Phương pháp tính tích phân và số phức phần 3

Hay so sanh dien tich tam giac AMN va dien tich hinh chin tren bdi D va phia dudi bdi P... THE TfCH VAT THE TRON XOAY 2... Tinh di?n tich mien D.. Tinh the tich vat the tron xoay dtroc t

267

531 Tin

h die

n tic

h hin

h than

g con

g gid

i han bdi (C) :

y = xln^x, tru

e hoan

h

va ha

i dudn

g thing

x =

1, x

= e

DH Xdy dung -1997

Gidi

Vi 1

< X

< e => xln^

X

u = h i^

.d

x

dv = x.dx V =

S=

—In

dx

X 2

6^

l)(dvdt)

Gidi

Ta

CO

: S =

X

- (

x + sin x)

1 f"

dx = sin^ xd

x

2 J

(1 cos2x)d

-x = -

X

-sin 2x

h phSn

g gid

i han bdi ca

4 — v

-a y =

4V2

DH khd 'i B

- 2002

b ) Tinh the tich vat thi tron xoay k h i quay hinh (D) xung quanh true Ox

DH Nong nghiep Hd Noi - 1999

269

SD-

l + x^

2 O

X =

±1

pi r

.2

3, (dvdt)

^ 27

1

(dvtt)

35 Tinh dien tic

h hin

h phan

g (D) gid

e

In

X

1 2

A/^

dx = Vx

^ = V

e-2V

^ ' = (2-

IS SG I

Tinh dien tich hin

h phSng

D )

gidi ha

n bd

i y = (

x + l)'^ v

a y = e

C

O :

f (x) =

5 (x+

1)'- e

=> f(x) >

^dx-eMx =

f69

- e (dvdt)

Tinh dien tich hin

1999

Phuong t r i n h hoanh do giao

diem ciia hai dudng

Gidi

Trirdc he

x t

ai 4 diem

X =

0,

X =

71, X = 271 , X

- 3n

Nhcf d

o t

hi ta c6 :

•371,

s = sin

r2

sin xdx +

= (-cosx) "

+(cosx) -(

cosx)

=6(dvdt)

2n

sin xdx

3ii 2n

401

Tin

h die

n tic

h hin

h phAn

g gid

i han bdi cac

d iT dn

g :

y = sin I

XI

va

y =

I x I -

: y = si

n I x

sin(-x) =

i be

n canh

Ta tha

y ha

i d

o th

x

X + 7i)

dx = 2

- cos

X + TtX

2

= (

4 + 7i

^)(dvdt)

|54l| Tin

h die

n tic

h hin

h phan

g gid

i han bdi ha

i dudn

g y

^ = x^ - x^ v

c din

h k

hi

Vay MXD : D = 10}

DH Qudc gia Ha Npi - 1997 Gidi

Phirong t r i n h hoanh do giao diem :

544 Tinh dien tich hin

g d : y

x v

a tru

e hoanh

y = f(x) l

< X <

0

cos

X

neu 0 < x < —

2

Di tha

y f(x ) lie

0

2 + s m

n kin

h R

=

±VR'

VR^

- x

^ v

a y = gu ) =

T^

-(

-V R^

t =

> d

x = Rcostdt

s = = 2R2 f2 V l - s i n H c o s t d t = 2R2 f 2 cos^ tdt

2 ~2

275

a) Tin

h S khi a

1) sa

o ch

o S dat gi

1998

Gidi

a) Khi

X >

0, a > 1 thi

2 2(

V2

^)

VI Vydy =

3V

I

S = ^(5-3

S„a

x = -

x

6 v

a 2

y = -x^ + 3x +

6

DH Hang hdi -

< => x = -

0

-2 3

Xet y = X Inx vdi x > 0

d :

y = k(x -

xo ) + y

o

Phifdng trin

P l

a :

X- = k(x

- X

o ) + y

o

o k

-x + kxo -

yo =

0 (*

)

Goi Xi, X2 la nghiem

ciia phiiom

g trin

h (*), t

a c

6 :

S =

x, + X2 = k, P = xi X2 = kx

o

yo

"kx^ = )dx X yo - kxo + (kx - "2 f = : S CO Ta

2 +(

yo-l

o kx

-o) (x

2

-Xi) (

x^

x?

-)

=

- Xi ) 3

k (x

2 +

X i ) +

6 (

y kxg) -

2 (x

g + X

o -kxo)-

2(

S2

P)

-= JV k'

4kxo +4yo

k2

-4kxo+

y r

a

< =>

k 2xo =

0

< =>

k = 2xo

3 3

Khid

o S„

,„

= i8

(y

x2 )2 = l(

o-yo

-x 2)

55l| Cho (P) : y^ = 2x va di/dng thin

g D : x

Tin

h die

n tic

h hin

h phan

g gid

i han boi (D) v

a P

DH Kink

te Quoc dan Ha

Noi

1997

Gidi

Taco: (-2)

ll = 2.1.2

c:> B'^

P = 2AC

ai A{2

; 2

)

73 V4

- d

x

Da

t X =

2sint =

> d

x = 2costdt

Doi ca

n :

X = V3

(4t + 2sin2t)

3

t =

0

3 47I +

3V3

^ ^ 2V3

4 7: +

3V3

4 7: +

V3

Do d

o S = + = (dvdt)

I5 54 I

Trong ma

t phin

g Oxy, tin

h die

n tic

h hin

h phin

g D gidi ha

n bd

i ca

c

dudng y = xe"

, y = 0, x = -

1, x = 2

Hoc vien BiCu chinh

a ham

so do

n die

u tan

g tre

n [-1

; 2] v

a y (0) =

0

= e^

Cx-l)

dx = xe" d

x xe" d

2

0 ( 2

: y

^ = 2x v

a (C2 ) :

27y^ = 8(x -

\

Gidi

Phuong trin

) va

( C2 ) :

54x =

8(x -\f

« 8x

^ 24x^ - aOx -

8 =

0

o (

x 4){2x + 1)^ =

0

x =

4

1 y'

X =

— loa

i v

i X = — >

Vay giao diem cua (Ci) va ( C 2 ) la : A(4; 2A/2), B(4; -2^2)

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 AMN va dien tich hinh chin tren bdi (D) va phia dudi bdi (P)

DH Kinh te Quoc dan Ha Ngi - 1996

Gidi

Goi A (a; pa^) e P

S = X

M + X

N = ^ =

2a

P X jj j^

Do do MN^

= (

XM

- XN)

^ + [ (2 pa xM

+ b ) -

( 2p ax

N +

b)]^

=

^ MN' =

(X

M - XN) ' + 4pV(x

M XN)^

-4P)(1 + 4pV) =

A

2

4b 4a + —

^a2

+ ^.V

l +

4 p2 a2

Vay

SAMN =

^M N.

d(

A, D) = Ja^

2

1 P

Vl+

4 p2

+ b ja^

+ = S

x +

b px^

)dx = pax^

^

: (X

N XM)

-:V S2

4P

-pa(xM + X

N)

+ b-|

(x

^ +x

2a2p + b -

4a + —

= 2

2 "

a + -

(2 2

S 2

= Si 3

-B THE TfCH VAT THE TRON XOAY

2 Cho h i n h p h i n g gidi h a n bdi

cac dirdng x = g(y), y = c, y = d,

x = 0, quay xung quanh Oy, tao

t h a n h vat the t r o n xoay t h i :

pd

g(y) dy

55?! Cho h i n h p h i n g (D) gidi h a n bdi cac dudng y = (x - 2)^ va y = 4

T i n h the tich ciia vat the t r o n xoay sinh ra bdi h i n h (D) k h i no quay xung quanh :

1 True Ox 2 True Oy

2 -V

^f ld

y = 87xJ

^'V^dy

=8 7t

g thin

+ x

^

Ta

CO : f '(x) = -

, C O

MXD : R

2x

(l + x

^)

^

X -ao

1

+ x

^ = -

1

, ca

c dudn

g thin

2 1 dy = 7iy

1

2 _ 7t

Do do the t i c h can t i m la V = V i + Va = 7iln2 (dvtt)

Goi (H) 1^ m i e n k i n gidi han 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 tich 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 diem ciia (L) va Ox

Goi D \k m i e n gidi h a n bdi (?) : y = 2x -x^ va true hoanh T i n h the

tich eua vat the V do ta quay xung quanh :

1 True Ox 2 True Oy

2

0

< X <

2 th

i y = 2

x x^ c=

-> x

^ 2x +

-1 -1

- y

, fx

= 1 + J

l

y vd i

yf d y+

f

(

l +^

yf d

2-

l-y)

2

dy +

7t

f

2

y+

2 (l -y

.-2

+ n

2y -^

(l -y

h V ciia va

t th

e dug

c ta

o ra k

hi qu ay h in

h gid

v

a ducfn

g th in

g x = 2 q ua nh t ru

e Ox

Gidi

Ta

CO :

-= -(

1)

x-^ 4

= (d vt t) 4 1

-562

Ti nh t he t ic

h kh

oi tr on x oa

y ducf

c ta

o th an

h kh

i qu ay q ua

g gi

di ha

n bd

i ca

c du 6n

g y = 0 , y

= -^xsinx

+

cos^

x, x = 0

, x

= 2

DH Bach khoa TP.HCM

- 1993

Gidi

Ta co : V

= n^^^fixjfdx

= n

2

(x si nx + co

s x)

dx

0

2 X sin X dx + 7t [2 cos^ x dx

* Tinh I2 = f2cos^ xdx = fx 1 + cos 2x

Vay V = Tcdi + I2) = - (4 + 71) ( d v d t )

4 563j Cho mien phing (D) gidi han bcfi cac duong :

y = tan'x, y = 0, x = : , x =

-4 -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

2. V

(D , =

n

V,D

) =

„(

-tan

'' X

4

)M

x + Tt

x (tan

^ x + 1) + (tan^

x + 1) -

1

\^ x)d

x

* (tan'' x

- tan

^ x + 1) d(ta

n x) -

TI

^ (tan'' x

- tan

^ x + 1

-7 i-

=

—(

dvtt

564| Tin

• 2

x = —

2

7 1 7

t X

V = —

si n2

X

dx = — x'

• n

sin2xd

x = —(dvtt)

h kh

oi tron xoay dua

Gidi

Ta

C O

: V =

TI r2

(h ix

n x => d

u = —

X

dv = I

n x.d

x =

> v = x(ln

x 1)

-V =

7t r2 (l nx rd

x = 7r X.

I

n x(l

n x

- 1)

V = 27iln2(

ln2-

l) -7 i

(lnx-l)-x]

1

= 27tln2(ln2-l

7t

)-[(

21n2-4) + 2]

(ln2) 21n

2 + l] = 2jx(ln2-l

)2

(dvtt)

X

=> du

j = —

-2)(

dvtt)

568| Trong mp Oxy, ch

o hin

h phan

g D gidi ha

h th

e tic

h kho

i tron xoay ta

o ne

n kh

i qua

y D

quanh

true Ox

BHKinh te TP.HCM -2001

Gidi

Ta

CO

Inx =

0

o x =

x

Dat

2

, 2 I

X

dx

e 2(x In

= 7t

(e 2) (dvtt)

-569J

Cho (P) :

y = x^ (

x >

0).

Gpi D la hin

g d c6 phuon

g trin

h y

- -3

x + 10,

y =

1.

Tin

i qua

y quan

h tru

e Ox

Gidi

Goi Vi

la the tic

h va

t th

e tro

n xoa

y d

o ABM

h va

t th

e tro

n xoa

y d

o NMC

h tru

do hin

h chii

f nha

gidi han bdi y = xe", x = 1, y = 0, vdi 0 < x < 1

bdi y = — va y = x^ k h i h i n h phang quay quanh Ox

)72| Ch

o hin

h phSn

g H gidi ha

n bcf

i ca

c dudn

g y = xlnx, y = 0, x

^ xd

x

Dat u = In^

x =

> d

u = 21nxdx

dv = x^dx

=> V =

Vay V =

n xd

x

Ta tin

h I = x^Inxdx Ji

dx x — , u = > d x = = In Dat u

dv = x^dx =

x -

-3 J

i x^dx = —In

x

-i x3

Tc

^-A(

2e 3.1

Tt(5e3

2)

27 (dvtt)

' x + sin'

* x , x = — , x = TI

e Ox

Gidi

Ta

CO

: V =

7t

^ (cos

* X + sin'' x)d

b) T i n h the t i c h vat the t r o n xoay duoc s i n h r a k h i D quay quanh Ox

DH Kien true Ha Ngi - 1998

Gidi

a) Dien t i c h h i n h ph^ng D

/•O - 4

575| T i n h the t i c h kho'i t r o n xoay tao t h a n h bdi h i n h t r o n

x^ + (y - b)^ = a^ (0 < a < b) quay quanh Ox

DH Kien true TP.HCM - 1991

Gidi

Vay phucfng t r i n h nijfa dudng t r o n A m B l a y i = b + Va ^ - x^

phuong t r i n h nufa du6ng t r o n A n B l a y 2 = b - V a ^ - x^

293

Do d

o V = T:

bV a2

h hin

h gid

i han

di (E) : — +i

Gidi

Taco: 4^

4 = ^ «

T rf '

^u^-^^)dx

= ^ f

^a

x2

^-)d x

h v

at th

e tro

n xoa

y sin

h r

a b

di hin

h eli

p + ^ <

1

quay quan

h Oy

DH Xdy dang -

1998

T DA NG THLfC

T RO NG T iC

H PH

AN

KIEN THLfC C

d BA

N

1.

Neu

f, g la ham

so lie

n tu

c tre

n doa

n [a

; b] v

a f(x) <

2 Ne

u f lien tuc tren [a; b] t

hi g(x).dx

a m < f(x) <

M V

x e

[a; b] t

hi

m(b

- a) <

(A)

(B)

Tron

g thiJ

c han

h t

a ca

n nh

d :

a) Muó

n chuTn

g m in h mot ba

t d in

g thuf c ciia tic

h p ha

n tic

h p ha n

da gio

'ng n ha

i, ta chi ca

y r a

giOa ha

i ham dudti da

u tic

h phan

nhau r

oi la

m nhu

trUdng hcfp tr en

b)

Trong truorng ho

h

bi cha

n cu

a ha

m na

y (tuf

c l

a ti

m Min, Ma

h cha

t (A), (B

) d

phan tren

578|

C hil ng m in

h rk ng : — < 2

2e='"

\dx

(do e < e < e

2ế"

\dx

< 2e

.dx

Suy r

a — < 2

Tie < dx \ 2e='"

58l| Chiing

min

h rk

ng : — <

dx

0 5 + 3cos

^ X

10

Xet ha

m s

o f(x) =

e

Ta CO :

dx

16 J

o 5 + 3cos

^

10

0 5 + 3cos^

x 1

0

Chufng min

m f(x) = x

- 4

x + 1; f '(x) =

0

X = 1,

+ 0 -

l-x) 2d x<

^

27

5831 Chufn

g min

h r^n

g : — <

6

dx 71

A/2

Gidi

Ta CO :

2X2 < 4

- x

^ x'^ <

4 x^

Ta xet T(x) = xcosx - sinx

T'(x) = cosx - xsinx - cosx = - x s i n x < 0 Vx e

dx <

n 3

thi 4

aVs

s in XQ

Zl vn

= dx

<

dx

0 2

dx

Hay:

Pha da

u b^n g

Xo =

—

th

i 1 <

^1-4"

A/I

0 <

x <

-= c [0; 1 ] =

> 0 < x

Chufngminh: 54V

dx

< 108

Gidi

Ta x

et f(x) =

t dan

g thiJ

c B C S ,

ta

c6

|f(x)| <

Vd^

x)

<6 (1)

Ta la

18

Vay f(x) >

Vl8 = 3 , /2 (2)

:

3 >/

2

< f(x) <

6

Su yr a:

3V2(ll +

7 )<

11

f(x).dx

< 6(1

1 + 7) = 108

Ghi chii : T

a c

6 th

e tim

in, Ma

x cu

a f{x) = Vx

11

1 1

Vll-x

- V

x +

2Vx + 7 2 V1 I-

X 2V

X + 7 V 1I -X ' f' (x

ih

f ^

Vl-x

^ co

s e

" + V3 + x

Gidi

Do ba

t din

g thuf

c B C S , t

x2 )

sin e" <

2

-2 f^

dx

<

f\

Vl-x

^ cose

" +73

^ (

Vl x^

cos e

" +

sTe^

sin e" )d

x <

2

(V

l x" co

-s e

" + V3 + e^sine

" )d

x <

2

|592| Chiin

g min

h rin

x

DH Tdy Nguyen -

2000

Vi

1

< X <

h ran

g : J

- >

V6 Jo

^ 1 r

-x =

f ^

1 + cos 2t

n ta c6 x

Ne

n 1-x

^ dx

0

V1 + x''

Ap dung ba

t dSn

g thiJ

c Bunhiacopski, t

a c

6 :

n rl

^)dx

1 — d

x <

- (dpcm)

1 x2

1 1

l<

(n-l)]

n (n - 1

dt <

(e" 1)

<(e^ -1)

1 1 e I 2

^'-}

(3)

T Cr (2)

va (3) su

^ (e"

- 1)

2J

(dpcm)

Ta CO : f [k^ + (4 - 4k)x + 4x^].dx = [k^x + 2(1 - + x" ] ^

= k^ + 6(1 - k) + 15 Vay theo d§ bai ta c6 : k^ - 6k + 9 < 0 <=> (k - 3)^ < 0 o k = 3

307

ChiJng min

h ran

Gidi

Xet

ham

so fix) =

Ta

c6

[f(x)]

2 = VcOS^

X + COS

7 1

cos

X

+ cos

X +

1

va da

t t = cosx DK

: 1

G'(t)

+ 0 -

^

^ 22

^

2 Vcos

^ x + co

s X + 1 ^

•''2A /3 0

04|

Chiing min

h : — <

e

^''-'''d

x <

21/^

BH Nong nghiep I

x x^

-vd

i x

e [0

; 2 ]

f '(x) =

1 2x,

f '(x) =

0

<=

> x = -

2

t f(x) = sin

X

2 1 ne

X

< f(x) Vx

-sinx

f(x

f (x)d

x =

4

f (x)d

x +

2

f (x)d

x >

4

f (x)d

X , d 4

x + X 7 1 - 0

•xsi

X

dx

SO TJ

Cho

X >

0 Chufn

g min

h rang v

di mo

i so' t

if nhie

n n > 1, t

a c

6 :

x2 X

^ X"

e

>

l + x+ — + — +

h hKng

phuon

g pha

p truy h

oi

* Vd

i n = 1 : Ta

CO

e'' >

1 Vy tho

a 0 < y < x

y ba

t ding thiJc diing v6i n = k, tiJ

^ >

1 +

y + — + + — vd

i 0 < y

x^ x

^ x'^^

e > l + x + — + — + +

2! 3! (k + D ! Vay bat dSng thdc diing vdri n = k + 1

Theo nguyen l i quy nap ta ket luan bat dang thufc dung Vn e N

Gidi

2 ế" "dx

311

Ddi ca

n

Nen Vay

, 1

• —

2 COS + "dx ^ Sm g 2 x = ế" M

X J

^

0

TCr (2) v

a (4) su

y r

a :

Tir (3) v

a (5) t

dx

= ln(

x + 1)|

a c

6 :

\2

1 =:

f

f

l ^ 2

ti l

e tre

n [0

; 1] ne

n tron

g ba

t

ding thuTc (2) c6 dau < nghiem nguyen)

= xe" |cos nx| dx < xe"

612] Chufn

g m in

h r^

ng : 27tV

7 <

j j"

V^2 + s in x)(6

- si

x

Do 0

<x

<2 Tt =>

- si nx ) = (2 + t)(

6 t) = -t^

+ 4

t + 12 V

t e [-1; 1 ]

f '(t ) = - 2t +

4, f '(t) =

0 =

> t = 2

t -«

15

=> A/7

< V(

2 + sin x)(6 - sin x) <

A/15

V7 dx

<

=> 27tV7 <

f^" V(

2 + s in x)(6

- si

n x)d

x <

V l5 2

Da

u kh on

g x ay r

a

vi X

Q =

n e [0; 27t]

, th

i :

7 <

(2 + sinx)(

6 s in x) =

12 <

15

{•2K

V(2 + s in x) t6 -s in x) dx < V l5

r2:t

dx

Do do 27IA/7 <

V(2 + s in x)(6

- si

n x)d

x <

2 nV l5

613| Chufng min

X

Gidi

Ta CO : sin''

x + cos''x =

1 2sin^x.cos'^x <

1

sin'* x + CDs'

x + cos''x >

2sin"xcos^x (cau 2)

< => 2(sin^x

+ cos''x ) >

sin^x + cos''

x + 2sin^xcos^x

o 2(sin^

x + cos''x) >

(sin'^x + cos^x)^

= 1 <=>

2 >

1 sin''

X

< 2

Xet h a m so f ( x ) = f l E i i day l a h a m so giam t r e n (0; 1]

Do d

o V

x e [0; 1 ] t

hi sin

x <

x

Vay I„

= sin°xdx

< f x"dx

+ 1

T Cr (1) v

a (2 ) : —

^ sin"^

^ 1 < L < ^

(sin 1 )

n +

1

< L <

n +

1

(2)

IG IS I

a) Chiln

g min

h :

x ^ <

x sinx

-Ta C

O

f '(x ) =

1 cosx >

0 V

x e

R

Vay fix ) l

x sinx >

1 + —

2

g"(x) = -sinx +

=> g'(x ) =

C OS

X

1 + — l

x sinx >

Tir (l) , (2

1 1

f

1^

f '(x ) = —

— + cotx

si n'

X X 2x

-(2x + sin2x) , g'(x

) = -(2 + 2cos2x ) <

0 V

x

Vay g(x ) l

x >

0 th

i g(x ) <

g(0

) =

0

Do do : f ' (X ) <

* xd

x

0 cos 2

x

Chijfng minh : ta

DH Quoc gia TP.HCM

- 1997

Dat u = tan

1 + tan^xjdx

=>

dx =

u = tan

t

u =

0

du u^+l

4

du

I(t) = tan

t U'' +1 j-trnit

u ^d

u f t

an

t ( u"

*

1) +

1

du

^ x)(

l + cos'* x)

2sin

3-^ x.cos

^ X

3

(1 + sin'* x)(

l + cos"* x) (

1 + sin^ x)(

l + cos* x)

3 si

n 2

,l + sin

*x

l + cos''xy

(1 + sin''x)(l + CDs'

(do x

e

1 1

1 + sin'' x

1 + cos*

x)(

l + sin" x)

2

sin 2x

X

1 + cos"

7 1 — > ) 6 sin" x l + x)( cos" (1 +

Dau "

="

khon

2

, 4

^ U i 4

sinx.cosxd

x ^

Jo

(1 + sin" x)(

l + cos" x) 1

2

20| Chiin

g min

h rkn

g f(x), g(x) l

a h

ai ha

m s6' lie

n tu

c tre

(x)f dx.JJg(x)f dx

Hoc vien Quan y

- 1995

Giai

Ta C O

: [X

f (x) - g(x)f >

0 V?

e

R ^ ?.¥(x) -

2Xf(x).g(x) + g'(x) >

0

Gidi

Ap d un

g ba

t d^

ng thiJc Bu nh ia co ps

ki ch

o ha

i ha

m f(x ) =

1 t re

n [0

f

I

Vl -f

^(

x) dx < ,1

-f2

(x )d

x

vi Id

g ba

t d^

ng thufc Bu nh ia co ps

ki ch

o ha

i ha

m F(x

1 t re

n [0

i

r

^d

x = x|), =

\

1-rl ^ ' 1

x > J 0 V • J 0

Vl -f

(x )d x<

o y = f(x ) c

6 da

o ha

m va dao ha

m cu

a n

o li en t

uc tr en [a;

0 va

M = m ax

f '(x ) Chiimg

m in

h rk ng :

xe [a ,b

l

(b -a

0, Vx

e [a

; b]

th

i f '(

Ta

CO f (x )

f '(t )d

t f'(

t)

dt

Ma |f '(t)

! <

M Vx

e [a

; b]