пособие 7 по матану (2 курс)

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978 5-91879-125-7

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¯¥ç

ª 䥤à ⥠ਨ äã-ªæ¨©

¯à ¡«¨ ¥-¨©

‘ à ⮢᪮£® £®áã-¨¢¥àáâ¨:â¥â

“„Š 517.262(075.3 )

ISBN

••Š 22.161ï722

978-5-91879-125-7

1.1. •¥®¡å®¤¨¬ë¥§ ⥮

¥â¨ç¥áª¨¥ ᢥ¤¥-¨ï.

3

b

• Ǒãáâì ¨¬¥¥âáï ¨-â¥£à « Za

f (x, y)dx. Ǒ®¢â®à-ë¬ ¨-â¥£à «®¬ - §ë¢ ¥âáï

d

b

f x, y dx!dy,

d

dy Za

b

Zc

ÃZa

®¡¨©£®-§¨-â¥-çâ£à¥£âáï«à 1« .

Zc

f

(

x, y

dx.

)

1.2Ǒਬ. Ǒਬ¥à¥(1à.1¢ëç¨á«.) ‚ëç¨á«¨âì¥-¨ï ¯®¢â®àª â®àë©--ë©

0

2−x

1

Z

1

Z

¯

1

•¥è¥-¨¥. ¢ëç¨á«‘-

¢ëç¨á« ¬

-ãâ ¥-- ¨-â¥£à «,d¯xà¨-¨¬(x +ïy)dy.

x

2

âã.१ã«ì‡1 ⥬

¨¬¥-¨ï-¥-èãâà- ©¥--¨-¥â£¥®£à¨-«,⥣थ«¯®¤ë:

-â¥£à «ì-®© äãx-ªæ¨§

¥®©-¡ã¤áâ ¥--â

−x

01

1

x55

x44

1

x3

1

31

0

Ã

2

−

x

01

2

¯

2

−

x

= dx1 (2 (x + y)dy =

(x + y)dy!dx =

µ

x · y +

y2

dx =

2

Z

Z

Z

Z

Z

¯

x

x

01

2 · x4

¯

0

2

¯

2

1

π

y

¶dx =

Z

x

− x − x2) + 2(4 − 4x + x2 −=x4)dx = Z µ−

− x3 − 2 · x

1.31 ‚ëç¨á«¨âì. ‡ ¤ -¨ï¯®¢â®à¤«ï á-묮áâ®ï⥠¨-⥣५ì«ë:-®£®µ−

·

−

−

·

3

¶¯x=0

−

à¥è2 ¥-¨ï.

2

¯

= 60

¯

¯

π4

π2

dy Z0 2y2exy dx

1.1. Z0

dx Z0

x · sin ydy

1.2. Z0

dy Zπ2

os(x − y)dx

1.3. Z0

01

−

0∞ 0

1e

x

1−y

Z

Z

Z

Z

Z

Z

1.4.

1dx 1

x + y

dy

1.5.

dy

3x +2 ydx

1.6.

dy

1 ex4−y dx

y

x

−∞

4

1

2

1

2 [1

4x

8

2 [1

y−

−∞

1+ os

∞

0π

x

π2

[1Žâ¢.7. Z

dy Z

y2 + x2y2

. . Z

dx

Z

y

xdy

. . Z

dx

Z

y dy

¥âë: 0

dx

1 8

sin

1 9

.1 1 [1.2 √

.3 e − 5 [1.4 2 (e − 1) [1.5 3 [1.6 1 [1.7 π

.8 3 [1.9 10π − 75

§2. „‚Ž‰•›… ˆ•’…ƒ•€‹›

«¥ • Ž¯à饥¥¢¤¥®¡««¥-¨á⨥.

Ǒãáâ쮯।¥-«¥-¯«®áª®á⨨ï äã-ªæ¨¨OXY § ¤ -® ¢ë¡-®¥à¥¥á⢮¬ D 楫 ª®¬

®¡®§- 稬 ¯«®é ¤ì

f (x, y). • §®¡ì

Dâ®çªã-

á⨠Ti,

‘®áâ ¢¨¬ á㬬ã

i-®© ç á⨠µTi ¨ -

¤®© ç áâ¨

(xi, yi).

X

…᫨ ã í⮩ áã¬¬ë ¯à¨

S =

f (xi, yi) µTi.

i

ªæ¨¨-0 áãé®â ¢ë¡®à¥áâ¢ã¥ââ®ç-¥¥ª,ç-â®ë©íâ¯à⥤¯à¥«,¥¤¥ª®â®àë©« - §ë-

¢-¥¥§âáá¨â¤¢®©--¨ë¬®â¨á¯®á®¡-â¥£à «®¬maxà §¡¨(®âµT¥äã-i)¨ï,→-

2.21..11.. ‚ëç¨á«•¥®¡å®¤¨¬ë¥-¨¥ ¤¢®©â¥ ¥®à-ëå¥ ZDZ f

f ¯® ¬-® ¥áâ¢ã D ¨ ®¡®§- ç ¥âáï

x, y

dx dy.

â¨ç-᪨â¥(£à¥ á¢)«®¢¥¤¥á¢-¨ï.¥¤¥-¨¥¬ ¨å ª ¯®¢â®à-ë¬

¥¥ •¬®•ã¤-¥¬¯à-¥¤áâ§ë¢ ¢¨âì1¢),®¡«

--¥à

D

¢¨¤áâì¥

¥£¢à¨à®¢¥-áâ¢:-¨ï

®¡« áâìî ¯¥à¢®£® ⨯ , ¥á«¨

(ᬠà¨áã-ª¨ 1 , 1¡,

£

D = {a ≤ x ≤ b, ϕ(x) ≤ y ≤ ψ(x)}

à¨á. 1

ϕ(x) ¨à¨áψ(x.)1¡{-¥¯à¥àë¢-ë¥ äã-ªæ¨¨à¨á. . 1¢

f (x, y)dxdy =

Z

b à ψ(x)f (x, y)dy!dx =

Z

b dx

ψ(x)f (x, y)dy.

Z Z

Z

Z

D

a

ϕ

x

a

ϕ

x

(

)

(

)

¥¥ •¬® -® ¯à¥¤áâ ¢¨âì2¢), ¢¨¤¥ -¥à ¢¥-áâ¢:

D

(á¬. à¨áã-ª¨ 2 , 2¡,

£

D = {c ≤ y ≤ d, ϕ(y) ≤ x ≤ ψ(y)}

à¨á. 2

ϕ(y) ¨ à¨áψ(y.)2¡{ -¥¯à¥àë¢-ë¥ äã-ªæ¨¨à¨á. . 2¢

d ψ(y)

d ψ(y)

⨯

- §ë¢( îâáï) í«¥=¬¥-â à-묨(®¡«) áâﬨ ¨-â¥

£Ž¡«à¨à®¢áâ¨-¯¨ï.¥à¢®f (£x,® ¨y)¢â®à®dxdy =£®

Ã

(

f x, y dx!dy

dy

f x, y dx.

Z Z

Z

)

Z

( )

Z

Z

D

c

ϕ y

c

ϕ y

áã-•®ª…᫨3), ®¡« áâì D -¥ ï¥âáï ®¡« áâìî ¯¥à¢®£® ¨«¨ ¢â®à®£® ⨯

(á¬. à¨-

à¨á. 3

â® á«¥¤ã¥â à §¡¨âì ®¡« áâì D ¯àï¬ë¬¨ x = constáâ¨(¨«¨) y = const -

®¡« áâ¨

D’®1£, D2, D3..., ª ¤ ï ¨§ ª®â®àëå ¨¬¥¥â ¢¨¤ ®¡«

¯¥à¢®£® ¨«¨ ¢â®à®£® ⨯ .

Z Z

n

f (x, y)dxdy = i=1 Z Z

f (x, y)dxdy.

D

X Di

2

ˆ§®¡à §

£ªà ªä¨ç®¬ã¥áª¨¨§®¡«âà¥åáâ좨¤

-

- ¡«. áâì - ¥£à¨à®¢ -¨ï.

31).

…᫨ ®¡« áâì -

-¨ï

á¨âáï ª

⨯ã, ®:

§ a ≤ xâì≤£bà. -¨æë ¨§¬¥- -¨ï

- © £à -¨æë ®¡« áâ¨

y. „«ï íâ® -ã - § ¯¨á âì ãà ¢-¥-¨¥ -¨

¥ å

D

£® y: y = ϕ(x). €- «®£¨ç-® ¤«ï

஢

¥© £à -¨æë

¢ëà §¨âì ¨§

¯®

y = ψ(x). ϕ(x) ψ(x) { -¨ -¨© ¨ ¢¥àå- © ¯à¥¤¥«ë

-…§á«¨¯¨á®¡«ìy áâì£à®®â¢-¨¨æë-¥ââá⢥£¨§¬¥--¥-.-¨ï ®â-®á¨âá⥨ï),ª£à¨à®¢¢â®à®¬ãâ®çª¨¯ã, â®:

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£¨ç-¥â¬¥£

y

- § c¯¨á≤ y âì≤ d£.à -¨æë

§¬¥-

¨ï

¥

¢®©

-¨æë ®¡« áâ¨â¥£à¨à-

x-.¨ï„«ï¢ëàí⮣§¨âì® -ã ¨§- -¥§£®¯¨á âì ãà ¢-¯à-¥¨¤¥¥«ë

-

¤«ï ¯à ¢®© £à -¨æë

x: x = ϕ(y). €- ®-

бв¨бв¢-б¡«¥¥-ª,--=¨п¨пбвмзв®¡л. «(ббв¤¢®©-¨¥з)-.¥п¢«пва¢¨вмª¥¥-(£§леа¨а®¢)¥¯®¢в®а¤¯авб¨ï¥-(¤¨§í«â-¥)¥-«ë¥¨ïç{£¬à¥©-áâ-¨¯àï¬ë¬¨,¨(¨«¨«®¢-¥â©¥-£-¡ë«¨©.à¨à®¢®©,á㬬ãâ®:¢¯¥¥àåà¢-¯®¢â®à¨ï-¨©««£.® ¥¨«¨«ì-ëå)-묨¢â®-

2¢ëç¨á«¨âìà®®áï¬-.1£4)-….3®á«¨-.⨯.ª‡Ǒਬ®à¤¨¡å®¡«¯¨á.¤¨¬®„«ï¥-£-áâ쮥¨ï.,ª¤¢®©¨á®®â¢àë-à-¢ëç¨á«§¡¨â줮©-¥

x

ψ y ϕ y

ψ y

Ǒਬ¥à 2.1.1. ‚ëç¨á«¨âìä¨ç¥-᪨⥣஡«ZDZ

xydxdy⥣à¨à®¢,¤¥ D: y = x + 3, y = 0,

x•¥=è−¥-2,¨¥x. 1)=. .ˆ§®¡à §¨¬ £à

áâì ¨-

-¨ï (à¨á. 4).

¨ â3)¥£.à¨à®¢‘- ç -«¨ï)§(ᯨá뢬®¥ «¥¥¢®¬¥¯à§-¥¤ç¥¥«ë-¨¥)-⥣à ஢ - ï ¯® x. •¨ -¨©

¤¥«

§- ç¥- ¥)

x = −2. ‚¥àå-¨© ¯à¥¤¥« (á ¬®¥ ¯à ¢®¥

“à

-¨¥ ¢ àå-¥© £à - æë ¨-⥣yà¨à®¢= 0 (íâ®-¨ï-¨ -¨© ¯à¥¤¥« ¨-⥣à¨à®¢ - .

4)

‡ ¯¨è¥¬ ¤¢®©.-®©

-â¥£à « ç¥à¥§ ¯®¢â®ày-ë©= x¯®+ä®à¬ã«3 (íâ® ¢¥¥àå

¥

¨ ¢ëç¨á«¨¬ ¥f£®:(x, y)dxdy =

Z

b à ψ(x)f (x, y)dy!dx =

b dx

ψ x)f (x, y)dy

Z Z

2

Z

Z

Z

D

x+3

a ϕ x 2

a

2ϕ x

( )

2

(

)

Z Z

xydxdy = Z2=dx21 Z2 xydy = Z2

xµ y2

x+3

Z2

x((x + 3)2 − 0)dx =

¶¯y=0dx = 2

0

3

2

¯

x44

1

3 + 9x22

2

D

−

−

¯

−

2

¯

1

+ 6x3

‚ë Z2

x

6¨x-â¥+£à9x«)dx 2

µ

¶¯x=

−

2

.

Ǒਬ¥à 2.1.2.

ç¨á«¨âì(+

=

¯

= 16

−

¯

á ¢¥à

¬¨ ¢ â®çª å

ZDZ

3xy + yâ¥dxdy£à¨à®¢, ¤¥ D { âà¥ã£®«ì-¨ª

• è -¨¥. 1). ˆ§®¡à A§¨¬(−1£,à0)ä¨ç, B(0¥,᪨1), C®¡«(1, 0)áâì. ¨-

-¨ï (à¨á. 5).

à . 5

Ž¡« áâì

-

¢â®à®¬ã ⨯ã.

-¨ï ®â-®á¨âáï ª

y = 1, ª¨¬ ®¡à §®¬ 0 ≤ y ≤ 1. ‡ ¯¨è¥¬ ¯à¥¤ «ë ¨§¬¥-¥

1.

-¨¢--¨©- ¯à« ¢®© - -¨æë:

=-¨ï+¯®1, ¢ëà §¨¬ ¨§ ãà ¢- -¨ï

:

=

y −

x•â®. “à

y

x

x

x

£à -¨æë:

x. ’¥¯¥àì

¬ ãà ¢â-¥£-à¨à®¢¨¥ ¯à ¢®©

¯®

y = 1−x, ¢ëà §¨¬ x : x = 1−y. •â® ¢¥àå-¨© ¯à ¤¥« -

-¨ï

x.

2

−

2

x

2

¯

1

¨ ¢ëç¨á«¨¬ ¥f£®:(x, y)dxdy =

d dy

dà ψ(y)f (x, y)dx!dy =

ψ(y)f (x, y)dx

Z Z

1

Z

Z

01

Z

Z

D

c

ϕ y

c ϕ y

44 + 3

1

1

( )

22 +

( )

0

1−y

3

1−y

Z Z

xy + y dxdy = Z

dy Z1

3xy + y dx

Z µ

y

y x¶¯x=y

−

1dy =

D

y

=

¯

= 1 3

¯

•¥è¥-¨¥. 1). ˆ§®¡ày = x2§

¬x2£à+ y2 = 2,

®¡««¥ áâìé ï¨-¢â¥¢£¥à¨à®¢å-¥© ¯-®¨ï«ã(á¬.¯«®áªà¨á®áâ¨.6). .

0

23

0

Z

y(1 − 2y + y2 − y2 + 2y − 1) + y2

(1 − y − y + 1)=dy2 = 2 Z −y3 + y2dy =

µ−y

3

¶¯0=

Ǒਬ¥à 2.1.3. ‚ëç¨á«¨âì ä¨ç-⥥£áª¨à «

y

6

¯

ZDZ

¯

ç¥-- ï ªà¨¢ë¬¨

2xydxdy, £¤¥ D {

¡« áâì, ®£à -¨-

à¨á. 6

•â

áâì

-

-¨ï { ®¡« áâì ¯ ࢮ£® ⨯ .

¯¨è£à â¥-¥¬¨ç¨¢£à¨à®¢-îé¨å¨æ먧¬®¡«¥-áâì¥-¨ï¨-â.¥„«ï£à¨à®¢íâ®-£®¨ï:- ©¤¥¬ â®çª¨ ¯¥à¥-

á¥ç32)¥-.¨ï‘-ªà¨¢ëå,管« § ®

x

ª®áâ¨,’ª ⮪â®çª¨-¯®¬

y = x2,

,

y2 + y

−

y

1,

½ x

2

y

2

2 = 0

½ y

− .

=¢ 2¢¥àå-¥© ¯®«ã¯«®á

¯®¤åãá«®¢¨î¤¨â+

®¡«ª®à= 2áâì¥- ¨-⥣à¨à®¢ -¨ï «¥ ¨â

«ã稫¨

¯¥à¥á¥ç¥-¨ï:

y = 1.

’®£¤ x = ±1.

’ ª¨¬ ®¡à §®¬, -

x : −1 ≤ x ≤ 1.

A(−1, 1), B(1, 1), â.¥. ¯à¥¤¥«ë ¨-⥣à¨à®¢ -¨ï ¯®

1

√2

1

2

1

¨ç¨¢ î騩 ®¡« áâì ¨-⥣à¨à®¢ -¨ï á y¨§ã:

4

1

3

5

¯

2

6

x4

x6

¨-⥣à¨à®¢ -¨ï. “à ¢-¥-¨¥ ª ¨¢®© ¢ àå-¥© ⥣y-à =æë:x2.

•â® -¨ -¨© ¯à¥¤¥«

-¥£®

x2

+ y2

= 2, ¢ëà §¨¬ ¨§

4). ‡: ¯¨è= √¥¬2

¤¢®©2-. ®©•â®¨-¢â¥¥àå£à-¨©« 篥।§¥¯®¢â®à« - -멨஢¨ ¢ëç¨á«¨¬-¨ï ¯® . ¥£®:

y y

− x

y

−

x

2

√

−x

2

x(2−x2 −x4)dx =

2xydxdy =

1

dx

2xydxdy= 1

=

1

2

¯y

dx =

1

2

2x · y

x2

Z Z

Z

Z

Z

Z

•¥è¥-¨¥. 1). ˆ§®¡àx§¨¬+ y®¡«= 4áâì¨ ¨-⥣à¨à®¢îé-¨ïïáï£à¢ «ä¨ç¥¢®©¥áª¨¯®«ã¯«®áª(á¬. à¨á®áâ¨.7).

D

x

¯

−

Z1

−

¯

µx −

−

¶¯x=−1

Ǒਬ¥à

x − x − x dx

−

.

2.1.4. ‚ëç¨á«¨âì ¨2-⥣ᯮ«à

=

¯

= 0

2 2

−

¯

®£à -¨ç¥-- ï ªà¨¢ ©

ZDZ (x − yx)dxdy, £¤¥ D { ®¡« áâì,

¢¥àå-¥© â®çª¨, £ y = −

-¨ -¨©¯à¥¤¥« ¨-⥣à¨à®¢ -¨ï) ¯® y), ¤® á §®¬©

−2•≤ ©¤y ≤¥¬2.

y = 2 (íâ®

å-

¨-

-

. ’ ª¨¬ ®¡à

¨-⥣à¨à®¢ -¨ï ¯®

«¥¢®© £à -¨æë. “à ¢ -¨¥ ®ªàã -®áâ¨:x, ¤«ï í⮣® á- ç «

- ©¤¥¬

¥

x2 + y2 = 4, ¢ëà ¥¬ ¨§ ãà ¢-¥-¨ï

¢¥àå- p

4

p

4

x

x

x : -x¨= ±

− y2. ’ ª¨¬ ®¡à §®¬ ãà ¢-¥-¨¥ «¥¢ © £à -¨æë: x = −

− y2

íâ®

¨© ¯à¥¤¥« ¨-

-¨ï ¯®

“à

- -¨¥ ¯à ¢®© £à -¨æë:

= 0,

x.

2

0

2

2

2

¯2

2

3

+ y33 + y4

1

16

Z Z