Phương pháp tính tích phân và số phức phần 3
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Gidi
>
Phirang t r i n h hoanh dp giao d i e m :
2" = 3 -
X
x = l
^
\
V T = 2" la h a m t a n g
1 :
Do do X = 1 la n g h i e m duy n h a t
S
=
fir
0
(3-x)-2''
3x-
1
.dx
2"
2
y
3
2
VP = 3 - X la h a m g i a m
Vay
/ y = 2'
Inx
--ln2
2
(dvdt).
53o| T i n h d i ^ n t i c h h i n h phfing gi6i h a n bdi cac diTdng
y=|x^-4x
+ 3|,
y = x + 3.
DH kiwi
A/2002
Gidi
Phirong t r i n h hoanh do giao d i e m cua h a i diTorng
x-4x
S
=
5r
0 L
+ 3| = x + 3
X =
0
X =
5
(x + 3) - (x'' - 4x + 3) d x - 2
_x^^5x2^
3
2
5
_iL + 2 x 2 - 3 x
3
(-x^ + 4x - 3)dx
109
(dvdt).
267
531
T i n h d i e n t i c h h i n h t h a n g cong gidi h a n b d i (C) : y = xln^x, true hoanh
va h a i dudng t h i n g x = 1, x = e.
DH Xdy dung
-1997
Gidi
Vi
1
Dat
=>
j^^lxln^ X
u = hi^ X
=:>
dv = x.dx
Uj
= In
xln^x > 0,
dx =
du
=
do do
I n ^ x.dx
X
21nx
,
.dx
S=
V =
X
=> d u j
=
—In^x
2
Ji
X
I n xdx
dx
X
Lai dat
.2
dvj = x.dx
Do do :
=
S =
•e
• hi X
1
2
2
e
2
V
-.dx
2
2
e
x
2• + 4
-l)(dvdt).
= -(6^
4
I532I T i n h dien t i c h m i e n gidi h a n bdi y = x, y = x + sin^x va hai dudng thSng
CO
Ta
0,
X =
X = 7t.
Gidi
:
S =
X
- (x + s i n x) dx =
sin^ xdx
1 f"
(1 - c o s 2 x ) d x = 2J
X
-
sin 2x
= - (dvdt).
2
T i n h d i e n t i c h cQa h i n h phSng gidi h a n b d i cac dudng :
I
x^
y = J4 - —
va
y = 4V2
DH
khd'i B -
2002
268
Gidi
y > 0
*
*
1/ —
y = 14
-
2
2
8
9
PhifOng t r i n h hoanh do giao d i e m :
2
4 -
<=>
4V2
x" - 8x - 1 2 8 =
x^ = 8
00
x^ = - 1 6 (loai)
X = ±2>/2
N h a n xet
S =
Vx e
4>/2
j-2%/2
4-
x2
x2
4
4V2
-2V2; 2V2
dx = 2
4 - —.dx -
r2^
x2
0
4V2
dx
dx
D a t X = 4sint
D o i can
Vay
I.=
h =
Vay
dx = 4costdt
X =
2V2
X =
0
4
^
1
* 4(1 + cos 2t)dt = 4 1 + - sin 2t
8 cos^ t d t =
r2j2
0
4
t = 0
x2
4V2
2
2J2
:dx =
S = 2(Ii-l2) =
I2V2
271 +
-
3;
=
71 +
2
4
3
(dvdt).
|534| Cho h i n h p h i n g (D) g i d i h a n bdi cac difdng y = —^—1
+ X
va y = — .
2
a) T i n h dien t i c h h i n h (D).
b ) T i n h the tich vat thi
tron xoay k h i quay h i n h (D) xung quanh true Ox.
DH Nong nghiep Hd Noi -
1999
269
PhiTOng t r i n h hoanh do :
a) Vay
b)
SD-
pi r
-1
1
^ dx
l l + x^;
Gidi
1^
^7t
2
l + x^
3,
.2
O
X =
±1
(dvdt)
Ca hai dirctng cong deu n^m tren Ox nen ta c6 :
VD/OX =
35
71
1
f1f
-1 V
1 + x'
^
dx - T:
4
-1
y
-dx =
V
^271
(dvtt).
Tinh dien tich hinh phang (D) gidi han bdi cac ducJng
X =
1,
X
= e,
y = 0,
y =
In X
2V^ '
DH Kien true Ha Noi - 1999
Gidi
Ta
CO
:
8,0,
In X
e
=
2A/^
1
• e Vxds
,
dx = Vx In X
X
• 1
= V^-2 f ' - ^
2V^
ISSGI
Tinh dien tich hinh phSng
= V e - 2 V ^ ' = (2-Ve)(dvdt)
1
gidi han bdi y = (x + l)'^ va y = e\ = 1.
( D )
DH Hue - 1998
Gidi
Phifdng t r i n h hoanh do giao diem (x + if
xet f(x) = (x + 1 / - e" C O :
= e" c6 nghiem x = 0 va ta
f (x) = 5 ( x + 1 ) ' - e" > 5 - e > 0
Vx G (0; 1) !=>
f dong bien tren (0; 1)
=>
Vay
f(x) > f(0) = 0
(x + l ) ^ d x -
S(D) =
^
+ i f > e"
(X
f69
eMx =
- e (dvdt).
Tinh dien tich hinh phing gidi han bdi true hoanh, y = x^ - 2x va dudng
thang x = - 1 , x = 2.
DH Thuang mai - 1999
270
Gidi
Ta
CO :
2x = 0
-
Vay
X =
0
X =
2
—CO
X
-
x^ - 2x
s
X
V
f
0
3
- X
-
3
2
+
J
X
X
2
+00
-
(2x-x^)dx
3 A
= -(dvdt).
3
3
1
2
0
(x^ - 2x)dx +
-1
f
0
+
(x^ - 2x)dx =
=
1
,2
38| T i n h dien t i c h h i n h p h i n g gi6i h a n bdi y =
x**
3
+
8x
7
3
3
HV Bau chinh
Viin
7
vay =
~
X
X —
thong
-
3
1997
Gidi
Phuong t r i n h
hoanh do
giao
d i e m ciia h a i dudng
7-x
_
x-3
"
.2
X-
8x
7
Y~
3
x = 0
<=>
X =
4
x = 7
Theo h i n h ve, ta c6 :
S
=
V
^
V
2
X-
8x
7
3
3
8x
-1 +
4
4
3'"^T~3~x-3
x^
4x^
4
3
3
3
iL + 2iL _ 1 X
-
4
ln(x -
dx
3)
= (9-41n4)(dvdt).
539I T i n h dien t i c h S gidi h a n bdi diidng y = sinx t r e n doan [0; 3 K ] va true
hoanh.
271
Gidi
J
y = sinx
0
Nhcf do t h i t a c6 :
1
Trirdc h e t t a t h a y di/dng cong
y = sinx cat x'Ox t a i 4 d i e m
0, X = 71, X = 271, X - 3n.
X =
s
=
•371,
0
1
sin x dx
sin xdx -
r27t
s i n xdx +
= (-cosx) " +(cosx)
2n
va
\
,n.M
'///"
1/
37t
X
sin xdx
3ii
-(cosx)
2n
=6(dvdt).
401 T i n h d i e n t i c h h i n h phAng gidi h a n b d i cac
y = sin IXI
diTdng :
y = IxI -
7t.
DHMaHd
N6i - A/2000
Gidi
sin x
CO
Ta
CO
Ta
:
:
y = sin I x
y = |x
I
neu x > 0
I = s i n ( - x ) = - sin X neu x
-
TI =
neu X
- X - 7t
neu x > 0
X - 71
<
<0
0
T a C O do t h i ben canh.
T a t h a y h a i do t h i do'i xufng
qua Oy n e n t a c6 :
/
S =
2
• 7t
sin X
V 0
— X
1
1
+ 7t
dx
J
S = 2 £ (sin x -
X + 7i)
dx = 2 - cos
X
= (4 + 7i^)(dvdt).
+ TtX
2
|54l| T i n h d i e n t i c h h i n h p h a n g gidi h a n b d i h a i dudng y^ = x^ - x^ va x = 2.
Gidi
H a m y^ = x^(x - 1) xac d i n h k h i
V a y M X D : D = 10}
u
1
X >
0
X =
[1,
272
Ta
CO
Vay
= x^(x - 1)
:
S
«
y
= ± xVx - 1 (x > 1)
= 2
f 2
Vay
"x = 2
u = 1
x = l
u = 0
1
S = 4
0
X \ X
-1
xVx - 1 dx
D a t u = Vx - 1 dx => dx = 2udu va
D o i can
1y =
dx
= j ^ ^ [ x V x - l - ( - x Vx - 1 )
0
+ 1 = x
2
\
(u^ + Dudu = 4
u
u
'x
= -x \ / x - l
= 3 (dvdt).
T ^ ' 2
T i n h dien t i c h h i n h p h ^ n g g i d i h a n bdi cac dudng x = 1, x = 2, true Ox
va difcfng cong y =
1
x ( l + x^)
Gidi
Taco
1
S
dx =
x ( l + x^)
'1
. dx
1 xd+x-^)
f2dx
f2
x
1
x d + x'^)
1
S = l n 2 - - I n 1 + x^
3^
54
vi 1 < X <
X^
- d x = ln2
1 + x-"
2
—
f2d(l +
1
x^)
1 + x^
= In 2 I n - ( d v d t ) .
3
4
T i n h dien t i c h h i n h phang gidi h a n bdi h a i dudng cong y = x^ va y = -x^.
DH Qudc gia Ha Npi - 1997
Gidi
Phirong t r i n h h o a n h do giao d i e m :
x'' = -x^
S =
<=>
x^ (x + 1) = 0
x ^ - C - x ^ ) dx
f
S=
o
J^(x^+x2)dx =
3
4
x
4
j
x_____
3
A
J
-1
= —(dvdt).
12
544
Tinh dien tich hinh phing gidi han bdi difdng t h i n g d : y = x + 1,
dudng cong y = cosx va true hoanh.
Gidi
Dien tich hinh phing phai t i m
chinh la dien tich gidi han bdi d6
t h i (C) : y = f(x) la true hoanh vdi
f(x) xac dinh boti :
X + 1 neu - 1 < X < 0
f(x) =
X
-1
1
V, 1
XJ
0
-1
(x + l)dx +
2
0
' / / / / /\
71
-
2
cos X neu 0 < x < —
2
D i thay f(x) lien tuc tren
S =
X
nen f(x) c6 tich phan tren doan do
cos xdx =
(x + 1)^
2
-1
+ sm X
^ =-(dvdt).
2
0
[545{ Tinh dien tich hinh tron O, ban kinh R.
Gidi
Ta CO phaong trinh diTdng tron tarn O, bain kinh R
+ y- = R^
y^ = R^ - x^
«
^
y
=
± V R ' - x^
Ta xem dudng tron (O; R) la hop cua hai difdng cong :
y = fix) = VR^ - x^
va
y = gu) = -VR^ - x^
Do do :
=
S
=
S
'VR^T^-(-VR^^)\
2
-R
Dat X = Rsint => dx = Rcostdt
Doi can
-R
X=
R
x =
t = il
2
'g(x)=-N/R^-X^
t = -^
2
274
s
=
= 2R2 f2 V l - s i n H c o s t d t = 2R2 f 2 cos^
2
= R2
tdt
~2
sin 2 t ^
2 (l + cos2t)dt = R2
= 7iR2(dvdt).
2
2
546| T i n h dien t i c h cua h i n h elip (E) : — +
= 1.
„2
.2
a
b
Gidi
Taco:
(E) : ^
a
2
+ ^
b
2
= 1
b
- a \.
a
<=>
>
0
b
y = + -Va^ a
-b
Ta xem (E) la hop cua h a i dudng cong :
f(x)= ^ V a ^ - x ^ ;
a
g(x)--
a
Suy ra dien t i c h ciia h i n h elip E
a
dx =
J-a L
D a t X = a sint => dx = acostdt
Vay
S=
b
fa
2Va^-x^dx
a J-a
X = a
D o i can
X = -a
t = ^
2
t = -^
2
— | \ V a ^ ( l - s i n H ) a cos t dt = 2ab J 2 cos^ t dt
2
S =
54?!
2ab
x ( l + cos2t)dt
I
1
A
t + - sin 2t
2
= ab
7t
= Trab (dvdt).
Goi S la dien t i c h h i n h ph&ng gidri h a n boti y = ax^ va y = — ax^, hai
2
ducfng t h i n g y = 1, y = 2 (vdi x > 0).
275
a) Tinh S khi a = 2.
b) Tinh tat ca cac gia t r i ciia a (a > 1) sao cho S dat gia t r i Idn nhat.
Tinh gia t r i Idn nhat do.
DH Hang hdi - 1998
Gidi
a) K h i
X >
0, a > 1 t h i
y = ax
1 2
y = -ax
dy =
S =
Va
va
V 2 - I f2
2(V2^)
Vydy =
3VI
VI
S = ^(5-3A^)
3VI
Khi a = 2 thi
S = — (5 - 3V2) (dvdt).
3
b) S = - ^ ( 5 - 3 V 2 )
3VI
Dodo
-
S„,a, o
Luc do
S„ax
=
a^i„
-
o
a=l
(5 - 3V2) (dvdt).
|548| Tinh dien tich hinh p h i n g gidfi han bdi :
2y = x^ + x - 6
va
2y = -x^ + 3x + 6.
DH Hang hdi - 1997
Gidi
PhifOng t r i n h hoanh do giao diem ciia hai dUcJng cong :
x^ + x - 6 = - x ^ + 3x + 6
c:>
x^-x-6 =0
<=>
S =
- ( x ^ + x-6)--(-x^
Ta c6 :
X
x^ -
X -
x= -2vx=3
+3x + 6)dx =
-2
-00
6
(•3
x^ -
-2
3
X -
6 dx
-foo
WwM -
276
Vay
(•3
S =
f
(-x^ +
X +
X
6)dx =
3
J-2
I549I Cho f(x) = <
X
-i
In
X
0
vdi
X >
vdi
X =
X
1
+ —
125
+ 6x
2
(dvdt).
-2
0
0'
T i n h dien t i c h h i n h p h 4 n g gidi h a n bdi y = f(x) va doan [0; 1] t r e n true
Ox.
BH Y duac TP.HCM
-
1994
Gidi
Xet
y =
X
Inx
vdi x > 0
y' = Inx + 1;
y' = 0
Inx = - 1 = Ine
X
= e-' =
1
0
X
<=>
+00
e
0
y'
+
1
y
e
Do do dien t i c h can t i m la
4
S =
[0 - x l n x ] d x = -
1
x In
X
dx
^
^
1/e
u = In x
du = — dx
dv = xdx
V
0
Dat
S = -
-1/e
x2
0
X
X
=
I n X dx = - — I n x
2
V
1
'0
1 pi
xdx
"2 .0
~
- In
—
X
-
= -(dvdt).
V
5501 Xet h i n h ch^n bdi (P) : y = x^ va dirdng t h a n g qua A(xo; yo) nSm t r o n g
(P) (nghia 1^ yo > XQ) va c6 he so goc k.
T i m k de dien t i c h nho n h a t .
Gidi
Phixang t r i n h dirdng t h i n g d qua A(xo; yo) c6 he so' goc k la :
277
55l|
d : y = k(x - xo ) + yo
Phifdng t r i n h h o a n h do giao d i e m
cua d va P l a :
X- = k ( x - Xo ) + yo
o
- k x + kxo - yo = 0
(*)
Goi X i , X2 l a n g h i e m ciia phiiomg t r i n h (*), t a c6 :
S = x, + X2 = k ,
Ta
CO
:
S =
f "2
P = x i X2 = kxo - yo
(kx - kxo + yo - X )dx =
"kx^
2
"2
+(yo-l«o)x-y
= ^(xl - X i ) + (yo - k x o ) ( x 2 - X i ) - - ( x ^ - x ? )
=
-
6
- X i ) 3 k (x2 + X i ) + 6 (y - k x g ) - 2 (xg + X i + X j X g )
I-
4^
3k2 + 6 ( y o - k x o ) - 2 ( S 2 - P )
= J V k ' - 4 k x o + 4 y o (k^ - 4kxo + 4yo)
b
= i(k2-4kxo+4yo)2
D
3
= ^ [ ( k - 2xo f + 4yo - 4x21i > 1 ^^^^ _ ^,^2
b
3
b
Dau " = " xay r a <=>
k - 2xo = 0
<=>
3
Khido
S„,„= i 8 ( y o - x 2 ) 2
b
k = 2xo
3
=l(yo-x2)2.
6
Cho (P) : y^ = 2x va di/dng t h i n g D : x - 2y + 2 = 0. Chufng m i n h (D) la
t i e p tuyen ciia (P).
T i n h dien t i c h h i n h phang g i d i h a n b o i (D) va P.
DH Kink te Quoc dan Ha Noi - 1997
Gidi
Taco:
( - 2 ) l l = 2.1.2
c:>
B'^P = 2AC
c=>
(D) tiep xuc v d i (P) t a i A{2; 2)
278
Ta
CO :
S
=
x +2
dx
-2
4
3
-2
= -(dvdt).
3
I552I T i n h dien tich hinh p h i n g gidi han bdi hai dudng cong :
= ax
va
= ay
(a > 0).
Giai
T a CO hai dudng cong (P) c i t nhau tai (0; 0) va A (a; a)
0
S
va
=
r
Jo
•I
y>0=>
y = ^[s^
,2
Vax
dx
a
r-
I -
— Vx a v x
3
3
3
y.
3 A
3a
3
553I T i n h di^n tich hinh p h i n g gidi han bdi cac dudng :
y = - V 4 - x^
va
x^ + 3y = 0.
BH Bach khoa Hd Noi
-2001
Gidi
PhiTctng trinh hoanh do giao diem ciia hai dudng :
..2
,
.4
= - V 4 - x^
—
9
=4-x2
o
x*+9x2-36 = 0
= ±V3
S =
f
,.2
;
dx = 2
p/3 /
^0
2
3
dx
279
I554I
S = 2
(
Tinh I = 2
73
V3
+ 2
V4 -
73
3
3
cos^ t dt = 4
^
S =
2V3
73
Jo
dx
D a t X = 2 s i n t => dx = 2costdt
I = 8
^
Do do
D o i can :
V3
X =
x = 0
3 ( l + c o s 2 t ) d t = (4t + 2 s i n 2 t )
47: +
+
3V3
47: +
=
V3
3
3
t = 0
47I +
3V3
(dvdt).
T r o n g m a t p h i n g Oxy, t i n h dien t i c h h i n h p h i n g D gidi h a n bdi cac
dudng y = xe", y = 0, x = - 1 , x = 2.
Hoc vien BiCu chinh Viin
thong
- 2001
Gidi
S
Nen
y = xe"
T a CO :
l a h a m so don dieu t a n g t r e n [ - 1 ; 2] va y (0) = 0
=
xe
dx =
= e^Cx-l)
555
2
0
Jo
-e"
xe" dx -
x-1)
0
-1
(
-1
xe" dx
2^
- (dvdt).
2
e^ + 2 -
T i n h dien t i c h m i e n gidi h a n bdi ( C i ) : y^ = 2x va ( C 2 ) : 27y^ = 8(x - \
Gidi
Phuong t r i n h hoanh do giao d i e m
cua
(Ci)
va
27y^ = ( x - l /
(C2) :
54x = 8(x
-\f
(x - 4){2x + 1)^ = 0
o
8x^ - 24x^ - aOx - 8 = 0
«
x = 4
1
X = —
2
y'
loai v i X = — > 0
2
280
Vay giao diem cua (Ci) va ( C 2 ) la : A(4; 2A/2), B ( 4 ; -2^2)
Ta
CO
:
27y' = 8(x -
if
o
y = ±
2A/2
3V3
(x - 1).A/X - 1 vdi X > 1
Do true hoanh la true do'i xilfng cua (Ci) va ( C 2 ) nen :
S
V2xdx+ r
=2
A/2I-^V(X-1)'
ix
J
= 2
0
V2^dx - ^
r
Ji
V3
^|{x-lfdx
4V2
.
5
32A/2
8 V2 „ /- 3 2 V 2
._^.1£9V3 =
3
15 Vs
3
72 /88
V2 = — V 2 ( d v d t ) .
15
15
5561 Cho diem A tuy y tren (P) : y = px^ (vdi p > 0). Goi (D) la dudng t h i n g
song song vdi tiep tuyen tai A va (P), (D) cAt (P) tai M , N .
Hay so sanh dien tich tam giac A M N va dien tich hinh c h i n tren bdi
(D) va phia dudi bdi (P).
DH Kinh te Quoc dan Ha Ngi - 1996
Gidi
Goi A (a; pa^) e P.
PhUcfng trinh ttA(P) : - (yA + y) =
2
PXAX
<=>
y - pa^ = 2pax
o
y = 2apx + pa^
(D) song song vdi tiep tuyen tai A cua (P) nen (D) cd phuong t r i n h :
(D) : y = 2pax + b
Vay
d(A, D) =
o
2pax - y + b = 0
2pa^ - pa^ + b
pa^ + b
V^pVTl
V4pVTl
Phucfng trinh hoanh do giao diem cua (D) va (P) la :
px^ = 2pax + b
o
px^ - 2pax - b = 0
(*)
Gpi X M , X N la 2 nghiem cua phuong t r i n h (*), ta c6 :
281
Do do
=>
. =^
S = XM + XN = ^
P - X j j . X j Np
^
=
2a
-
M N ^ = (XM - x ^ f + (YM
- YN)^
M N ^ = (XM - XN)^ + [ ( 2 p a x M + b) - ( 2 p a x N + b)]^
M N ' = (XM - XN)' + 4 p V ( x M - XN)^ = (XM - XN)1(1 + 4p'a')
= (S^-4P)(1 + 4pV) =
A
4a
2
4b
+ — (1 + 4p^a^)
P
«
MN
= 2 ^ a 2 + ^.Vl
)
+ 4p2a2
4p2a2
Vl+
Vay
SAMN = ^ M N . d ( A , D ) = Ja^
2
pa2
1
+b ja^+-
pa2 + b
Vl + 4 p^a^
P
= Si
M a t khac t a c6 :
( 2 p a x + b - px^ )dx =
pax^ + bx -
px
pa(xN^ - XM^) + b(xN - XM) - ^ (XN^ - XM^)
3
: (XN - X M )
:VS2
-4P
S2
a
2
pa(xM + X N ) + b - | ( x ^ + x ^ i
paS + b - ^ ( S 2
2A
" (2
—a p+- b
+ - 3
3 j
+XMXN)
-P)
3
2a2p + b -
2 V . ^ .
= 2
Do do
4a
+ —
=1
+ ^(pa^ + b)
3 \
=-Si.
3
282
B. THE TfCH VAT THE TRON XOAY
KIEN THLfC C d B A N
1. Cho h i n h p h i n g gidi h a n b d i cac dudng y = f(x), x = a, x = b, quay xung
quanh Ox, tao t h a n h v a t t h e t r o n xoay t h i :
f b
VT =
2. Cho h i n h
phing
gidi
n
[f(x)fd
han bdi
X
= g(y)
cac dirdng x = g(y), y = c, y = d,
x = 0, quay xung quanh Oy, tao
t h a n h v a t the t r o n xoay t h i :
pd
55?!
g(y)
dy
Cho h i n h p h i n g (D) gidi h a n b d i cac dudng y = (x - 2)^ va y = 4.
T i n h t h e t i c h ciia v a t t h e t r o n xoay s i n h r a b d i h i n h (D) k h i no quay
xung quanh :
1. True Ox
2. True Oy.
DH Hang
hdi - 2000
Gidi
1. D quay xung quanh true Ox
V
= n
0 -
42
-
(X -
2)'
dx
4
= 6 4 71 - 71 f ' ' ( x - 2 ) M x
Jo
V
= 6 4 71 -
256?:
Ttt^
-2
(dvtt).
0
w
2
F
4
X
283
2. D quay xung quanh true Oy
Ta
CO
:
y = (x - 2f
<=>
-2=
±Vy
<=>
3
+
V=7I
- ( 2 - V ^ f l d y = 87xJ^'V^dy = 8 7 t | y 2
12871
(dvtt).
558| Tinh the tieh cua vat the tron xoay do quay xung quanh Oy phan mat
phang hufu han duoc gidi han bai hai true toa dp, dudng t h i n g x = 1 va
diicfng cong y -
1
1 + x^
BH Hai Phong - 1997
Gidi
f(x) =
Xet
Ta
CO :
1 + x^
f '(x) =
,
CO
MXD : R
2x
-
(l + x^)^
X
0
-ao
0
+
y'
y
(Do t h i ben canh).
Ta
CO
:
1
y =
x^=l-l
1 + x^ = -
1 + x^
yy
y
Goi Vi la the tich khoi tron xoay do quay xung quanh Oy cua
(C) : x^ = — - 1, cac dudng t h i n g x = 0, y = - va y = 1 thi :
1
-'II
- - 1 dy = 7c(lny - y)
.y
=
T:
1
f, 1
I 2
(0 - 1) - In
1^
2,
ln2-2J
Goi V2 la the tich khoi tron xoay do quay xung quanh Oy cua hinh gidi
han boi cac di^dng x = l , x = 0, y = 0, y = : i
284
1
2
1 dy =
2
7iy
0
Do do the t i c h can t i m la
_
7t
^2
V = V i + Va = 7iln2 (dvtt).
Goi ( H ) 1^ m i e n k i n g i d i h a n bdi dudng cong (L) : y = x - ^ l n ( l + x ^ ) ,
true Ox, va diTdng t h a n g x = 1. T i n h the t i c h ciia vait the tao r a k h i cho
(H) quay xung quanh true Ox.
Hoc vien Ngan hang TP.HCM
-
1999
Gidi
Phucfng t r i n h hoanh do giao d i e m ciia (L) va Ox
•
fx = 0
x J l n d + x^) = 0
<=>
[x > 0
(l + x^)>l
Vay
V = 7t f ^ x 2 l n ( l + x^)d>
Jo
u = l n ( l + x^)
=>
du =
dv = x^dx
=>
v =
Dat
Nen
x = 0
V = - ( x ^ + l ) l n ( x ^ +1)
Sx'^ dx
1 + x^
+1
- 71
^x^dx = - 2 1 n 2 - 7 t —
V = - ( 2 In 2 - 1 ) (dvtt).
3
Goi D \k m i e n g i d i h a n b d i (?) : y = 2x - x ^ va true hoanh. T i n h t h e
t i c h eua v a t t h e V do t a quay xung quanh :
1. True Ox
2. True Oy.
Giai
1. Phucfng t r i n h h o a n h do giao d i e m ciia P va Ox :
2x - x = 0
Vay
V
X =
0
x = 2
dx
285
2.
0
V
=
V =
2
< X <
r2
71
71
r4
( 2 x - x ^ ) ^ d x = 7i C(4x'^ - 4 x ^ H-x"* )dx
Jo
— X
3
3
thi
C5
X -
-
X
4
+
y = 2 x - x^
,
1 = ±
16TC
c=>
«
(dvtt).
x^ - 2 x + 1
V
71
=
T i n h the
fVl-Vl-yf
rl
71
7T
dy+
2 - y - 2 ( l - y ) 2
f (l
7t
f
2
1
- Vl -
+ ^ l - y f
dy +
i
y
vdi
vdi
X
X
dugc tao
e [ 1 ; 2]
e [0; l ]
dy
2-y+
2(l-y)2
dy
+ n 2 y - ^ - - ( l - y ) 2
2 y - ^ . - a - y ) 2
t i c h V ciia v a t t h e
1 - y
fx = 1 + J l - y
,
[x = 1
Vay
V
4
y2
=
0
=
^(dvtt).
3
r a k h i quay h i n h gidi h a n
diicfng y^ = (x - 1 ) ^ v a ducfng t h i n g x = 2 q u a n h t r u e
bdi
Ox.
Gidi
Ta
CO :
V
=
71
=
f
2
y'dx
(x -
71
=
562
D^dx
= -
-(x-1)^
4
(dvtt).
4
1
T i n h t h e t i c h k h o i t r o n x o a y ducfc t a o t h a n h k h i q u a y q u a n h O x
p h a n g g i d i h a n b d i cac d u 6 n g y = 0, y = -^xsinx
DH
hinh
+ cos^ x , x = 0, x = 2.
Bach
khoa
TP.HCM
-
1993
Gidi
Taco :
V
= n^^^fixjfdx
= n
2 ( x s i n x + cos
x)dx
0
286
2 X sin
X
0
Tinh I i =
2 X sin
X
dx +
dx.
Dat
Tcdi + I2)
= -
2 cos
fx 1 + cos 2x
* Tinh I2 = f2cos^ xdx =
V =
JO
2 xsin xdx = - x cos X 2
Ii =
Vay
[2 cos^ x dx
7t
0
u = x
dv = sin
X
dx =
dx =
2
dx
X
cos
X
+ sin x)
sin2x
2 _n
(-X
X +
du = dx
v = - cos x
0
= 1
^4
(4 + 71) ( d v d t ) .
4
563j Cho mien p h i n g (D) gidi han bcfi cac duong :
y = tan-'x,
y = 0,
x=:-,
4
x = -4
1. Tinh di?n tich mien (D).
2. Tinh the tich vat the tron xoay dtroc tao thanh khi quay (D) xung
quanh true Ox.
DH Nong nghicp - A/1999
Gidi
1.
Dat
S=
F
- tan^ x dx +
S =
t =
-X
0
X
tan^ x dx
\ tan^ xdx = f ° - tan^ - t(-dt) = f * tan^ t dt
J-Jo
thi :
2 " tan^
Jo
dx = 2
4
0
tan
S = 2 * tan x.d(tan x) + 2
0
+ 21nv
J2
X
(tan x + 1) dx - 2
4
0
tan
f - d(cos x) „ f 1
= 2 r u.du + 2
Jo
0
cos X
= l + 21n
V
0
X
dx
V
2 ,
S = (1 - ln2)(dvdt).
287
2.
V(D,
=
n
„ ( - t a n ' ' X ) M x + Tt
(tan^ xfdx
= Ti \^ x)dx
4
tan* X (tan^ x + 1) - tan^ x (tan^ x + 1) + (tan^ x + 1) - 1 dx
V,D)
* (tan'' x - t a n ^ x + 1) d(tan x) - TI
=
4
-1
^ (tan'' x - tan^ x + 1) - n
tdx
~4
(t" - t ^ + l ) d t - 7 i - = — ( d v t t ) .
2
2
564| T i n h the t i c h vat the t r o n xoay s i n h ra bdi phep quay xung quanh true
Ox cua h i n h phSng g i 6 i h a n b d i Ox va dudng y = Vx sin x (0 < x < TT).
DHDubi
1
-AI2004
Gidi
•
sin
X
71
7t
V =
2
J
2
^
x dx = —
2
X
. —sin2x
4
+
71
—
4
2
(1 - cos 2x)
•n
0
X
dx = — x'
2 J0
x cos 2x dx
sin2xdx = —(dvtt).
4
|565i T i n h the t i c h k h o i t r o n xoay duac tao r a k h i quay quanh true Ox mien
(D) g i d i h a n b d i y = I n x , y = 0, x = 2.
Gidi
Ta
:
CO
r2
V =
TI
(hix)2dx
u = I n x => du = —
Dat
X
dv = I n x.dx => v = x ( l n x - 1)
=
V
=
V
r2
7t
( l n x r d x = 7r
X.
I n x ( l n x - 1)
27iln2(ln2-l)-7i[x(lnx-l)-x]
1
V
27i(ln 2f
=
27tln2(ln2-l)-7t[(21n2-4)
=
=
+ 2]
- 271 I n 2 - 271 I n 2 + 271
27ir(ln2)--21n2 + l ] = 2jx(ln2-l)2(dvtt).
288
C h o ( D ) l a m i e n g i d i h a n b d i cac dLTcrng y = V x , y = 2 - x v a y = 0.
1. T i n h d i e n t i c h ciia m i e n ( D ) .
2. T i n h
t h e t i c h k h o i t r o n xoay
dirge t a o t h a n h
k h i t a quay
xung
quanh true Oy.
BH
Quoc gia TP.HCM
-
A/2000
Gidi
P h u a n g t r i n h h o a n h do giao d i e m
t =
0
A/X >
t^ + t - 2 = 0
ft =
>0
t = 1 V t = - 2 (loai)
Vay
1=
x = 1
S =
A/^dx +
S^HK
=
-xV^
0
2 . T a CO :
Do do
y
o
= Vx
V
=
V =
T:
71
=>
f1
x =
+ - .1.1 = - ( d v d t ) .
2
6
- x
y = 2
y^;
( 2 - y ) 2 d y - 7 t f (y^fdy
Jo
=
2 - y
= n{ (4 - 4y + y ^ - y ^ j d y
Jo
3271
4 y - 2 y ^ . ^ - > ^ ' '
X
15
(dvtt).
T i n l i t h e t i c h k h o i t r o n xoay g i d i h a n b a i difdng y = x h i x , y = 0, x = 1,
X = e k h i t a quay xung q u a n h true Ox.
DHXaydUng
-1997
Gidi
T a eo :
Dat
V =
7:
f ( x I n x)^ d x =
u = ( I n x)2
dv = x^dx
Vay
TI
f x^ I n x^ d x
du = — I n x dx
V
=
x2ln2 X dx = — (lnx)2
3
Ji
x^ ha
X
dx
289
Ui
= In X
Lai dat
dvj =
Do do
X
=> duj = —
x
dx =>
V = 71 — ( l n x ) 2
3
Tte'^
27t
V =
3
e
+
= —
271
271
9
x^ I n
X
X'^
3
27
271
9 J
x^dx
(5e^ - 2 ) ( d v t t ) .
568| T r o n g mp Oxy, cho h i n h p h a n g D gidi h a n bdi cac diTdng y = Inx va
y = 0, X = e. T i n h the t i c h k h o i t r o n xoay tao nen k h i quay D quanh
true Ox.
BHKinh
te TP.HCM
-2001
Gidi
V =
Vay
Inx = 0
Ta
CO
o
x = 1
(In x f d x
7c
0
2
,
2 Inx
u = I n x => du =
dx
x
Dat
dv = dx
V
=
71
Jl
71
=
71
-
=>
'1
2
e
= x
V
(hix) dx =
xln^x
71
V
hi
e-2
X
e
" 21nxdx
1
1
)
(ti^ng p h a n 1 I a n nufa)
dx
e - 2(x I n X - x)| ^ = 7t(e - 2) (dvtt).
569J Cho (P) : y = x^ (x > 0). Gpi D la h i n h gidi h a n bdi m i e n ngoai cua (P),
dudng t h a n g d c6 phuong t r i n h y - - 3 x + 10, y = 1. T i n h the t i c h vat
the t r o n xOay k h i quay quanh true Ox.
Gidi
Goi V i la t h e t i c h v a t the t r o n xoay do A B M N quay quanh Ox
V2 la the t i c h v a t the t r o n xoay do N M C D quay quanh Ox
V 3 l a the t i c h h i n h t r u do h i n h chiif n h a t A B C D quay quanh Ox
V l a the t i c h c^n t i m .
290
Ta
V = V i + V2 -
CO
•2
V=
71
„ „
(x^fdx
+n
J2
1
V =
V3
r3
(-3x + 10)2dx-7i
^ xMx + Tt C(9x^
1
J2
\ n —
5
+100x)
1
I
_
„
+ 7n-2n
5
- 60x + 100)dx - 71
+ 7i(3x^ - 3 0 x ^
SITI
V =
5671
=
1-3
l^dx
f 3
dx
- 1)
- 71(3
2
,,
(dvtt).
5
57o| T i n h the t i c h k h o i t r o n xoay tao nen k h i quay quanh true Ox h i n h S
gidi h a n bdi y = xe", x = 1, y = 0, v d i 0 < x < 1.
Gidi
Ta
V =
CO
u =
Dat
7t
1
' ( x e ' ' ) M x = 71
x^e^Mx
0
du - 2 x d x
X
dv = e2Mx ^
1
v = -e2''
2
Uj = X => duj = dx
Vay
V=
-x^e^
2
- n
0
xe^^dx
^
1
0
L a i dat
4
=
2
7ie
2
0
^ 2
1
7t
o
71
o„
e"^ + - e
2
4
0
= -(e^ - I X d v t t ) .
4
57l| T i n h the t i c h v a t the t r o n xoay duoc s i n h r a bdi h i n h p h a n g g i d i h a n
bdi y = — va y = x^ k h i h i n h p h a n g quay quanh Ox.
3
DH Hang hdi - 1999
Gidi
/
Ta
CO
:
V =
TT
.3^2
(x2)2dx-7i
dx =
v3y
V=
71.-
7t
X
9'T
486
35
Tt
X
dx
f 3
9 J
xMx
(dvtt).
291
