Bài tập toán cao cấp tập 3 part 4 pot

11.4. T´ıch phˆan suy rˆo
.
ng 99
v`a
+∞

a
(αf(x)+βg(x))dx = α
+∞

a
f(x)dx + β
+∞

a
g(x)dx.
2) Cˆong th´u
.
c Newton-Leibnitz.Nˆe
´
u trˆen khoa
’
ng [a, +∞) h`am f(x)
liˆen tu
.
cv`aF (x), x ∈ [a, +∞) l`a nguyˆen h`am n`ao d
´ocu
’
a n´o th`ı
+∞


a
f(x)dx = F(x)


+∞
a
= F (+∞) −F(a)
trong d
´o F (+∞) = lim
x→+∞
F (x).
3) Cˆong th´u
.
cd
ˆo
’
ibiˆe
´
n. Gia
’
su
.
’
f(x), x ∈ [a,+∞) l`a h`am liˆen tu
.
c,
ϕ(t), t ∈ [α, β] l`a kha
’
vi liˆen tu
.

cv`aa = ϕ(α)  ϕ(t) < lim
t→β−0
ϕ(t)=
+∞. Khi d
´o:
+∞

a
f(x)dx =
β

α
f(ϕ(t))ϕ

(t)dt. (11.27)
4) Cˆong th´u
.
c t´ıch phˆan t`u
.
ng phˆa
`
n. Nˆe
´
u u(x)v`av(x), x ∈ [a, +∞)
l`a nh˜u
.
ng h`am kha
’
vi liˆen tu
.

c v`a lim
x→+∞
(uv)tˆo
`
nta
.
i th`ı:
+∞

a
udv = uv


+∞
a
−
+∞

a
vdu (11.28)
trong d
´o uv


+∞
a
= lim
x→+∞
(uv) −u(a)v(a).
3. C´ac d

iˆe
`
ukiˆe
.
nhˆo
.
itu
.
1) Tiˆeu chuˆa
’
n Cauchy.T´ıch phˆan
+∞

a
f(x)dx hˆo
.
itu
.
khi v`a chı
’
khi
∀ε>0, ∃b = b(ε)  a sao cho ∀b
1
>bv`a ∀b
2
>bta c´o:



b

2

b
1
f(x)dx



<ε.
100 Chu
.
o
.
ng 11. T´ıch phˆan x´ac d
i
.
nh Riemann
2) Dˆa
´
uhiˆe
.
u so s´anh I. Gia
’
su
.
’
g(x)  f(x)  0 ∀x  a v`a f(x),
g(x) kha
’
t´ıch trˆen mo

.
id
oa
.
n[a, b], b<+∞. Khi d´o :
(i) Nˆe
´
u t´ıch phˆan
+∞

a
g(x)dx hˆo
.
itu
.
th`ı t´ıch phˆan
+∞

a
f(x)dx hˆo
.
itu
.
.
(ii) Nˆe
´
u t´ıch phˆan
+∞

a

f(x)dx phˆan k`y th`ı t´ıch phˆan
+∞

a
g(x)dx phˆan
k`y.
3) Dˆa
´
uhiˆe
.
u so s´anh II. Gia
’
su
.
’
f(x)  0, g(x) > 0 ∀x  a v`a
lim
x→+∞
f(x)
g(x)
= λ.
Khi d
´o:
(i) Nˆe
´
u0<λ<+∞ th`ı c´ac t´ıch phˆan
+∞

a
f(x)dx v`a

+∞

a
g(x)dx
d
ˆo
`
ng th`o
.
ihˆo
.
itu
.
ho˘a
.
cd
ˆo
`
ng th`o
.
i phˆan k`y.
(ii) Nˆe
´
u λ = 0 v`a t´ıch phˆan
+∞

a
g(x)dx hˆo
.
itu

.
th`ı t´ıch phˆan
+∞

a
f(x)dx hˆo
.
itu
.
.
(iii) Nˆe
´
u λ =+∞ v`a t´ıch phˆan
+∞

a
f(x)dx hˆo
.
itu
.
th`ı t´ıch phˆan
+∞

a
g(x)dx hˆo
.
itu
.
.
D

ˆe
’
so s´anh ta thu
.
`o
.
ng su
.
’
du
.
ng t´ıch phˆan
+∞

a
dx
x
α


hˆo
.
itu
.
nˆe
´
u α>1,
phˆan k`y nˆe
´
u α  1.

(11.29)
11.4. T´ıch phˆan suy rˆo
.
ng 101
D
-
i
.
nh ngh˜ıa. T´ıch phˆan
+∞

a
f(x)dx du
.
o
.
.
cgo
.
i l`a hˆo
.
itu
.
tuyˆe
.
td
ˆo
´
inˆe
´

u
t´ıch phˆan
+∞

a
|f(x)|dx hˆo
.
itu
.
v`a du
.
o
.
.
cgo
.
i l`a hˆo
.
itu
.
c´o d
iˆe
`
ukiˆe
.
nnˆe
´
u
t´ıch phˆan
+∞


a
f(x)dx hˆo
.
itu
.
nhu
.
ng t´ıch phˆan
+∞

a
|f(x)|dx phˆan k`y.
Mo
.
i t´ıch phˆan hˆo
.
itu
.
tuyˆe
.
td
ˆo
´
idˆe
`
uhˆo
.
itu
.

.
3) T`u
.
dˆa
´
uhiˆe
.
u so s´anh II v`a (11.29) r´ut ra
Dˆa
´
uhiˆe
.
u thu
.
.
c h`anh. Nˆe
´
u khi x → +∞ h`am du
.
o
.
ng f(x) l`a vˆo
c`ung b´e cˆa
´
p α>0 so v´o
.
i
1
x
th`ı

(i) t´ıch phˆan
+∞

a
f(x)dx hˆo
.
itu
.
khi α>1;
(ii) t´ıch phˆan
+∞

a
f(x)dx phˆan k`y khi α  1.
C
´
AC V
´
IDU
.
V´ı d u
.
1. T´ınh t´ıch phˆan
I =
+∞

2
dx
x
2

√
x
2
− 1
·
Gia
’
i. Theo d
i
.
nh ngh˜ıa ta c´o
+∞

2
dx
x
2
√
x
2
− 1
= lim
b→+∞
b

2
dx
x
2
√

x
2
− 1
·
D
˘a
.
t x =
1
t
, ta thu d
u
.
o
.
.
c
102 Chu
.
o
.
ng 11. T´ıch phˆan x´ac d
i
.
nh Riemann
I(b)=
b

2
dx

x
2
√
x
2
− 1
=
1/b

1/2
−dt
t
2
·
1
t
2

1
t
2
− 1
= −
1/b

1/2
tdt
√
1 − t
2

=
√
1 − t
2



1/b
1/2
=

1 −
1
b
2
−

1 −
1
4
.
T`u
.
d
´o suy r˘a
`
ng I = lim
b→+∞
I(b)=
2 −

√
3
2
.Nhu
.
vˆa
.
y t´ıch phˆan d
˜acho
hˆo
.
itu
.
. 
V´ı du
.
2. Kha
’
o s´at su
.
.
hˆo
.
itu
.
cu
’
a t´ıch phˆan
+∞


1
2x
2
+1
x
3
+3x +4
dx.
Gia
’
i. H`am du
.
´o
.
idˆa
´
u t´ıch phˆan > 0 ∀x  1. Ta c´o
f(x)=
2x
2
+1
x
3
+3x +4
=
2+
1
x
2
x +

3
x
+
4
x
2
·
V´o
.
i x d
u
’
l´o
.
n h`am f(x) c´o d´ang d
iˆe
.
unhu
.
2
x
.Dod
´otalˆa
´
y h`am
ϕ(x)=
1
x
d
ˆe

’
so s´anh v`a c´o
lim
x→+∞
f(x)
ϕ(x)
= lim
x→+∞
(2x
2
+1)x
x
2
+3x +4
=2=0.
V`ı t´ıch phˆan
∞

1
dx
x
phˆan k`y nˆen theo dˆa
´
uhiˆe
.
u so s´anh II t´ıch phˆan d
˜a
cho phˆan k`y. 
V´ı du
.

3. Kha
’
o s´at su
.
.
hˆo
.
itu
.
cu
’
a t´ıch phˆan
∞

2
dx
3
√
x
3
− 12
·
11.4. T´ıch phˆan suy rˆo
.
ng 103
Gia
’
i. Ta c´o bˆa
´
td˘a

’
ng th ´u
.
c
1
3
√
x
3
− 1
>
1
x
khi x>2.
Nhu
.
ng t´ıch phˆan
∞

2
dx
x
phˆan k `y, do d
´o theo dˆa
´
uhiˆe
.
u so s´anh I t´ıch
phˆan d
˜a cho phˆan k`y. 

V´ı d u
.
4. Kha
’
o s´at su
.
.
hˆo
.
itu
.
v`a d
˘a
.
c t´ınh hˆo
.
itu
.
cu
’
a t´ıch phˆan
+∞

1
sin x
x
dx.
Gia
’
i. D

ˆa
`
u tiˆen ta t´ıch phˆan t`u
.
ng phˆa
`
nmˆo
.
t c´ach h`ınh th´u
.
c
+∞

1
sin x
x
dx = −
cos x
x



+∞
1
−
+∞

1
cos x
x

2
dx = cos 1 −
+∞

1
cos x
x
2
dx.
(11.30)
T´ıch phˆan
+∞

1
cos x
x
2
dx hˆo
.
itu
.
tuyˆe
.
tdˆo
´
i, do d´on´ohˆo
.
itu
.
.Nhu

.
vˆa
.
y
ca
’
hai sˆo
´
ha
.
ng o
.
’
vˆe
´
pha
’
i (11.30) h˜u
.
uha
.
n. T`u
.
d
´o suy ra ph´ep t´ıch
phˆan t `u
.
ng phˆa
`
nd

˜a thu
.
.
chiˆe
.
nl`aho
.
.
pl´yv`avˆe
´
tr´ai cu
’
a (11.30) l`a t´ıch
phˆan hˆo
.
itu
.
.
Ta x´et su
.
.
hˆo
.
itu
.
tuyˆe
.
td
ˆo
´

i. Ta c´o
|sin x|  sin
2
x =
1 − cos 2x
2
v`a do vˆa
.
y ∀b>1 ta c´o
b

1
|sin x|
x
dx 
1
2
b

1
dx
x
−
1
2
b

1
cos 2x
x

dx. (11.31)
104 Chu
.
o
.
ng 11. T´ıch phˆan x´ac d
i
.
nh Riemann
T´ıch phˆan th´u
.
nhˆa
´
to
.
’
vˆe
´
pha
’
icu
’
a (11.31) phˆan k`y. T´ıch phˆan th ´u
.
hai o
.
’
vˆe
´
pha

’
id
´ohˆo
.
itu
.
(diˆe
`
ud´odu
.
o
.
.
c suy ra b˘a
`
ng c´ach t´ıch phˆan t`u
.
ng
phˆa
`
nnhu
.
(11.30)). Qua gi´o
.
iha
.
n (11.31) khi b → +∞ ta c´o vˆe
´
pha
’

i
cu
’
a (11.31) dˆa
`
nd
ˆe
´
n ∞ v`a do d´o t´ıch phˆan vˆe
´
tr´ai cu
’
a (11.31) phˆan
k`y, t ´u
.
c l`a t´ıch phˆan d
˜a cho hˆo
.
itu
.
c´o diˆe
`
ukiˆe
.
n (khˆong tuyˆe
.
tdˆo
´
i). 
B

`
AI T
ˆ
A
.
P
T´ınh c´ac t´ıch phˆan suy rˆo
.
ng cˆa
.
n vˆo ha
.
n
1.
∞

0
xe
−x
2
dx (DS.
1
2
)
2.
∞

0
dx
x

√
x
2
− 1
.(D
S.
π
6
)
3.
∞

0
dx
(x
2
+1)
2
.(DS.
π − 2
8
)
4.
∞

0
x sin xdx.(DS. Phˆan k`y)
5.
∞


−∞
2xdx
x
2
+1
.(D
S. Phˆan k`y)
6.
∞

0
e
−x
sin xdx.(DS.
1
2
)
7.
+∞

2

1
x
2
−1
+
2
(x +1)
2


dx.(D
S.
2
3
+
1
2
ln 3)
8.
+∞

−∞
dx
x
2
+4x +9
.(D
S.
π
√
5
5
)
11.4. T´ıch phˆan suy rˆo
.
ng 105
9.
+∞


√
2
xdx
(x
2
+1)
3
.(DS.
1
36
). Chı
’
dˆa
˜
n. D
˘a
.
t x =
√
t.
10.
+∞

1
dx
x
√
x
2
+ x +1

.(D
S. ln

1+
2
√
3

). Chı
’
dˆa
˜
n. D
˘a
.
t x =
1
t
.
11.
+∞

1
arctgx
x
2
dx.(DS.
π
4
+

ln 2
2
)
12.
+∞

3
2x +5
x
2
+3x − 10
dx.(D
S. Phˆan k`y)
13.
∞

0
e
−ax
sin bxdx, a>0. (DS.
b
a
2
+ b
2
)
14.
+∞

0

e
−ax
cos bxdx, a>0. (DS.
a
a
2
+ b
2
)
Kha
’
o s´at su
.
.
hˆo
.
itu
.
cu
’
a c´ac t´ıch phˆan suy rˆo
.
ng cˆa
.
n vˆo ha
.
n
15.
∞


1
e
−x
x
dx.(D
S. Hˆo
.
itu
.
)
Chı
’
dˆa
˜
n.
´
Ap du
.
ng bˆa
´
td
˘a
’
ng th´u
.
c
e
−x
x
 e

−x
∀x  1.
16.
+∞

2
xdx
√
x
4
+1
.(D
S. Phˆan k`y)
Chı
’
dˆa
˜
n.
´
Ap du
.
ng bˆa
´
td
˘a
’
ng th´u
.
c
x

√
x
4
+1
>
x
√
x
4
+ x
4
∀x  2.
17.
+∞

1
sin
2
3x
3
√
x
4
+1
dx.(D
S. Hˆo
.
itu
.
)

106 Chu
.
o
.
ng 11. T´ıch phˆan x´ac d
i
.
nh Riemann
18.
+∞

1
dx
√
4x +lnx
.(D
S. Phˆan k `y)
19.
+∞

1
ln

1+
1
x

x
α
dx.(DS. Hˆo

.
itu
.
nˆe
´
u α>0)
20.
+∞

0
xdx
3
√
x
5
+2
.(D
S. Hˆo
.
itu
.
)
21.
+∞

1
cos 5x − cos 7x
x
2
dx.(DS. Hˆo

.
itu
.
)
22.
+∞

0
xdx
3
√
1+x
7
.(DS. Hˆo
.
itu
.
)
23.
+∞

0
√
x +1
1+2
√
x + x
2
dx.(DS. Hˆo
.

itu
.
)
24.
∞

1
1
√
x
(e
1/x
−1)dx.(DS. Hˆo
.
itu
.
)
25.
∞

1
x +
√
x +1
x
2
+2
5
√
x

4
+1
dx.(D
S. Phˆan k`y)
26.
∞

3
dx

x(x − 1)(x −2)
.(D
S. Hˆo
.
itu
.
)
27
∗
.
∞

0
(3x
4
− x
2
)e
−x
2

dx.(DS. Hˆo
.
itu
.
)
Chı
’
dˆa
˜
n. So s´anh v´o
.
i t´ıch phˆan hˆo
.
itu
.
+∞

0
e
−
x
2
2
dx (ta
.
i sao ?) v`a ´ap
du
.
ng dˆa
´

uhiˆe
.
u so s´anh II.
11.4. T´ıch phˆan suy rˆo
.
ng 107
28
∗
.
+∞

5
ln(x −2)
x
5
+ x
2
+1
dx.(D
S. Hˆo
.
itu
.
)
Chı
’
dˆa
˜
n.
´

Ap du
.
nng hˆe
.
th ´u
.
c
lim
t→+∞
ln t
t
α
=0∀α>0 ⇒ lim
x→+∞
ln(x −2)
x
α
=0∀α>0.
T`u
.
d
´o so s´anh t´ıch phˆan d˜a c h o v ´o
.
i t´ıch phˆan hˆo
.
itu
.
+∞

5

dx
x
α
, α>1.
Tiˆe
´
pd
ˆe
´
n´apdu
.
ng dˆa
´
uhiˆe
.
u so s´anh II.
11.4.2 T´ıch phˆan suy rˆo
.
ng cu
’
a h`am khˆong bi
.
ch˘a
.
n
1. Gia
’
su
.
’

h`am f(x) x´ac d
i
.
nh trˆen khoa
’
ng [a, b) v`a kha
’
t´ıch trˆen mo
.
i
d
oa
.
n[a, ξ], ξ<b.Nˆe
´
utˆo
`
nta
.
i gi´o
.
iha
.
nh˜u
.
uha
.
n
lim
ξ→b−0

ξ

0
f(x)dx (11.32)
th`ı gi´o
.
iha
.
nd
´odu
.
o
.
.
cgo
.
i l`a t´ıch phˆan suy rˆo
.
ng cu
’
a h`am f(x) trˆen [a, b)
v`a k´yhiˆe
.
u l`a:
b

a
f(x)dx. (11.33)
Trong tru
.

`o
.
ng ho
.
.
p n`ay t´ıch phˆan suy rˆo
.
ng (11.33) d
u
.
o
.
.
cgo
.
il`at´ıch
phˆan hˆo
.
itu
.
.Nˆe
´
u gi´o
.
iha
.
n (11.32) khˆong tˆo
`
nta
.

ith`ıt´ıch phˆan suy
rˆo
.
ng (11.33) phˆan k `y.
D
i
.
nh ngh˜ıa t´ıch phˆan suy rˆo
.
ng cu
’
a h`am f(x)x´acdi
.
nh trˆen khoa
’
ng
(a, b]d
u
.
o
.
.
c ph´at biˆe
’
utu
.
o
.
ng tu
.

.
.
Nˆe
´
u h`am f(x) kha
’
t´ıch theo ngh˜ıa suy rˆo
.
ng trˆen c´ac khoa
’
ng [a, c)
v`a (c, b] th`ı h`am d
u
.
o
.
.
cgo
.
i l`a h`am kha
’
t´ıch theo ngh˜ıa suy rˆo
.
ng trˆen
108 Chu
.
o
.
ng 11. T´ıch phˆan x´ac d
i

.
nh Riemann
doa
.
n[a, b] v`a trong tru
.
`o
.
ng ho
.
.
p n`ay t´ıch phˆan suy rˆo
.
ng d
u
.
o
.
.
c x´ac d
i
.
nh
bo
.
’
id
˘a
’
ng th ´u

.
c:
b

a
f(x)dx =
c

a
f(x)dx +
b

c
f(x)dx.
2. C´ac cˆong th´u
.
cco
.
ba
’
n
1) Nˆe
´
u c´ac t´ıch phˆan
b

a
f(x)dx v`a
b


a
g(x)dx hˆo
.
itu
.
th`ı ∀α, β ∈ R
ta c´o t´ıch phˆan
b

a
[αf(x)+βg(x)]dx hˆo
.
itu
.
v`a
b

a
[αf(x)+βg(x)]dx = α
b

a
f(x)dx + β
b

a
g(x)dx.
2) Cˆong th´u
.
c Newton-Leibnitz. Nˆe

´
u h`am f(x), x ∈ [a, b)liˆen tu
.
c
v`a F (x) l`a mˆo
.
t nguyˆen h`am n`ao d
´ocu
’
a f trˆen [a, b) th`ı:
b

a
f(x)dx = F (x)


b−0
a
= F(b − 0) − F (a),
F (b −0) = lim
x→b−0
F (x).
3) Cˆong th´u
.
cd
ˆo
’
ibiˆe
´
n. Gia

’
su
.
’
f(x) liˆen tu
.
c trˆen [a, b) c`on ϕ(t),
t ∈ [α, β) kha
’
vi liˆen tu
.
cv`aa = ϕ(α)  ϕ(t) < lim
t→β−0
ϕ(t)=b. Khi
d
´o :
b

a
f(x)dx =
β

α
f[ϕ(t)]ϕ

(t)dt.
11.4. T´ıch phˆan suy rˆo
.
ng 109
4) Cˆong th´u

.
c t´ıch phˆan t`u
.
ng phˆa
`
n. Gia
’
su
.
’
u(x), x ∈ [a, b)v`av(x),
x ∈ [a, b) l`a nh˜u
.
ng h`am kha
’
vi liˆen tu
.
c v`a lim
x→b−0
(uv)tˆo
`
nta
.
i. Khi d´o;
b

a
udv = uv



b
a
−
b

a
vdu
uv


b
a
= lim
x→b−0
(uv) −u(a)v(a).
3. C´ac d
iˆe
`
ukiˆe
.
nhˆo
.
itu
.
1) Tiˆeu chuˆa
’
n Cauchy. Gia
’
su
.

’
h`am f(x) x´ac d
i
.
nh trˆen khoa
’
ng
[a, b), kha
’
t´ıch theo ngh˜ıa thˆong thu
.
`o
.
ng trˆen mo
.
id
oa
.
n[a, ξ], ξ<b
v`a khˆong bi
.
ch˘a
.
n trong lˆan cˆa
.
nbˆen tr´ai cu
’
ad
iˆe
’

m x = b. Khi d´o
t´ıch phˆan
b

a
f(x)dx hˆo
.
itu
.
khi v`a chı
’
khi ∀ε>0, ∃η ∈ [a, b) sao cho
∀η
1
,η
2
∈ (η,b)th`ı



η
2

η
1
f(x)dx



<ε.

2) Dˆa
´
uhiˆe
.
u so s´anh I. Gia
’
su
.
’
g(x)  f(x)  0 trˆen khoa
’
ng [a, b)
v`a kha
’
t´ıch trˆen mˆo
˜
id
oa
.
n con [a, ξ], ξ<b. Khi d´o:
(i) Nˆe
´
u t´ıch phˆan
b

a
g(x)dx hˆo
.
itu
.

th`ı t´ıch phˆan
b

a
f(x)dx hˆo
.
itu
.
.
(ii) Nˆe
´
u t´ıch phˆan
b

a
f(x)dx phˆan k`y th`ı t´ıch phˆan
b

a
g(x)dx phˆan
k`y.
3) Dˆa
´
uhiˆe
.
u so s´anh II. Gia
’
su
.
’

f(x)  0, g(x) > 0, x ∈ [a, b)v`a
lim
x→b−0
f(x)
g(x)
= λ.
Khi d
´o:
110 Chu
.
o
.
ng 11. T´ıch phˆan x´ac d
i
.
nh Riemann
(i) Nˆe
´
u0<λ<+∞ th`ı c´ac t´ıch phˆan
b

a
f(x)dx v`a
b

a
g(x)dx dˆo
`
ng
th`o

.
ihˆo
.
itu
.
ho˘a
.
cd
ˆo
`
ng th`o
.
i phˆan k`y.
(ii) Nˆe
´
u λ = 0 v`a t´ıch phˆan
b

a
g(x)dx hˆo
.
itu
.
th`ı t´ıch phˆan
b

a
f(x)dx
hˆo
.

itu
.
.
(iii) Nˆe
´
u λ =+∞ v`a t´ıch phˆan
b

a
f(x)dx hˆo
.
itu
.
th`ı t´ıch phˆan
b

a
g(x)dx hˆo
.
itu
.
.
D
ˆe
’
so s´anh ta thu
.
`o
.
ng su

.
’
du
.
ng t´ıch phˆan:
b

a
dx
(b − x)
α


hˆo
.
itu
.
nˆe
´
u α<1
phˆan k`y nˆe
´
u α  1
ho˘a
.
c
b

a
dx

(x − a)
α


hˆo
.
itu
.
nˆe
´
u α<1
phˆan k`y nˆe
´
u α  1.
D
-
i
.
nh ngh˜ıa. T´ıch phˆan
b

a
f(x)dx du
.
o
.
.
cgo
.
i l`a hˆo

.
itu
.
tuyˆe
.
td
ˆo
´
inˆe
´
u
t´ıch phˆan
b

a
|f(x)|dx hˆo
.
itu
.
v`a du
.
o
.
.
cgo
.
i l`a hˆo
.
itu
.

c´o d
iˆe
`
ukiˆe
.
nnˆe
´
ut´ıch
phˆan
b

a
f(x)dx hˆo
.
itu
.
nhu
.
ng
b

a
|f(x)|dx phˆan k`y.
4) Tu
.
o
.
ng tu
.
.

nhu
.
trong 11.4.1 ta c´o
11.4. T´ıch phˆan suy rˆo
.
ng 111
Dˆa
´
uhiˆe
.
u thu
.
.
c h`anh. Nˆe
´
u khi x → b − 0 h`am f(x)  0 x´ac d
i
.
nh
v`a liˆen tu
.
c trong [a, b) l`a vˆo c `ung l´o
.
ncˆa
´
p α so v´o
.
i
1
b − x

th`ı
(i) t´ıch phˆan
b

a
f(x)dx hˆo
.
itu
.
khi α<1;
(ii) t´ıch phˆan
b

a
f(x)dx phˆan k`y khi α  1.
C
´
AC V
´
IDU
.
V´ı d u
.
1. X´et t´ıch phˆan
1

0
dx
√
1 − x

2
.
Gia
’
i. H`am f(x)=
1
√
1 − x
2
liˆen tu
.
c v`a do d´o n´o kha
’
t´ıch trˆen mo
.
i
d
oa
.
n[0, 1 − ε], ε>0, nhu
.
ng khi x → 1 − 0th`ıf(x) → +∞.Tac´o
lim
ε→0
1−ε

0
dx
√
1 − x

2
= lim
ε→0
arc sin(1 − ε) = asrc sin 1 =
π
2
·
Nhu
.
vˆa
.
y t´ıch phˆan d
˜a cho hˆo
.
itu
.
. 
V´ı d u
.
2. Kha
’
o s´at su
.
.
hˆo
.
itu
.
cu
’

a t´ıch phˆan
1

0
√
xdx
√
1 − x
4
·
Gia
’
i. H`am du
.
´o
.
idˆa
´
u t´ıch phˆan c´o gi´an d
oa
.
nvˆoc`ung ta
.
idiˆe
’
m
x = 1. Ta c´o
√
x
√

1 − x
4

1
√
1 − x
∀x ∈ [0, 1).
Nhu
.
ng t´ıch phˆan
1

0
dx
√
1 − x
hˆo
.
itu
.
,nˆen theo dˆa
´
uhiˆe
.
u so s´anh I
t´ıch phˆan d
˜a cho hˆo
.
itu
.

.
112 Chu
.
o
.
ng 11. T´ıch phˆan x´ac d
i
.
nh Riemann
V´ı du
.
3. Kha
’
o s´at su
.
.
hˆo
.
itu
.
cu
’
a t´ıch phˆan
1

0
dx
e
x
− cos x

·
Gia
’
i. O
.
’
d
ˆay h`am du
.
´o
.
idˆa
´
u t´ıch phˆan c´o gi´an d
oa
.
nvˆoc`ung ta
.
i
d
iˆe
’
m x = 0. Khi x ∈ (0, 1] ta c´o
1
e
x
− cos x

1
xe

v`ır˘a
`
ng xe  e
x
−cos x (ta
.
i sao ?). Nhu
.
ng t´ıch phˆan
1

0
1
xe
dx phˆan k`y
nˆen t´ıch phˆan d
˜a cho phˆan k`y. 
V´ı du
.
4. Kha
’
o s´at su
.
.
hˆo
.
itu
.
cu
’

a t´ıch phˆan
+∞

0
arctgx
x
α
dx, α  0.
Gia
’
i. Ta chia khoa
’
ng lˆa
´
y t´ıch phˆan l`am hai sao cho khoa
’
ng th´u
.
nhˆa
´
t h`am c´o bˆa
´
tthu
.
`o
.
ng ta
.
id
iˆe

’
m x = 0. Ch˘a
’
ng ha
.
n ta chia th`anh hai
nu
.
’
a khoa
’
ng (0, 1] v`a [1, +∞). Khi d
´o ta c´o
+∞

0
arctgx
x
α
dx =
1

0
arctgx
x
α
dx +
+∞

0

arctgx
x
α
dx. (11.34)
D
ˆa
`
utiˆenx´et t´ıch phˆan
1

0
arctgx
x
α
dx,Tac´o
f(x)=
arctgx
x
α
∼
(x→0)
x
x
α
=
1
x
α−1
= ϕ(x)
11.4. T´ıch phˆan suy rˆo

.
ng 113
T´ıch phˆan
1

0
ϕ(x)dx hˆo
.
itu
.
khi α − 1 < 1 ⇒ α<2. Do d´ot´ıch
phˆan
1

0
f(x)dx c˜ung hˆo
.
itu
.
khi α<2 theo dˆa
´
uhiˆe
.
u so s´anh II.
X´et t´ıch phˆan
∞

1
f(x)dx.
´

Ap du
.
ng dˆa
´
uhiˆe
.
u so s´anh II trong 1
◦
ta
d
˘a
.
t ϕ(x)=
1
x
α
v`a c´o
lim
x→+∞
f(x)
ϕ(x)
= lim
x→+∞
x
α
arctgx
x
α
=
π

2
·
V`ı t´ıch phˆan
∞

0
dx
x
α
hˆo
.
itu
.
khi α>1 nˆen v´o
.
i α>1 t´ıch phˆan d
u
.
o
.
.
c
x´et hˆo
.
itu
.
.Nhu
.
vˆa
.

yca
’
hai t´ıch phˆan o
.
’
vˆe
´
pha
’
i (11.34) chı
’
hˆo
.
itu
.
khi
1 <α<2.
D
´och´ınh l`a diˆe
`
ukiˆe
.
nhˆo
.
itu
.
cu
’
a t´ıch phˆan d˜a cho. 
V´ı d u

.
5. Kha
’
o s´at su
.
.
hˆo
.
itu
.
cu
’
a t´ıch phˆan
1

0
ln(1 +
3
√
x
2
)
√
x sin
√
x
dx.
Gia
’
i. H`am du

.
´o
.
idˆa
´
u t´ıch phˆan khˆong bi
.
ch˘a
.
n trong lˆan cˆa
.
n pha
’
i
cu
’
ad
iˆe
’
m x = 0. Khi x → 0 + 0 ta c´o
ln(1 +
3
√
x
2
)
√
x sin
√
x

∼
(x→0+0)
3
√
x
2
x
=
1
3
√
x
= ϕ(x).
V`ı t´ıch phˆan
1

0
dx
3
√
x
hˆo
.
itu
.
nˆen theo dˆa
´
uhiˆe
.
u so s´anh II, t´ıch phˆan

d
˜a c h o h ˆo
.
itu
.
. 
114 Chu
.
o
.
ng 11. T´ıch phˆan x´ac d
i
.
nh Riemann
B
`
AI T
ˆ
A
.
P
T´ınh c´ac t´ıch phˆan suy rˆo
.
ng sau.
1.
6

2
dx
3


(4 − x)
2
.(DS. 6
3
√
2)
2.
2

0
dx
3

(x −1)
2
.(DS. 6)
3.
e

1
dx
x ln x
.(D
S. Phˆan k`y)
4.
2

0
dx

x
2
− 4x +3
.(D
S. Phˆan k`y)
5.
1

0
x ln xdx.(DS. −0, 25)
6.
3

2
xdx
4
√
x
2
− 4
.(D
S.
2
3
4
√
125)
7.
2


0
dx
(x − 1)
2
.(DS. Phˆan k `y)
8.
2

−2
xdx
x
2
− 1
.(D
S. Phˆan k`y)
9.
2

0
x
3
dx
√
4 − x
2
.(DS.
16
3
). Chı
’

dˆa
˜
n. D
˘a
.
t x = 2 sin t.
10.
0

−1
e
1/x
x
3
dx.(DS. −
2
e
)
11.4. T´ıch phˆan suy rˆo
.
ng 115
11.
1

0
e
1/x
x
3
dx.(DS. Phˆan k `y)

12.
1

0
dx

x(1 − x)
.(D
S. π)
13.
b

a
dx

(x − a)(b −x)
; a<b.(D
S. π)
14.
1

0
x ln
2
xdx.(DS.
1
4
)
Kha
’

o s´at su
.
.
hˆo
.
itu
.
cu
’
a c´ac t´ıch phˆan suy rˆo
.
ng sau d
ˆay.
15.
1

0
cos
2
x
3
√
1 − x
2
dx.(DS. Hˆo
.
itu
.
)
16.

1

0
ln(1 +
3
√
x
e
sin x
− 1
dx.(D
S. Hˆo
.
itu
.
)
17.
1

0
dx
e
√
x
−1
.(D
S. Hˆo
.
itu
.

)
18.
1

0
√
xdx
e
sinx
− 1
.(D
S. Hˆo
.
itu
.
)
19.
1

0
x
2
dx
3

(1 − x
2
)
5
.(DS. Phˆan k `y)

20.
1

0
x
3
dx
3

(1 − x
2
)
5
.(DS. Phˆan k `y)
116 Chu
.
o
.
ng 11. T´ıch phˆan x´ac d
i
.
nh Riemann
21.
1

0
dx
e
x
− cos x

.(D
S. Phˆan k`y)
22.
π/4

0
ln(sin 2x)
5
√
x
dx.(D
S. Hˆo
.
itu
.
)
23.
1

0
ln x
√
x
dx.(D
S. Hˆo
.
itu
.
)
Chı

’
dˆa
˜
n. Su
.
’
du
.
ng hˆe
.
th ´u
.
c lim
x→0+0
x
α
ln x =0∀α>0 ⇒ c´o thˆe
’
lˆa
´
y
α =
1
4
ch˘a
’
ng ha
.
n ⇒
|lnx|

√
x
<
1
x
3/4
.
24.
1

0
sin x
x
2
dx.(DS. Phˆan k `y)
25.
2

0
dx
√
x − x
3
.(DS. Hˆo
.
itu
.
)
26.
2


1
(x − 2)
x
2
− 3x
2
+4
dx.(D
S. Phˆan k`y)
27.
1

0
dx

x(e
x
−e
−x
)
.(D
S. Hˆo
.
itu
.
)
28.
2


0

16 + x
4
16 − x
4
dx.(DS. Hˆo
.
itu
.
)
29.
1

0
√
e
x
−1
sin x
dx.(D
S. Hˆo
.
itu
.
)
30.
1

0

3

ln(1 + x)
1 − cos x
dx.(D
S. Phˆan k`y)
Chu
.
o
.
ng 12
T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
12.1 T´ıch phˆan 2-l´o
.
p 118
12.1.1 Tru
.
`o
.
ng ho
.
.
pmiˆe
`
nch˜u
.

nhˆa
.
t 118
12.1.2 Tru
.
`o
.
ng ho
.
.
pmiˆe
`
ncong 118
12.1.3 Mˆo
.
tv`ai´u
.
ng du
.
ng trong h`ınh ho
.
c 121
12.2 T´ıch phˆan 3-l´o
.
p 133
12.2.1 Tru
.
`o
.
ng ho

.
.
pmiˆe
`
n h`ınh hˆo
.
p 133
12.2.2 Tru
.
`o
.
ng ho
.
.
pmiˆe
`
ncong 134
12.2.3 136
12.2.4 Nhˆa
.
nx´etchung 136
12.3 T´ıch phˆan d
u
.
`o
.
ng 144
12.3.1 C´ac d
i
.

nh ngh˜ıa co
.
ba
’
n 144
12.3.2 T´ınh t´ıch phˆan d
u
.
`o
.
ng 146
12.4 T´ıch phˆan m˘a
.
t 158
12.4.1 C´ac d
i
.
nh ngh˜ıa co
.
ba
’
n 158
12.4.2 Phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan m˘a
.
t 160
118 Chu

.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
12.4.3 Cˆong th´u
.
c Gauss-Ostrogradski . . . . . . . 162
12.4.4 Cˆong th´u
.
cStokes 162
12.1 T´ıch phˆan 2-l´o
.
p
12.1.1 Tru
.
`o
.
ng ho
.
.
pmiˆe
`
nch˜u
.
nhˆa
.

t
Gia
’
su
.
’
D =[a, b] × [c, d]={(x, y):a  x  b, c  y  d}
v`a h`am f(x,y) liˆen tu
.
c trong miˆe
`
n D. Khi d
´o t´ıch phˆan 2-l´o
.
pcu
’
a
h`am f(x, y) theo miˆe
`
nch˜u
.
nhˆa
.
t
D = {(x, y):a  x  b; c  y  d}
d
u
.
o
.

.
c t´ınh theo cˆong th ´u
.
c

D
f(M)dxdy =
b

a
dx
d

c
f(M)dy; (12.1)

D
f(M)dxdy =
d

c
dy
b

a
f(M)dx, M =(x, y). (12.2)
Trong (12.1): d
ˆa
`
u tiˆen t´ınh t´ıch phˆan trong I(x) theo y xem x l`a h˘a

`
ng
sˆo
´
, sau d
´o t´ıch phˆan kˆe
´
t qua
’
thu du
.
o
.
.
c I(x) theo x.D
ˆo
´
iv´o
.
i (12.2) ta
c˜ung tiˆe
´
n h`anh tu
.
o
.
.
ng tu
.
.

nhu
.
ng theo th´u
.
tu
.
.
ngu
.
o
.
.
cla
.
i.
12.1.2 Tru
.
`o
.
ng ho
.
.
pmiˆe
`
n cong
Gia
’
su
.
’

h`am f(x, y)liˆen tu
.
c trong miˆe
`
nbi
.
ch˘a
.
n
D = {(x, y):a  x  b; ϕ
1
(x)  y  ϕ
2
(x)}
12.1. T´ıch phˆan 2-l´o
.
p 119
trong d´o y = ϕ
1
(x) l`a biˆen du
.
´o
.
i, y = ϕ
2
(x) l`a biˆen trˆen, ho˘a
.
c
D = {(x, y):c  y  d; g
1

(y)  x  g
2
(y)}
trong d
´o x = g
1
(y)l`abiˆen tr´ai c`on x = g
2
(y) l`a biˆen pha
’
i, o
.
’
d
ˆay
ta luˆon gia
’
thiˆe
´
t c´ac h`am ϕ
1
,ϕ
2
,g
1
,g
2
dˆe
`
u liˆen tu

.
c trong c´ac khoa
’
ng
tu
.
o
.
ng ´u
.
ng. Khi d
´o t´ıch phˆan 2-l´o
.
p theo miˆe
`
n D luˆon luˆon tˆo
`
nta
.
i.
D
ˆe
’
t´ınh t´ıch phˆan 2-l´o
.
p ta c´o thˆe
’
´ap du
.
ng mˆo

.
t trong hai phu
.
o
.
ng
ph´ap sau.
1
+
Phu
.
o
.
ng ph´ap Fubini du
.
.
atrˆend
i
.
nh l´y Fubini vˆe
`
viˆe
.
cdu
.
at´ıch
phˆan 2-l´o
.
pvˆe
`

t´ıch phˆan l˘a
.
p. Phu
.
o
.
ng ph´ap n`ay cho ph´ep ta d
u
.
at´ıch
phˆan 2-l´o
.
pvˆe
`
t´ıch phˆan l˘a
.
p theo hai th´u
.
tu
.
.
kh´ac nhau:

D
f(M)dxdy =
b

a

ϕ

2
(x)

ϕ
1
(x)
f(M)dy

dx =
b

a
dx
ϕ
2
(x)

ϕ
1
(x)
f(M)dy, (12.3)

D
f(M)dxdy =
d

c

g
2

(y)

g
1
(y)
f(M)dx

dy =
d

c
dy
g
2
(y)

g
1
(y)
f(M)dx. (12.4)
T`u
.
(12.3) v`a (12.4) suy r˘a
`
ng cˆa
.
ncu
’
a c´ac t´ıch phˆan trong biˆe
´

n thiˆen
v`a phu
.
thuˆo
.
c v`ao biˆe
´
n m`a khi t´ınh t´ıch phˆan trong, n´o d
u
.
o
.
.
c xem l`a
khˆong d
ˆo
’
i. Cˆa
.
ncu
’
a t´ıch phˆan ngo`ai luˆon luˆon l`a h˘a
`
ng sˆo
´
.
Nˆe
´
u trong cˆong th´u
.

c (12.3) (tu
.
o
.
ng ´u
.
ng: (12.4)) phˆa
`
nbiˆen du
.
´o
.
i
hay phˆa
`
n biˆen trˆen (tu
.
o
.
ng ´u
.
ng: phˆa
`
n biˆen tr´ai hay pha
’
i) gˆo
`
mt`u
.
mˆo

.
t
sˆo
´
phˆa
`
nv`amˆo
˜
i phˆa
`
n c´o phu
.
o
.
ng tr`ınh riˆeng th`ı miˆe
`
n D cˆa
`
n chia th`anh
nh˜u
.
ng miˆe
`
n con bo
.
’
i c´ac d
u
.
`o

.
ng th˘a
’
ng song song v´o
.
i tru
.
c Oy (tu
.
o
.
ng
´u
.
ng: song song v´o
.
i tru
.
c Ox) sao cho mˆo
˜
imiˆe
`
n con d
´o c´ac phˆa
`
n biˆen
du
.
´o
.

i hay trˆen (tu
.
o
.
ng ´u
.
ng: phˆa
`
nbiˆen tr´ai, pha
’
i) d
ˆe
`
uchı
’
du
.
o
.
.
cbiˆe
’
u
diˆe
˜
nbo
.
’
imˆo
.

tphu
.
o
.
ng tr`ınh.
2
+
Phu
.
o
.
ng ph´ap d
ˆo
’
ibiˆe
´
n. Ph´ep dˆo
’
ibiˆe
´
n trong t´ıch phˆan 2-l´o
.
p
d
u
.
o
.
.
c thu

.
.
chiˆe
.
n theo cˆong th´u
.
c

D
f(M)dxdy =

D
∗
f[ϕ(u, v),ψ(u, v)]



D(x, y)
D(u, v)



dudv (12.5)
120 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe

´
n
trong d´o D
∗
l`a miˆe
`
nbiˆe
´
n thiˆen cu
’
ato
.
adˆo
.
cong (u, v)tu
.
o
.
ng ´u
.
ng
khi c´ac d
iˆe
’
m(x, y)biˆe
´
n thiˆen trong D: x = ϕ(u, v), y = ψ(u, v);
(u, v) ∈ D
∗
,(x,y) ∈ D; c`on

J =
D(x, y)
D(u, v)
=







∂x
∂u
∂x
∂v
∂y
∂u
∂y
∂v







= 0 (12.6)
l`a Jacobiˆen cu
’
a c´ac h`am x = ϕ(u, v), y = ψ(u, v).

To
.
ad
ˆo
.
cong thu
.
`o
.
ng d `ung ho
.
nca
’
l`a to
.
ad
ˆo
.
cu
.
.
c(r, ϕ). Ch´ung
liˆen hˆe
.
v´o
.
ito
.
ad
ˆo

.
Dˆecac bo
.
’
i c´ac hˆe
.
th ´u
.
c x = r cos ϕ, y = r sin ϕ,
0  r<+∞,0 ϕ<2π.T`u
.
(12.6) suy ra J = r v`a trong to
.
ad
ˆo
.
cu
.
.
c (12.5) c´o da
.
ng

D
f(M)dxdy =

D
∗
f(r cos ϕ, r sin ϕ)rdrdϕ. (12.7)
K´yhiˆe

.
uvˆe
´
pha
’
icu
’
a (12.7) l`a I(D
∗
). C´o c´ac tru
.
`o
.
ng ho
.
.
pcu
.
thˆe
’
sau
d
ˆa y .
(i) Nˆe
´
ucu
.
.
ccu
’

ahˆe
.
to
.
ad
ˆo
.
cu
.
.
cn˘a
`
m ngo`ai D th`ı
I(D
∗
)=
ϕ
2

ϕ
1
dϕ
r
2
(ϕ)

r
1
(ϕ)
f(r cos ϕ, r sin ϕ)rdr. (12.8)

(ii) Nˆe
´
ucu
.
.
cn˘a
`
m trong D v`a mˆo
˜
i tia d
irat`u
.
cu
.
.
cc˘a
´
t biˆen ∂D
khˆong qu´a mˆo
.
td
iˆe
’
mth`ı
I(D
∗
)=
2π

0

dϕ
r(ϕ)

0
f(r cos ϕ, r sin ϕ)rdr. (12.9)
(iii) Nˆe
´
ucu
.
.
cn˘a
`
m trˆen biˆen ∂D cu
’
a D th`ı
I(D
∗
)=
ϕ
2

ϕ
1
dϕ
r(ϕ)

0
f(r cos ϕ, r sin ϕ)rdr. (12.10)
12.1. T´ıch phˆan 2-l´o
.

p 121
12.1.3 Mˆo
.
tv`ai´u
.
ng du
.
ng trong h`ınh ho
.
c
1
+
Diˆe
.
nt´ıchS
D
cu
’
amiˆe
`
n ph˘a
’
ng D du
.
o
.
.
c t´ınh theo cˆong th´u
.
c

S
D
=

D
dxdy ⇒ S
D
=

D
∗
rdrdϕ. (12.11)
2
+
Thˆe
’
t´ıch vˆa
.
tthˆe
’
h`ınh tru
.
th˘a
’
ng d´u
.
ng c´o d
´ay l`a miˆe
`
n D (thuˆo

.
c
m˘a
.
t ph˘a
’
ng Oxy) v`a gi´o
.
iha
.
n ph´ıa trˆen bo
.
’
im˘a
.
t z = f(x, y) > 0d
u
.
o
.
.
c
t´ınh theo cˆong th´u
.
c
V =

D
f(x, y)dxdy. (12.12)
3

+
Nˆe
´
um˘a
.
t(σ)du
.
o
.
.
cchobo
.
’
iphu
.
o
.
ng tr`ınh z = f(x, y) th`ı diˆe
.
n
t´ıch cu
’
an´od
u
.
o
.
.
cbiˆe
’

udiˆe
˜
nbo
.
’
i t´ıch phˆan 2-l´o
.
p
S
σ
=

D(x,y)

1+(f

x
)
2
+(f

y
)
2
dxdy, (12.13)
trong d
´o D(x, y)l`ah`ınh chiˆe
´
u vuˆong g´oc cu
’

am˘a
.
t(σ)lˆen m˘a
.
t ph˘a
’
ng
to
.
ad
ˆo
.
Oxy.
C
´
AC V
´
IDU
.
V´ı d u
.
1. T´ınh t´ıch phˆan

D
xydxdy, D = {(x, y):1 x  2; 1  y  2}.
Gia
’
i. Theo cˆong th´u
.
c (12.2):


D
xydxdy =
2

1
dy
2

1
xydx.
122 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
T´ınh t´ıch phˆan trong (xem y l`a khˆong dˆo
’
i) ta c´o
I(x)=
2

1
xydx = y
x
2

2



2
1
=2y −
1
2
y.
Bˆay gi`o
.
t´ınh t´ıch phˆan ngo`ai:

D
xydxdy =
2

1

2y −
1
2
y

dy =
9
4
· 
V´ı d u

.
2. T´ınh t´ıch phˆan

D
xydxdy nˆe
´
u D du
.
o
.
.
c gi´o
.
iha
.
nbo
.
’
i c´ac
d
u
.
`o
.
ng cong y = x − 4, y
2
=2x.
Gia
’
i. B˘a

`
ng c´ach du
.
.
ng c´ac d
u
.
`o
.
ng gi˜u
.
a c´ac giao d
iˆe
’
m A(8,4) v`a
B(2, −2) cu
’
ach´ung, ba
.
nd
o
.
cs˜ethudu
.
o
.
.
cmiˆe
`
nlˆa

´
y t´ıch phˆan D.
Nˆe
´
ud
ˆa
`
u tiˆen lˆa
´
y t´ıch phˆan theo x v`a tiˆe
´
pdˆe
´
nlˆa
´
y t´ıch phˆan theo
y th`ı t´ıch phˆan theo miˆe
`
n D d
u
.
o
.
.
cbiˆe
’
udiˆe
˜
nbo
.

’
imˆo
.
t t´ıch phˆan bˆo
.
i
I =

D
xydxdy =
4

−2
ydy
y
4

y
2
/2
xdx,
trong d
´odoa
.
n[−2, 4] l`a h`ınh chiˆe
´
ucu
’
amiˆe
`

n D lˆen tru
.
c Oy.T`u
.
d
´o
I =
4

−2
y

x
2
2



y
4
y
2
/2

dy =
1
2
4

−2

y

(y +4)
2
−
y
4
4

dy =90.
Nˆe
´
u t´ınh t´ıch phˆan theo th´u
.
tu
.
.
kh´ac: d
ˆa
`
u tiˆen theo y, sau d´o theo
x th`ı cˆa
`
n chia miˆe
`
n D th`anh hai miˆe
`
n con bo
.
’

id
u
.
`o
.
ng th˘a
’
ng qua B v`a
song song v´o
.
i tru
.
c Oy v`a thu d
u
.
o
.
.
c
I =

D
1
+

D
2
=
2


0
xdx
√
2x

−
√
2x
ydy +
8

2
xdx
√
2x

x−4
ydy
=
2

0
xdx · 0+
8

2
x

y
2

2



√
2x
x−4

dx =90.
12.1. T´ıch phˆan 2-l´o
.
p 123
Nhu
.
vˆa
.
y t´ıch phˆan 2-l´o
.
pd
˜a cho khˆong phu
.
thuˆo
.
cth´u
.
tu
.
.
t´ınh t´ıch
phˆan. Do vˆa

.
y, cˆa
`
ncho
.
nmˆo
.
tth´u
.
tu
.
.
t´ıch phˆan d
ˆe
’
khˆong pha
’
i chia
miˆe
`
n. 
V´ı du
.
3. T´ınh t´ıch phˆan

D
(y − x)dxdy. trong d´omiˆe
`
n D du
.

o
.
.
c
gi´o
.
iha
.
nbo
.
’
ic´acd
u
.
`o
.
ng th˘a
’
ng y = x +1,y = x − 3, y = −
1
3
x +
7
3
,
y = −
1
3
x +5.
Gia

’
i. D
ˆe
’
tr´anh su
.
.
ph´u
.
cta
.
p, ta su
.
’
du
.
ng ph´ep d
ˆo
’
ibiˆe
´
n u = −y −x;
v = y +
1
3
x v`a ´ap du
.
ng cˆong th´u
.
c (12.5). Qua ph´ep d

ˆo
’
ibiˆe
´
nd˜acho
.
n,
d
u
.
`o
.
ng th˘a
’
ng y = x +1biˆe
´
n th`anh d
u
.
`o
.
ng th˘a
’
ng u = 1; c`on y = x −3
biˆe
´
n th`anh u = −3 trong m˘a
.
t ph˘a
’

ng Ouv;tu
.
o
.
ng tu
.
.
, c´ac d
u
.
`o
.
ng th˘a
’
ng
y = −
1
3
x+
7
3
, y = −
1
3
x+5 biˆe
´
n th`anh c´ac d
u
.
`o

.
ng th˘a
’
ng v =
7
3
, v =5.
Do d
´omiˆe
`
n D
∗
tro
.
’
th`anh miˆe
`
n D
∗
=[−3, 1] ×

7
3
, 5

.Dˆe
˜
d`ang thˆa
´
y

r˘a
`
ng
D(x, y)
D(u, v)
= −
3
4
.Dod
´o theo cˆong th´u
.
c (12.5):

D
(y − x)dxdy =

D
∗

1
4
u +
3
4
v

−

−
3

4
u +
3
4
v

3
4
dudv
=

D
∗
3
4
ududv =
5

7/3
dv
4

−3
3
4
udu = −8. 
Nhˆa
.
n x´et. Ph´ep d
ˆo

’
ibiˆe
´
n trong t´ıch phˆan hai l´o
.
p nh˘a
`
mmu
.
cd
´ı c h
d
o
.
n gia
’
n h´oa miˆe
`
nlˆa
´
y t´ıch phˆan. C´o thˆe
’
l´uc d
´o h`am du
.
´o
.
idˆa
´
ut´ıch

phˆan tro
.
’
nˆen ph´u
.
cta
.
pho
.
n.
V´ı d u
.
4. T´ınh t´ıch phˆan

D
(x
2
+ y
2
)dxdy, trong d´o D l`a h`ınh tr`on
gi´o
.
iha
.
nbo
.
’
id
u
.

`o
.
ng tr`on x
2
+ y
2
=2x.
Gia
’
i. Ta chuyˆe
’
n sang to
.
ad
ˆo
.
cu
.
.
c v`a ´ap du
.
ng cˆong th´u
.
c (12.7).
Cˆong th´u
.
cliˆen hˆe
.
(x, y)v´o
.

ito
.
ad
ˆo
.
cu
.
.
c(r, ϕ)v´o
.
icu
.
.
cta
.
id
iˆe
’
m O(0, 0)