DS.
1
6
)
55. z = x
2
+ y
2
, y = x
2
, y =1,z = 0. (DS.
88
105
)
56. z =

x
2
+ y
2
, x
2
+ y
2
= a

2
, z = 0. (DS.
2
3
πa
3
)
57. z = x
2
+ y
2
, x
2
+ y
2
= a
2
, z = 0. (DS.
πa
4
2
)
58. z = x, x
2
+ y
2
= a
2
, z = 0. (DS.
4a

3
3
)
59. z =4−x
2
− y
2
, x = ±1, y = ±1. (DS. 13
1
3
)
60. 2 − x − y −2z =0,y = x
2
, y = x.(DS.
11
120
)
61. x
2
+ y
2
=4x, z = x, z =2x.(DS. 4π)
T´ınh diˆe
.
n t´ıch c´ac phˆa
`
nm˘a
.
td
˜achı

’
ra.
62. Phˆa
`
nm˘a
.
t ph˘a
’
ng 6x +3y +2z = 12 n˘a
`
m trong g´oc phˆa
`
n t´am I.
(D
S. 14)
63. Phˆa
`
nm˘a
.
t ph˘a
’
ng x + y + z =2a n˘a
`
m trong m˘a
.
t tru
.
x
2
+ y

2
= a
2
.
(D
S. 2a
2
√
3)
12.2. T´ıch phˆan 3-l´o
.
p 133
64. Phˆa
`
nm˘a
.
t paraboloid z = x
2
+ y
2
n˘a
`
m trong m˘a
.
t tru
.
x
2
+ y
2

=4.
(D
S.
π
6
(17
√
17 − 1))
65. Phˆa
`
nm˘a
.
t2z = x
2
+ y
2
n˘a
`
m trong m˘a
.
t tru
.
x
2
+ y
2
=1.
(D
S.
2

3
(2
√
2 − 1)π)
66. Phˆa
`
nm˘a
.
t n´on z =

x
2
+ y
2
n˘a
`
m trong m˘a
.
t tru
.
x
2
+ y
2
= a
2
.
(D
S. πa
2

√
2)
67. Phˆa
`
nm˘a
.
tcˆa
`
u x
2
+y
2
+z
2
= R
2
n˘a
`
m trong m˘a
.
t tru
.
x
2
+y
2
= Rx.
(D
S. 2R
2

(π − 2))
68. Phˆa
`
nm˘a
.
t n´on z
2
= x
2
+ y
2
n˘a
`
m trong m˘a
.
t tru
.
x
2
+ y
2
=2x.
(D
S. 2
√
2π)
69. Phˆa
`
nm˘a
.

t tru
.
z
2
=4x n˘a
`
m trong g´oc phˆa
`
n t´am th´u I v`a gi´o
.
iha
.
n
bo
.
’
im˘a
.
t tru
.
y
2
=4x v`a m˘a
.
t ph˘a
’
ng x = 1. (DS.
4
3
(2

√
2 − 1))
70. Phˆa
`
nm˘a
.
tcˆa
`
u x
2
+ y
2
+ z
2
= R
2
n˘a
`
m trong m˘a
.
t tru
.
x
2
+ y
2
= a
2
(a  R). (DS. 4πa(a −
√

a
2
− R
2
))
12.2 T´ıch phˆan 3-l´o
.
p
12.2.1 Tru
.
`o
.
ng ho
.
.
pmiˆe
`
n h`ınh hˆo
.
p
Gia
’
su
.
’
miˆe
`
n D ⊂ R
3
:

D =[a, b] × [c, d] ×[e, g]={(x, y, z):a  x  b, c  y  d, e  z  g}
v`a h`am f(x,y, z)liˆen tu
.
c trong D. Khi d
´o t´ıch phˆan 3-l´o
.
pcu
’
a h`am
f(x,y, z) theo miˆe
`
n D d
u
.
o
.
.
c t´ınh theo cˆong th ´u
.
c

D
f(x,y, z)dxdydz =
b

a

d

c


g

e
f(x,y, z)dz

dy

dx
=
b

a
dx
d

c
dy
g

e
f(M)dx. (12.15)
134 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´

n
T`u
.
(12.15) suy ra c´ac giai d
oa
.
n t´ınh t´ıch phˆan 3-l´o
.
p:
(i) D
ˆa
`
u tiˆen t´ınh I(x, y)=
g

e
f(M)dz;
(ii) Tiˆe
´
p theo t´ınh I(x)=
d

c
I(x, y)dy;
(iii) Sau c`ung t´ınh t´ıch phˆan I =
b

a
I(x)dx.
Nˆe

´
u t´ıch phˆan (12.15) d
u
.
o
.
.
c t´ınh theo th ´u
.
tu
.
.
kh´ac th`ı c´ac giai d
oa
.
n
t´ınh vˆa
˜
ntu
.
o
.
ng tu
.
.
:d
ˆa
`
u tiˆen t´ınh t´ıch phˆan trong, tiˆe
´

pdˆe
´
n t´ınh t´ıch
phˆan gi˜u
.
a v`a sau c`ung l`a t´ınh t´ıch phˆan ngo`ai.
12.2.2 Tru
.
`o
.
ng ho
.
.
pmiˆe
`
n cong
1
+
Gia
’
su
.
’
h`am f(M)liˆen tu
.
c trong miˆe
`
nbi
.
ch˘a

.
n
D =

(x, y, z):a  x  b, ϕ
1
(x)  y  ϕ
2
(x),g
1
(x, y)  z  g
2
(x, y)

.
Khi d
´o t´ıch phˆan 3-l´o
.
pcu
’
a h`am f(M) theo miˆe
`
n D d
u
.
o
.
.
c t´ınh theo
cˆong th´u

.
c

D
f(M)dxdydz =
b

a

ϕ
2
(x)

ϕ
1
(x)

g
2
(x,y)

g
1
(x,y)
f(M)dx

dy

dx (12.16)
ho˘a

.
c

D
f(M)dxdydz =

D(x,y)
dxdy
g
2
(x,y)

g
1
(x,y)
f(M)dz, (12.17)
trong d
´o D(x, y)l`ah`ınh chiˆe
´
u vuˆong g´oc cu
’
a D lˆen m˘a
.
t ph˘a
’
ng Oxy.
Viˆe
.
ct´ınh t´ıch phˆan 3-l´o
.

pd
u
.
o
.
.
c quy vˆe
`
t´ınh liˆen tiˆe
´
p ba t´ıch phˆan thˆong
12.2. T´ıch phˆan 3-l´o
.
p 135
thu
.
`o
.
ng theo (12.16) t `u
.
t´ıch phˆan trong, tiˆe
´
pd
ˆe
´
nt´ıch phˆan gi˜u
.
av`a
sau c`ung l`a t´ınh t´ıch phˆan ngo`ai. Khi t´ınh t´ıch phˆan 3-l´o
.

p theo cˆong
th ´u
.
c (12.17): d
ˆa
`
u tiˆen t´ınh t´ıch phˆan trong v`a sau d´o c ´o t h ˆe
’
t´ınh t´ıch
phˆan 2-l´o
.
p theo miˆe
`
n D(x, y) theo c´ac phu
.
o
.
ng ph´ap d
˜a c´o trong 12.1.
2
+
Phu
.
o
.
ng ph´ap d
ˆo
’
ibiˆe
´

n. Ph´ep dˆo
’
ibiˆe
´
n trong t´ıch phˆan 3-l´o
.
p
d
u
.
o
.
.
ctiˆe
´
n h`anh theo cˆong th ´u
.
c

D
f(M)dxdydz =

D
∗
f

ϕ(u, v, w),ψ(u, v, w),χ(u, v, w)

×
×




D(x, y, z)
D(u, v, w)



dudvdw, (12.18)
trong d
´o D
∗
l`a miˆe
`
nbiˆe
´
n thiˆen cu
’
ato
.
adˆo
.
cong u, v, w tu
.
o
.
ng ´u
.
ng khi
c´ac d

iˆe
’
m(x, y, z)biˆe
´
n thiˆen trong D: x = ϕ(u, v, w), y = ψ(u, v, w),
z = χ(u, v, w),
D(x, y, z)
D(u, v, w)
l`a Jacobiˆen cu
’
a c´ac h`am ϕ, ψ, χ
J =
D(x, y, z)
D(u, v, w)
=











∂ϕ
∂u
∂ϕ
∂v

∂ϕ
∂w
∂ψ
∂u
∂ψ
∂v
∂ψ
∂w
∂χ
∂u
∂χ
∂v
∂χ
∂w











=0. (12.19)
Tru
.
`o
.

ng ho
.
.
pd
˘a
.
cbiˆe
.
tcu
’
ato
.
adˆo
.
cong l`a to
.
adˆo
.
tru
.
v`a to
.
adˆo
.
cˆa
`
u.
(i) Bu
.
´o

.
c chuyˆe
’
nt`u
.
to
.
ad
ˆo
.
Dˆec´ac sang to
.
adˆo
.
tru
.
(r, ϕ, z)du
.
o
.
.
c thu
.
.
c
hiˆe
.
n theo c´ac hˆe
.
th ´u

.
c x = r cos ϕ, y = r sin ϕ, z = z;0 r<+∞,
0  ϕ<2π, −∞ <z<+∞.T`u
.
(12.19) suy ra J = r v`a trong to
.
a
d
ˆo
.
tru
.
ta c´o

D
f(M)dxdydz =

D
∗
f

r cos ϕ, r sin ϕ, z

rdrdϕdz, (12.20)
trong d
´o D
∗
l`a miˆe
`
nbiˆe

´
n thiˆen cu
’
ato
.
adˆo
.
tru
.
tu
.
o
.
ng ´u
.
ng khi d
iˆe
’
m
(x, y, z)biˆe
´
n thiˆen trong D.
136 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´

n
(ii) Bu
.
´o
.
c chuyˆe
’
nt`u
.
to
.
ad
ˆo
.
Dˆec´ac sang to
.
adˆo
.
cˆa
`
u(r, ϕ, θ)du
.
o
.
.
c
thu
.
.
chiˆe

.
n theo c´ac hˆe
.
th ´u
.
c x = r sin θ cos ϕ, y = r sin θ sin ϕ, z =
r cos θ,0 r<+∞,0 ϕ<2π,0 θ  π.T`u
.
(12.19) ta c´o
J = r
2
sin θ v`a trong to
.
adˆo
.
cˆa
`
u ta c´o

D
f(M)dxdydz =
=

D
∗
f

r sin θ cos ϕ, r sin θ sin ϕ, r cos θ

r

2
sin θdrdϕdθ, (12.21)
trong d
´o D
∗
l`a miˆe
`
nbiˆe
´
n thiˆen cu
’
ato
.
adˆo
.
cˆa
`
utu
.
o
.
ng ´u
.
ng khi d
iˆe
’
m
(x, y, z)biˆe
´
n thiˆen trong D.

12.2.3
Thˆe
’
t´ıch cu
’
avˆa
.
tthˆe
’
cho´an hˆe
´
tmiˆe
`
n D ⊂ R
3
du
.
o
.
.
c t´ınh theo cˆong
th ´u
.
c
V
D
=

D
dxdydz. (12.22)

12.2.4 Nhˆa
.
n x´et chung
B˘a
`
ng c´ach thay dˆo
’
ith´u
.
tu
.
.
t´ınh t´ıch phˆan trong t´ıch phˆan 3-l´o
.
ptas˜e
thu d
u
.
o
.
.
c c´ac cˆong th´u
.
ctu
.
o
.
ng tu
.
.

nhu
.
cˆong th´u
.
c (12.16) d
ˆe
’
t´ınh t´ıch
phˆan. Viˆe
.
c t`ım cˆa
.
n cho t´ıch phˆan d
o
.
n thˆong thu
.
`o
.
ng khi chuyˆe
’
nt´ıch
phˆan 3-l´o
.
pvˆe
`
t´ıch phˆan l˘a
.
pd
u

.
o
.
.
c thu
.
.
chiˆe
.
nnhu
.
d
ˆo
´
iv´o
.
i tru
.
`o
.
ng ho
.
.
p
t´ıch phˆan 2-l´o
.
p.
C
´
AC V

´
IDU
.
V´ı du
.
1. T´ınh t´ıch phˆan l˘a
.
p
I =
1

−1
dx
1

x
2
dy
2

0
(4 + z)dx.
12.2. T´ıch phˆan 3-l´o
.
p 137
Gia
’
i. Ta t´ınh liˆen tiˆe
´
p ba t´ıch phˆan x´ac di

.
nh thˆong thu
.
`o
.
ng b˘a
´
t
d
ˆa
`
ut`u
.
t´ıch phˆan trong
I(x, y)=
2

0
(4 + z)dz =4z


2
0
+
z
2
2




2
0
= 10;
I(x)=
1

x
2
I(x, y)dy =10
1

x
2
dy = 10(1 − x
2
);
I =
1

−1
I(x)dx =
1

−1
10(1 − x
2
)dx =
40
3
· 

V´ı du
.
2. T´ınh t´ıch phˆan
I =

D
(x + y + z)dxdydz,
trong d
´omiˆe
`
n D du
.
o
.
.
c gi´o
.
iha
.
nbo
.
’
i c´ac m˘a
.
t ph˘a
’
ng to
.
ad
ˆo

.
v`a m˘a
.
t
ph˘a
’
ng x + y + z =1.
Gia
’
i. Miˆe
`
n D d
˜a cho l`a mˆo
.
tt´u
.
diˆe
.
nc´oh`ınh chiˆe
´
u vuˆong g´oc trˆen
m˘a
.
t ph˘a
’
ng Oxy l`a tam gi´ac gi´o
.
iha
.
nbo

.
’
i c´ac d
u
.
`o
.
ng th˘a
’
ng x =0,
y =0,x + y = 1. R˜o r`ang l`a x biˆe
´
n thiˆen t`u
.
0d
ˆe
´
n1(doa
.
n[0, 1] l`a
h`ınh chiˆe
´
ucu
’
a D lˆen tru
.
c Ox). Khi cˆo
´
d
i

.
nh x,0 x  1th`ıy biˆe
´
n
thiˆen t`u
.
0d
ˆe
´
n1−x.Nˆe
´
ucˆo
´
di
.
nh ca
’
x v`a y (0  x  1, 0  y  1 −x)
th`ı d
iˆe
’
m(x, y, z)biˆe
´
n thiˆen theo du
.
`o
.
ng th˘a
’
ng d

´u
.
ng t`u
.
m˘a
.
t ph˘a
’
ng
z =0d
ˆe
´
nm˘a
.
t ph˘a
’
ng x + y + z =1,t´u
.
cl`az biˆe
´
n thiˆen t`u
.
0d
ˆe
´
n
1 − x −y. Theo cˆong th´u
.
c (12.16) ta c´o
I =

1

0
dx
1−x

0
dy
1−x−y

0
(x + y + z)dz.
138 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
Dˆe
˜
d`ang thˆa
´
yr˘a
`
ng
I =
1


0
dx
1−x

0

xz + yz +
z
2
2




1−x−y
0
dy
=
1
2
1

0

y − yx
2
− xy
2
−

y
3
3




1−x
0

dx
=
1
6
1

0
(2 − 3x + x
3
)dx =
1
8
· 
V´ı d u
.
3. T´ınh I =

D
dxdydz
(x + y + z)

3
, trong d´omiˆe
`
n D du
.
o
.
.
c gi´o
.
i
ha
.
nbo
.
’
i c´ac m˘a
.
t ph˘a
’
ng x + z =3,y =2,x =0,y =0,z =0.
Gia
’
i. Miˆe
`
n D d
˜a cho l`a mˆo
.
th`ınh l˘ang tru
.

c´o h`ınh chiˆe
´
u vuˆong
g´oc lˆen m˘a
.
t ph˘a
’
ng Oxy l`a h`ınh ch˜u
.
nhˆa
.
t D(x, y)=

(x, y):0
x  3, 0  y  2

.V´o
.
id
iˆe
’
m M(x, y)cˆo
´
di
.
nh thuˆo
.
c D(x, y)diˆe
’
m

(x, y, z) ∈ D biˆe
´
n thiˆen trˆen d
u
.
`o
.
ng th˘a
’
ng d
´u
.
ng t`u
.
m˘a
.
t ph˘a
’
ng Oxy
(z =0)d
ˆe
´
nm˘a
.
t ph˘a
’
ng x + z =3,t´u
.
cl`az biˆe
´

n thiˆen t`u
.
0d
ˆe
´
n3−x:
0  z  3 − x.T`u
.
d
´o theo (12.17) ta c´o

D
f(M)dxdydz =

D(x,y)
dxdy
z=3−x

z=0
(x + y + z +1)
−3
dz
=

D(x,y)

(x + y + z +1)
−2
−2




3−x
0

dxdy = ···=
4ln2− 1
8
· 
V´ı du
.
4. T´ınh t´ıch phˆan

D
(x
2
+ y
2
+ z
2
)dxdydz, trong d´omiˆe
`
n
D d
u
.
o
.
.
c gi´o

.
iha
.
nbo
.
’
im˘a
.
t3(x
2
+ y
2
)+z
2
=3a
2
.
Gia
’
i. Phu
.
o
.
ng tr`ınh m˘a
.
tbiˆen cu
’
a D c´o thˆe
’
viˆe

´
tdu
.
´o
.
ida
.
ng
x
2
a
2
+
y
2
b
2
+
z
2
(a
√
3)
2
=1.
12.2. T´ıch phˆan 3-l´o
.
p 139
D´o l`a m˘a
.

t elipxoid tr`on xoay, t´u
.
cl`aD l`a h`ınh elipxoid tr`on xoay.
H`ınh chiˆe
´
u vuˆong g´oc D(x, y)cu
’
a D lˆen m˘a
.
t ph˘a
’
ng Oxy l`a h`ınh tr`on
x
2
+ y
2
 a
2
.Dod´o ´ap du
.
ng c´ach lˆa
.
p luˆa
.
nnhu
.
trong c´ac v´ıdu
.
2
v`a 3 ta thˆa

´
yr˘a
`
ng khi d
iˆe
’
m M(x, y) ∈ D(x, y)du
.
o
.
.
ccˆo
´
d
i
.
nh th`ı diˆe
’
m
(x, y, z)cu
’
amiˆe
`
n D biˆe
´
n thiˆen trˆen d
u
.
`o
.

ng th˘a
’
ng d
´u
.
ng M(x,y)t`u
.
m˘a
.
tbiˆen du
.
´o
.
icu
’
a D
z = −

3(a
2
− x
2
− y
2
)
d
ˆe
´
nm˘a
.

tbiˆen trˆen
z =+

3(a
2
− x
2
− y
2
).
T`u
.
d
´o theo (12.17) ta c´o
I =

D(x,y)
dxdy
+
√
3(a
2
−x
2
−y
2
)

−
√

3(a
2
−x
2
−y
2
)
(x
2
+ y
2
+ z
2
)dz
=2a
2
√
3

x
2
+y
2
a
2

a
2
− x
2

− y
2
dxdy = |chuyˆe
’
n sang to
.
adˆo
.
cu
.
.
c|
=2a
2
√
3

ra
√
a
2
−r
2
rdrdϕ = a
2
√
3
2π

0

dϕ
a

0
(a
2
− r
2
)
1/2
rdr
=
4πa
5
√
3
· 
V´ı d u
.
5. T´ınh thˆe
’
t´ıch cu
’
avˆa
.
tthˆe
’
gi´o
.
iha

.
nbo
.
’
i c´ac m˘a
.
t ph˘a
’
ng
x + y + z =4,x =3,y =2,x =0,y =0,z =0.
Gia
’
i. Miˆe
`
n D d
˜a cho l`a mˆo
.
th`ınh lu
.
cdiˆe
.
n trong khˆong gian. N´o
c´o h`ınh chiˆe
´
u vuˆong g´oc D(x, y) lˆen m˘a
.
t ph˘a
’
ng Oxy l`a h`ınh thang
vuˆong gi´o

.
iha
.
nbo
.
’
i c´ac d
u
.
`o
.
ng th˘a
’
ng x =0,y =0,x =3,y =2v`a
140 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
x + y = 4. Do d´o´apdu
.
ng (12.17) ta c´o
V
D
=


D
dxdydz =

D(x,y)
dxdy
4−x−y

0
dz =

D(x,y)
(4 − x − y)dxdy
=
1

0
dy
3

0
(4 − x − y)dx +
2

1
dy
4−y

0
(4 − x −y)dx
=

1

0

(4 − y)x −
x
2
2




3
0

dy +
2

1

(4 − y)x −
x
2
2




4−y
0


dy
=
1

0

15
2
−3y

dy +
1
2
2

1
(4 − y)
2
dy =
55
6
· 
V´ı du
.
6. T´ınh t´ıch phˆan
I =

D
z


x
2
+ y
2
dxdydz,
trong d
´omiˆe
`
n D gi´o
.
iha
.
nbo
.
’
im˘a
.
t ph˘a
’
ng y =0,z =0,z = a v`a m˘a
.
t
tru
.
x
2
+ y
2
=2x (x  0, y  0, a>0).

Gia
’
i. Chuyˆe
’
n sang to
.
ad
ˆo
.
tru
.
ta thˆa
´
yphu
.
o
.
ng tr`ınh m˘a
.
t tru
.
x
2
+
y
2
=2x trong to
.
adˆo
.

tru
.
c´o da
.
ng r = 2 cos ϕ,0 ϕ 
π
2
(h˜ay v˜e h`ınh
!). Do d
´o theo cˆong th´u
.
c (12.20) ta c´o
I =
π/2

0
dϕ
2 cosϕ

0
r
2
dr
a

0
zdz =
a
2
2

π/2

0
dϕ
2 cosϕ

0
r
2
dr
=
4a
2
3
π/2

0
cos
3
ϕdϕ =
8
9
a
2
. 
V´ı du
.
7. T´ınh t´ıch phˆan
I =


D
(x
2
+ y
2
)dxdydz,
12.2. T´ıch phˆan 3-l´o
.
p 141
nˆe
´
umiˆe
`
n D l`a nu
.
’
a trˆen cu
’
a h`ınh cˆa
`
u x
2
+ y
2
+ z
2
 R
2
, z  0.
Gia

’
i. Chuyˆe
’
n sang to
.
ad
ˆo
.
cˆa
`
u, miˆe
`
nbiˆe
´
n thiˆen D
∗
cu
’
a c´ac to
.
adˆo
.
cˆa
`
utu
.
o
.
ng ´u
.

ng khi d
iˆe
’
m(x, y, z)biˆe
´
n thiˆen trong D l`a c´o da
.
ng
D
∗
:0 ϕ<2π, 0  θ 
π
2
, 0  r  R.
T`u
.
d
´o
I =

D
∗
r
2
sin
2
θ · r
2
sin θdrdϕdθ =
2π


0
dϕ
π/2

0
sin
3
θdθ
R

0
r
4
dr
=
4
15
πR
5
. 
B
`
AI T
ˆ
A
.
P
T´ınh c´ac t´ıch phˆan l˘a
.

p sau
1.
1

0
dx
√
x

0
ydy
2−2x

1−x
dz.(DS.
1
12
)
2.
a

0
ydy
h

0
dx
a−y

0

dz.(DS.
a
3
h
6
)
3.
2

0
dy
2

√
2y−y
2
xdx
3

0
z
2
dz.(DS. 30)
4.
1

0
dx
1−x


0
dy
1−x−y

0
dz
(1 + x + y + z)
3
.(DS.
ln 2
2
−
5
16
)
5.
c

0
dz
b

0
dy
a

0
(x
2
+ y

2
+ z
2
)dx.(DS.

abc
3
(a
2
+ b
2
+ c
2
)

)
142 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
6.
a

0
dx

a−x

0
dy
a−x−y

0
(x
2
+ y
2
+ z
2
)dz.(DS.
a
5
20
)
T´ınh c´ac t´ıch phˆan 3-l´o
.
p theo miˆe
`
n D gi´o
.
iha
.
nbo
.
’
i c´ac m˘a

.
td
˜achı
’
ra.
7.

D
(x + y − z)dxdydz; x = −1, x =1;y =0,y =1;
z =0,z = 2. (D
S. −2)
8.

D
xydxdydz; x =1,x =2;y = −2, y = −1; z =0,z =
1
2
.
(D
S. −
8
9
)
9.

D
dxdydz
(x + y + z)
2
; x =1,x =2;y =1,y =2;z =1,z =2.

(D
S.
1
2
ln
128
125
)
10.

D
(x +2y +3z +4)dxdydz; x =0,x =3;y =0,y =2;
z =0,z = 1. (D
S. 54)
11.

D
zdxdydz; x =0,y =0,z =0;x + y + z = 1. (DS.
1
24
)
12.

D
xdxdydz; x =0. y =0,z =0,y =1;x + z = 1. (DS.
1
6
)
13.


D
yzdxdydz; x
2
+ y
2
+ z
2
=1,z  0. (DS. 0)
14.

D
xydxdydz; x
2
+ y
2
=1,z =0,z =1(x  0, y  0).
(D
S.
1
8
)
15.

D
xyzdxdydz; x =0,y =0,z =0,x
2
+ y
2
+ z
2

=1
12.2. T´ıch phˆan 3-l´o
.
p 143
(x  0, y  0, z  0). (DS.
1
48
)
16.

D

x
2
+ y
2
dxdydz; x
2
+ y
2
= z
2
, z =0,z = 1. (DS. π/6)
17.

D
(x
2
+ y
2

+ z
2
)dxdydz; x =0,x = a, y =0,y = b,
z =0,z = c.(D
S.
abc
3
(a
2
+ b
2
+ c
2
))
18.

D
ydxdydz; y =
√
x
2
+ z
2
, y = h, h>0. (DS.
πh
4
4
)
T´ınh c´ac t´ıch phˆan 3-l´o
.

p sau b˘a
`
ng phu
.
o
.
ng ph´ap d
ˆo
’
ibiˆe
´
n.
19.

D
(x
2
+ y
2
+ z
2
)dxdydz; x
2
+ y
2
+ z
2
 R
2
.(DS.

4πR
5
5
)
20.

D
(x
2
+ y
2
)dxdydz; z = x
2
+ y
2
, z = 1. (DS.
π
6
)
21.

D

x
2
+ y
2
+ z
2
dxdydz; x

2
+ y
2
+ z
2
 R
2
.(DS. πR
4
)
22.

D
z

x
2
+ y
2
dxdydz; x
2
+ y
2
=2x, y =0,z =0,z =3.
(D
S. 8)
23.

D
zdxdydz; x

2
+ y
2
+ z
2
 R
2
, x  0, y  0, z  0.
(D
S.
πR
4
16
)
24.

D
(x
2
−y
2
)dxdydz; x
2
+ y
2
=2z, z = 2. (DS.
16π
3
)
25.


D
z

x
2
+ y
2
dxdydz; y
2
=3x − x
2
, z =0,z = 2. (DS. 24)
144 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
T´ınh thˆe
’
t´ıch cu
’
a c´ac vˆa
.
tthˆe
’

gi´o
.
iha
.
nbo
.
’
i c´ac m˘a
.
td
˜achı
’
ra.
26. x =0,y =0,z =0,x +2y + z − 6=0. (D
S. 36)
27. 2x +3y +4z = 12; x =0,y =0,z = 0. (D
S. 12)
28.
x
a
+
y
b
+
z
c
=1,x =0,y =0,z = 0. (D
S.
abc
6

)
29. ax = y
2
+ z
2
, x = a.(DS.
πa
3
2
)
30. 2z = x
2
+ y
2
, z = 2. (DS. 4π)
31. z = x
2
+ y
2
, x
2
+ y
2
+ z
2
= 2. (DS.
π
6
[8
√

2 − 7])
32. z =

x
2
+ y
2
, z = x
2
+ y
2
.(DS.
π
6
)
33. x
2
+ y
2
−z =1,z = 0. (DS.
π
2
)
34. 2z = x
2
+ y
2
, y + z = 4. (DS.
81π
4

)
35.
x
2
a
2
+
y
2
b
2
+
z
2
c
2
= 1. (DS.
4
3
πabc)
12.3 T´ıch phˆan du
.
`o
.
ng
12.3.1 C´ac di
.
nh ngh˜ıa co
.
ba

’
n
Gia
’
su
.
’
h`am f(M), P (M)v`aQ(M), M =(x, y)liˆen tu
.
cta
.
imo
.
id
iˆe
’
m
cu
’
ad
u
.
`o
.
ng cong d
odu
.
o
.
.

c L = L(A, B)v´o
.
id
iˆe
’
mdˆa
`
u A v`a diˆe
’
m cuˆo
´
i B.
Chia mˆo
.
t c´ach t`uy ´y L(A, B) th`anh n cung nho
’
v´o
.
id
ˆo
.
d`ai tu
.
o
.
ng ´u
.
ng
l`a ∆s
0

,∆s
1
,∆s
2
, ,∆s
n−1
.D˘a
.
t d = max
0in−1
(∆s
i
). Trong mˆo
˜
i cung
nho
’
,lˆa
´
ymˆo
.
t c´ach t`uy ´y d
iˆe
’
m N
0
,N
1
, ,N
n−1

. t´ınh gi´a tri
.
f(N
i
),
P (N
i
)v`aQ(N
i
)ta
.
idiˆe
’
m N
i
d´o .
X´et hai phu
.
o
.
ng ph´ap lˆa
.
ptˆo
’
ng t´ıch phˆan sau d
ˆay.
12.3. T´ıch phˆan d u
.
`o
.

ng 145
Phu
.
o
.
ng ph´ap I. Lˆa
´
y gi´a tri
.
f(N
i
) nhˆan v´o
.
id
ˆo
.
d`ai cung ∆s
i
tu
.
o
.
ng
´u
.
ng v`a lˆa
.
ptˆo
’
ng t´ıch phˆan

σ
1
=
n−1

i=0
f(N
i
)∆s
i
. (*)
Phu
.
o
.
ng ph´ap II. Kh´ac v´o
.
i c´ach lˆa
.
ptˆo
’
ng t´ıch phˆan (∗), trong
phu
.
o
.
ng ph´ap n`ay ta lˆa
´
y gi´a tri
.

P (N
i
), Q(N
i
) nhˆan khˆong pha
’
iv´o
.
i
d
ˆo
.
d`ai cu
’
a c´ac cung nho
’
m`a l`a nhˆan v´o
.
i h`ınh chiˆe
´
u vuˆong g´oc cu
’
a c´ac
cung nho
’
d
´o trˆen c´ac tru
.
cto
.

adˆo
.
,t´u
.
c l`a lˆa
.
ptˆo
’
ng
σ
x
=
n−1

i=0
P (N
i
)∆x
i
;∆x
i
= pro
Ox
∆s
i
,
σ
y
=
n−1


i=0
Q(N
i
)∆y
i
;∆y
i
= pro
Oy
∆s
i
.
Mˆo
˜
i c´ach lˆa
.
ptˆo
’
ng t´ıch phˆan trˆen d
ˆa y s ˜e d ˆa
˜
ndˆe
´
nmˆo
.
tkiˆe
’
ut´ıch
phˆan d

u
.
`o
.
ng.
D
-
i
.
nh ngh˜ıa 12.3.1. Nˆe
´
utˆo
`
nta
.
i gi´o
.
iha
.
nh˜u
.
uha
.
n lim
d→0
σ
1
khˆong phu
.
thuˆo

.
c v`ao ph´ep phˆan hoa
.
ch d
u
.
`o
.
ng cong L th`anh c´ac cung nho
’
v`a
khˆong phu
.
thuˆo
.
c v`ao viˆe
.
ccho
.
n c´ac d
iˆe
’
m trung gian N
i
trˆen mˆo
˜
i cung
nho
’
th`ı gi´o

.
iha
.
nd
´odu
.
o
.
.
cgo
.
i l`a t´ıch phˆan d
u
.
`o
.
ng theo d
ˆo
.
d`ai (hay t´ıch
phˆan d
u
.
`o
.
ng kiˆe
’
u I) cu
’
a h`am f(x, y) theo d

u
.
`o
.
ng cong L = L(A, B).
K´yhiˆe
.
u:

L
f(x, y)ds. (12.23)
D
-
i
.
nh ngh˜ıa 12.3.2. Ph´at biˆe
’
utu
.
o
.
ng tu
.
.
nhu
.
trong d
i
.
nh ngh˜ıa 12.3.1:

1
+
. lim
d→0
σ
x
= lim
d→0
n−1

i=0
P (N
i
)∆x
i
=

L(A,B)
P (x, y)dx
(12.24)
146 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
go

.
i l`a t´ıch phˆan du
.
`o
.
ng theo ho`anh d
ˆo
.
(nˆe
´
u (12.24) tˆo
`
nta
.
ih˜u
.
uha
.
n)
2
+
. lim
d→0
σ
y
= lim
d→0
n−1

i=0

Q(N
i
)∆y
i
=

L(A,B)
Q(x, y)dy
(12.25)
go
.
i l`a t´ıch phˆan d
u
.
`o
.
ng theo tung d
ˆo
.
(nˆe
´
u (12.25) tˆo
`
nta
.
ih˜u
.
uha
.
n)

Thˆong thu
.
`o
.
ng ngu
.
`o
.
i ta lˆa
.
ptˆo
’
ng t´ıch phˆan da
.
ng
Σ=
n−1

i=0
P (N
i
)∆x
i
+
n−1

o=0
Q(N
i
)∆y

i
v`a nˆe
´
u ∃ lim
d→0
Σ th`ı gi´o
.
iha
.
nd
´odu
.
o
.
.
cgo
.
i l`a t´ıch phˆan d
u
.
`o
.
ng theo to
.
a
d
ˆo
.
da
.

ng tˆo
’
ng qu´at:

L(A,B)
P (x, y)dx + Q(x, y)dy. (12.26)
D
-
i
.
nh l´y. Nˆe
´
u c´ac h`am f(x, y), P (x, y), Q(x, y) liˆen tu
.
c theo d
u
.
`o
.
ng
cong L(A, B)=L th`ı c´ac t´ıch phˆan d
u
.
`o
.
ng (12.23) - (12.26) tˆo
`
nta
.
i

h˜u
.
uha
.
n.
T`u
.
d
i
.
nh ngh˜ıa 12.3.1 v`a kh´ai niˆe
.
mdˆo
.
d`ai cung (khˆong phu
.
thuˆo
.
c
hu
.
´o
.
ng cu
’
a cung) v`a d
i
.
nh ngh˜ıa 12.3.2 v`a t´ınh chˆa
´

tcu
’
ah`ınh chiˆe
´
ucu
’
a
cung (h`ınh chiˆe
´
ud
ˆo
’
idˆa
´
u khi dˆo
’
ihu
.
´o
.
ng cu
’
a cung) suy ra t´ınh chˆa
´
t
quan tro
.
ng cu
’
a t´ıch phˆan d

u
.
`o
.
ng: t´ıch phˆan d
u
.
`o
.
ng theo d
ˆo
.
d`ai khˆong
phu
.
thuˆo
.
c v`ao hu
.
´o
.
ng cu
’
ad
u
.
`o
.
ng cong; t´ıch phˆan d
u

.
`o
.
ng theo to
.
ad
ˆo
.
d
ˆo
’
idˆa
´
ukhidˆo
’
ihu
.
´o
.
ng d
u
.
`o
.
ng cong.
12.3.2 T´ınh t´ıch phˆan du
.
`o
.
ng

Phu
.
o
.
ng ph´ap chung d
ˆe
’
t´ınh t´ıch phˆan du
.
`o
.
ng l`a d
u
.
aviˆe
.
c t´ınh t´ıch
phˆan d
u
.
`o
.
ng vˆe
`
t´ıch phˆan x´ac d
i
.
nh. Cu
.
thˆe

’
l`a: xuˆa
´
t ph´at t`u
.
phu
.
o
.
ng
12.3. T´ıch phˆan d u
.
`o
.
ng 147
tr`ınh cu
’
adu
.
`o
.
ng lˆa
´
y t´ıch phˆan L = L(A, B) ta biˆe
´
nd
ˆo
’
ibiˆe
’

uth´u
.
cdu
.
´o
.
i
dˆa
´
u t´ıch phˆan d
u
.
`o
.
ng th`anh biˆe
’
uth´u
.
cmˆo
.
tbiˆe
´
n m`a gi´a tri
.
cu
’
abiˆe
´
nd
´o

ta
.
id
iˆe
’
mdˆa
`
u A v`a diˆe
’
m cuˆo
´
i B s˜e l`a cˆa
.
ncu
’
a t´ıch phˆan x´ac di
.
nh thu
d
u
.
o
.
.
c.
1
+
Nˆe
´
u L(A, B)du

.
o
.
.
cchobo
.
’
i c´ac phu
.
o
.
ng tr`ınh tham sˆo
´
x = ϕ(t),
y = ψ(t), t ∈ [a,b] (trong d
´o ϕ, ψ kha
’
vi liˆen tu
.
cv`aϕ

2
+ ψ

2
> 0) th`ı
ds =

ϕ


2
+ ψ

2
dt

L(A,B)
f(x, y)ds =
b

a
f[ϕ(t),ψ(t)]

ϕ

2
+ ψ

2
dt (12.27)
v`a

L(A,B)
P (x, y)dx + Q(x, y)dy =
=
b

a

P


ϕ(t),ψ(t)

ϕ

(t)+Q

ϕ(t),ψ(t)

ψ

(t)

dt. (12.28)
2
+
Nˆe
´
u L(A, B)du
.
o
.
.
cchobo
.
’
iphu
.
o
.

ng tr`ınh y = g(x), x ∈ [a, b]
(trong d
´o g(x) kha
’
vi liˆen tu
.
ctrˆen[a, b]) th`ı
ds =

1+g

2
(x)dx

L(A,B)
f(x, y)ds =
b

a
f[x, g(x)]

1+g

2
(x)dx. (12.29)
v`a

L(A,B)
Pdx+ Qdy =
b


a

P (x, g(x)) + Q(x, g(x))g

(x)

dx. (12.30)
148 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
3
+
Nˆe
´
u L(A, B)du
.
o
.
.
cchodu
.
´o
.

ida
.
ng to
.
ad
ˆo
.
cu
.
.
c ρ = ρ(ϕ) α  ϕ 
β th`ı
ds =

ρ
2
+ ρ

ϕ
2
dϕ

L(A,B)
f(x, y)ds =
β

α
f[ρ cos ϕ, ρ sin ϕ]

ρ

2
+ ρ

2
dϕ. (12.31)
4
+
T´ıch phˆan du
.
`o
.
ng theo to
.
ad
ˆo
.
c´o thˆe
’
t´ınh nh`o
.
cˆong th´u
.
c Green.
Nˆe
´
u P (x, y), Q(x, y) v`a c´ac d
a
.
o h`am riˆeng
∂Q

∂x
,
∂P
∂y
c`ung liˆen tu
.
c
trong miˆe
`
n D gi´o
.
iha
.
nbo
.
’
id
u
.
`o
.
ng cong khˆong tu
.
.
c˘a
´
t tro
.
nt`u
.

ng kh´uc
L = ∂D th`ı

L
+
Pdx+ Qdy =

D

∂Q
∂x
−
∂P
∂y

dxdy. (12.32)
Cˆong th´u
.
c (12.32) go
.
i l`a cˆong th´u
.
c Green, trong d
´o

L
+
l`a t´ıch phˆan
theo d
u

.
`o
.
ng cong k´ın c´o hu
.
´o
.
ng du
.
o
.
ng L
+
.
Hˆe
.
qua
’
. Diˆe
.
nt´ıch miˆe
`
n D gi´o
.
iha
.
nbo
.
’
id

u
.
`o
.
ng cong L d
u
.
o
.
.
c t´ınh
theo cˆong th´u
.
c
S
D
=
1
2

L
xdy −ydx. (12.33)
5
+
Nhˆa
.
nx´et vˆe
`
t´ıch phˆan du
.

`o
.
ng trong khˆong gian. Gia
’
su
.
’
L =
L(A, B)l`ad
u
.
`o
.
ng cong khˆong gian; f,P,Q,R l`a nh˜u
.
ng h`am ba biˆe
´
n
liˆen tu
.
c trˆen L. Khi d
´o t u
.
o
.
ng tu
.
.
nhu
.

tru
.
`o
.
ng ho
.
.
pd
u
.
`o
.
ng cong ph˘a
’
ng
ta c´o thˆe
’
d
i
.
nh ngh˜ıa t´ıch phˆan du
.
`o
.
ng theo d
ˆo
.
d`ai

L(A,B)

f(x, y, z)ds v`a
t´ıch phˆan d
u
.
`o
.
ng theo to
.
ad
ˆo
.

L
P (x, y, z)dx,

L
Q(x, y, z)dy,

L
R(x, y, z)dz
12.3. T´ıch phˆan d u
.
`o
.
ng 149
v`a

L
Pdx+ Qdy + Rdz.
Vˆe

`
thu
.
.
cchˆa
´
tk˜y thuˆa
.
t t´ınh c´ac t´ıch phˆan n`ay khˆong kh´ac biˆe
.
tg`ı
so v´o
.
i tru
.
`o
.
ng ho
.
.
pd
u
.
`o
.
ng cong ph˘a
’
ng.
C
´

AC V
´
IDU
.
V´ı du
.
1. T´ınh t´ıch phˆan d
u
.
`o
.
ng

L
x
y
ds, trong d
´o L l`a cung parabˆon
y
2
=2x t`u
.
d
iˆe
’
m(1,
√
2) dˆe
´
ndiˆe

’
m(2, 2).
Gia
’
i. Ta t`ım vi phˆan d
ˆo
.
d`ai cung. Ta c´o
y =
√
2x, y

=
1
√
2x
,
ds =

1+y

2
dx =

1+
1
2x
dx =
√
1+2x

√
2x
dx.
Tu
.
d
´o suy ra

L
x
y
ds =
2

1
x
√
2x
·
√
1+2x
√
2x
dx =
1
6
[5
√
5 − 3
√

3]. 
V´ı du
.
2. T´ınh d
ˆo
.
d`ai cu
’
adu
.
`o
.
ng astroid x = acos
3
t, y = a sin
3
t,
t ∈ [0,2π].
Gia
’
i. Ta ´ap du
.
ng cˆong th´u
.
c: d
ˆo
.
d`ai (L)=

L

ds. Trong tru
.
`o
.
ng
ho
.
.
p n`ay ta c´o
x

= −3a cos
2
t sin t, y

=3a sin
2
t cos t, ds =
3a
2
sin 2tdt.
V`ıd
u
.
`o
.
ng cong d
ˆo
´
ix´u

.
ng v´o
.
i c´ac tru
.
cto
.
ad
ˆo
.
nˆen
d
ˆo
.
d`ai(L)=4
π/2

0
3a
2
sin 2tdt =6a

−cos 2t
2




π/2
0

=6a. 
150 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
V´ı du
.
3. T´ınh

L
(x − y)ds, trong d´o L : x
2
+ y
2
=2ax.
Gia
’
i. Chuyˆe
’
n sang to
.
ad
ˆo
.
cu

.
.
c x = r cos ϕ, y = r sin ϕ. Trong to
.
a
d
ˆo
.
cu
.
.
cphu
.
o
.
ng tr`ınh d
u
.
`o
.
ng tr`on c´o da
.
ng r =2a cos ϕ, −
π
2
 ϕ 
π
2
.
Vi phˆan d

ˆo
.
d`ai cung
ds =

r
2
+ r

ϕ
2
dϕ =

4a
2
cos
2
ϕ +4a
2
sin
2
ϕdϕ =2adϕ.
Do d
´o
I =

L
(x −y)ds =
π/2


−π/2

(2a cos ϕ) cos ϕ −(2a sin ϕ) sin ϕ

2adϕ
=4a
2
π/2

−π/2
cos
2
ϕdϕ =2πa
2
. 
V´ı du
.
4. T´ınh t´ıch phˆan

L
(3x
2
+ y)dx +(x −2y
2
)dy, trong d´o L l`a
biˆen cu
’
a h`ınh tam gi´ac v´o
.
id

ı
’
nh A(0, 0), B(1, 0), C(0, 1).
Gia
’
i. Theo t´ınh chˆa
´
tcu
’
a t´ıch phˆan d
u
.
`o
.
ng ta c´o

L
=

AB
+

BC
+

CA
.
a) Trˆen ca
.
nh AB ta c´o y =0⇒ dy = 0. Do d

´o

AB
=
1

0
3x
2
dx =1.
b) Trˆen ca
.
nh BC ta c´o x+y =1⇒ y = −x+1, dy = −dx.Dod
´o

BC
=
0

1
[3x
2
+(1− x) − x + 2(1 − x
2
)]dx = −
5
3
·
12.3. T´ıch phˆan d u
.

`o
.
ng 151
c) Trˆen ca
.
nh CA ta c´o x =0⇒ dx =0v`adod´o

CA
= −
0

1
2y
2
dy =
2
3
·
Nhu
.
vˆa
.
y

L
=1−
5
3
+
2

3
=0. 
V´ı du
.
5. T´ınh t´ıch phˆan

L
(x+y)dx−(x−y)dy, trong d´o L l`a du
.
`o
.
ng
elip
x
2
a
2
+
y
2
b
2
=1c´odi
.
nh hu
.
´o
.
ng du
.

o
.
ng.
Gia
’
i. 1
+
Ta c´o thˆe
’
t´ınh tru
.
.
ctiˆe
´
p t´ıch phˆan d
˜a cho b˘a
`
ng c´ac phu
.
o
.
ng
ph´ap d
˜a nˆeu (ch˘a
’
ng ha
.
nb˘a
`
ng c´ach tham sˆo

´
h´oa phu
.
o
.
ng tr`ınh elip).
2
+
Nhu
.
ng d
o
.
n gia
’
nho
.
nca
’
l`a su
.
’
du
.
ng cˆong th´u
.
c Green. Ta c´o
P = x + y, Q = −(x −y) ⇒
∂Q
∂x

−
∂P
∂y
= −2.
Do d
´o theo cˆong th´u
.
c Green ta c´o

L
=

x
2
a
2
+
y
2
b
2
1
(−2)dxdy = −2πab,
v`ıdiˆe
.
n t´ıch h`ınh elip b˘a
`
ng πab. 
V´ı d u
.

6. T´ınh t´ıch phˆan

L
2(x
2
+ y
2
)dx + x(4y +3)dy, trong d´o L l`a
d
u
.
`o
.
ng gˆa
´
pkh´uc ABC v´o
.
id
ı
’
nh A(0, 0), B(1, 1) v`a C(0, 2).
Gia
’
i. Nˆe
´
u ta nˆo
´
i A v´o
.
i C th`ı thu d

u
.
o
.
.
cd
u
.
`o
.
ng gˆa
´
pkh´uc k´ın L
∗
gi´o
.
iha
.
n∆ABC. Trˆen ca
.
nh CA ta c´o x =0nˆen dx =0v`at`u
.
d
´o

CA
2(x
2
+ y
2

)dx + x(4y +3)dy =0.
152 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
Do d´o

L
+

CA
=

L
∗
⇒

L
=

L
∗
.
´
Ap du

.
ng cˆong th´u
.
c Green ta c´o

L
=

∆ABC
[(4y +3)− 4y]dxdy =3

∆ABC
dxdy
=3S
∆ABC
=3. 
B
`
AI T
ˆ
A
.
P
T´ınh c´ac t´ıch phˆan d
u
.
`o
.
ng theo d
ˆo

.
d`ai sau dˆay
1.

C
(x + y)ds, C l`a doa
.
n th˘a
’
ng nˆo
´
i A(9, 6) v´o
.
i B(1, 2). (D
S. 36
√
5)
2.

C
xyds, C l`a biˆen h`ınh vuˆong |x| + |y| = a, a>0. (DS. 0)
3.

C
(x + y)ds, C l`a biˆen cu
’
a tam gi´ac dı
’
nh A(1, 0), B(0, 1), C(0, 0).
(D

S. 1 +
√
2)
4.

C
ds
x − y
, C l`a d
oa
.
n th˘a
’
ng nˆo
´
i A(0, 2) v´o
.
i B(4, 0). (D
S.
√
5 ln 2)
5.

C

x
2
+ y
2
ds, C l`a du

.
`o
.
ng tr`on x
2
+ y
2
= ax.(DS. 2a
2
)
6.

C
(x
2
+ y
2
)
n
ds, C l`a du
.
`o
.
ng tr`on x
2
+ y
2
= a
2
.(DS. 2πa

2n+1
)
7.

C
e
√
x
2
+y
2
ds, C l`a biˆen h`ınh qua
.
t tr`on
12.3. T´ıch phˆan d u
.
`o
.
ng 153

(r, ϕ):0 r  a, 0  ϕ 
π
4

.
(D
S. 2(e
a
− 1) +
πae

a
4
)
8.

C
xyds, C l`a mˆo
.
t phˆa
`
ntu
.
elip n˘a
`
m trong g´oc phˆa
`
ntu
.
I.
(D
S.
ab
3
·
a
2
+ ab + b
2
a + b
)

Chı
’
dˆa
˜
n. Su
.
’
du
.
ng phu
.
o
.
ng tr`ınh tham sˆo
´
cu
’
ad
u
.
`o
.
ng elip: x =
a cos t, y = b sin t.
9.

C
ds

x

2
+ y
2
+4
, C l`a d
oa
.
n th˘a
’
ng nˆo
´
idiˆe
’
m O(0, 0) vo
.
i A(1, 2).
(D
S. ln
√
5+3
4
)
10.

C
(x
2
+ y
2
+ z

2
)ds, C l`a cung du
.
`o
.
ng cong x = a cos t, y = a sin t,
z = bt;0 t  2π, a>0, b>0.
(D
S.
2π
3
√
a
2
+ b
2
(3a
2
+4π
2
b
2
))
11.

C
x
2
ds, C l`a du
.

`o
.
ng tr`on



x
2
+ y
2
+ z
2
= a
2
x + y + z =0
(D
S.
2πa
3
3
)
Chı
’
dˆa
˜
n. Ch ´u
.
ng to
’
r˘a

`
ng

C
x
2
ds =

C
y
2
ds =

C
z
2
ds v`a t`u
.
d
´o suy
ra
I =
1
3

C
(x
2
+ y
2

+ z
2
)ds.
154 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
12.

C
(x + y)ds, C l`a mˆo
.
t phˆa
`
ntu
.
d
u
.
`o
.
ng tr`on




x
2
+ y
2
+ z
2
= R
2
y = x
n˘a
`
m trong g´oc phˆa
`
n t´am I. (D
S. R
2
√
2)
13. T´ınh

C
xyzds, C l`a mˆo
.
t phˆa
`
ntu
.
d
u
.

`o
.
ng tr`on



x
2
+ y
2
+ z
2
= R
2
x
2
+ y
2
=
R
2
4
n˘a
`
m trong g´oc phˆa
`
n t´am I.
T´ınh c´ac t´ıch phˆan d
u
.

`o
.
ng theo to
.
ad
ˆo
.
sau dˆay
14.

C
y
2
dx + x
2
dy, C l`a du
.
`o
.
ng t `u
.
d
iˆe
’
m(0, 0) dˆe
´
ndiˆe
’
m(1, 1):
1) C l`a d

oa
.
n th˘a
’
ng.
2) C l`a cung parabol y = x
2
.
3) C l`a cung parabol y =
√
x.
(D
S. 1)
2
3
;2)
7
10
;3)
7
10
)
15.

C
y
2
dx − x
2
dy, C l`a du

.
`o
.
ng tr`on b´an k´ınh R =1v`ac´ohu
.
´o
.
ng
ngu
.
o
.
.
cchiˆe
`
ukimd
ˆo
`
ng hˆo
`
v`a:
1) v´o
.
i tˆam ta
.
igˆo
´
cto
.
ad

ˆo
.
.
2) v´o
.
i tˆam ta
.
id
iˆe
’
m(1, 1).
(D
S. 1) 0; 2) −4π)
16.

C
xdy −ydx, C l`a du
.
`o
.
ng gˆa
´
pkh´uc d
ı
’
nh ta
.
ic´acdiˆe
’
m(0, 0), (1, 0)

12.3. T´ıch phˆan d u
.
`o
.
ng 155
v`a (1, 2). (DS. 2)
17.

C
cos ydx − sin xdy, C l`a doa
.
n th˘a
’
ng t`u
.
d
iˆe
’
m(2, −2) dˆe
´
ndiˆe
’
m
(−2, 2). (D
S. −2 sin 2)
18.

C
(x
2

+ y
2
)dx +(x
2
− y
2
)dy, C l`a du
.
`o
.
ng cong y =1−|1 − x|,
0  x  2. (D
S.
4
3
)
19.

C
(x + y)dx +(x −y)dy, C l`a elip c´o hu
.
´o
.
ng du
.
o
.
ng
x
2

a
2
+
y
2
b
2
=1.
(D
S. 0)
20.

C
(2a − y)dx + xdy, C l`a mˆo
.
t v`om cuˆo
´
ncu
’
adu
.
`o
.
ng xicloid
x = a(t − sin t), y = a(1 − cos t), 0  t  2π.(D
S. −2πa
2
)
21.


C
dx + dy
|z|+ |y|
, C l`a biˆen c´o hu
.
´o
.
ng du
.
o
.
ng cu
’
a h`ınh vuˆong v´o
.
id
ı
’
nh
ta
.
id
iˆe
’
m A(1, 0), B(0, 1), C(−1, 0) v`a D(0, −1). (DS. 0)
22.

C
(x
2

− y
2
)dx +(x
2
+ y
2
)dy, C l`a elip c´o hu
.
´o
.
ng du
.
o
.
ng
x
2
a
2
+
y
2
b
2
= 1. (DS. 0)
23.

C
(x
2

+ y
2
)dx + xydy, C l`a cung cu
’
adu
.
`o
.
ng y = e
x
t`u
.
d
iˆe
’
m
(0, 1) d
ˆe
´
ndiˆe
’
m(1,e). (DS.
3e
2
4
+
1
2
)
24.


C
(x
3
− y
2
)dx + xydy, C l`a cung cu
’
adu
.
`o
.
ng y = a
x
t`u
.
d
iˆe
’
m
(0, 1) d
ˆe
´
ndiˆe
’
m(1,a). (DS.
1
4
+
a

2
2
+
3(1 − a
2
)
4lna
)
25.

C
y
2
dx + x
2
dy, C l`a v`om th´u
.
nhˆa
´
tcu
’
ad
u
.
`o
.
ng xicloid
156 Chu
.
o

.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
x = a(t −sin t), y = a(1 −cos t), a>0c´odi
.
nh hu
.
´o
.
ng theo hu
.
´o
.
ng
t˘ang cu
’
a tham sˆo
´
.(D
S. a
3
π(5 −2π))
´
Ap du
.
ng cˆong th´u
.

c Green d
ˆe
’
t´ınh t´ıch phˆan du
.
`o
.
ng
26.

C
xy
2
dy − x
2
dx, C l`a du
.
`o
.
ng tr`on x
2
+ y
2
= a
2
.(DS.
πa
4
4
)

27.

C
(x + y)dx − (x −y)dy, C l`a elip
x
2
a
2
+
y
2
b
2
= 1. (DS. −2πab)
28.

C
e
−x
2
+y
2
(cos 2xydx + sin 2xydy), C l`a du
.
`o
.
ng tr`on x
2
+ y
2

= R
2
.
(D
S. 0)
29.

C
(xy + e
x
sin x + x + y)dx +(xy − e
−y
+ x − sin y)dy,
C l`a d
u
.
`o
.
ng tr`on x
2
+ y
2
=2x.(DS. −π)
30.

C
(1 + xy)dx + y
2
dy, C l`a biˆen cu
’

anu
.
’
a trˆen cu
’
a h`ınh tr`on
x
2
+ y
2
 2x (y  0). (DS. −
π
2
)
31.

C
(x
2
+ y
2
)dx +(x
2
− y
2
)dy, C l`a biˆen cu
’
a tam gi´ac ∆ABC v´o
.
i

A =(0, 0), B =(1, 0), C =(0, 1), Kiˆe
’
m tra kˆe
´
t qua
’
b˘a
`
ng c´ach
t´ınh tru
.
.
ctiˆe
´
p. (D
S. 0)
32.

C
(2xy −x
2
)dx +(x + y
3
)dy, C l`a biˆen cu
’
amiˆe
`
nbi
.
ch˘a

.
n gi´o
.
iha
.
n
bo
.
’
i hai d
u
.
`o
.
ng y = x
2
v`a y
2
= x.Kiˆe
’
m tra kˆe
´
t qua
’
b˘a
`
ng c´ach t´ınh
tru
.
.

ctiˆe
´
p. (D
S.
1
30
)
33.

C
e
x
[(1 − cos y)dx − (y − sin y)dy], C l`a biˆen cu
’
a tam gi´ac ABC
v´o
.
i A =(1, 1), B =(0, 2) v`a C =(0, 0). (D
S. 2(2 − e))