# Кратные, криволинейные и поверхностные интегралы: Учебное пособие

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• ,

2008

• 2

517.373, 514.742.4, 512.623

( ),

.

.

,

,

.

: . . , .. .

: ..-.., ...

• 3

.

,

,

.

: ( ) ,

,

, , ,

, -.

.

,

,

,

,

,

.

, .

,

,

, - .

.

• 4

I.

1. , .

D,

L. -

1 2, , ..., nS S SD D D ( 1 2, , ..., nS S SD D D

),

d1, d2, ..., dn. di

iSD . iSD i (.1).

.1.

D z = f(x, y). f(P1),

f(P2),, f(Pn)

f(Pi)Si :

1

( )n

n i ii

V f P S=

= D . (1)

1. 1

( )n

n i ii

V f P S=

= D

f(x, y) D.

• 5

. ( ( , ) 0f x y )

(1)

Si f(Pi).

2.

(1) n max 0id ,

D , Pi ,

f(x, y) D

max 0 1

( , ) lim ( )i

n

i id iD

f x y dxdy f P S =

= D . (2)

f (x,y)

D, D , -

, dxdy = dS .

1.

,

D ,

. ,

f(x, y)

, -, D, -,

max 0

lim ( ) 0,id

S st t - = (3)

, S s

.

, .

2.

: f(x, y)

D, .

• 6

, :

1. f(x, y) D, kf(x, y), k = const,

,

( , ) ( , ) .D D

kf x y dxdy k f x y dxdy= (4)

2. D f(x, y) g(x, y),

f(x, y) g(x, y),

( )( , ) ( , ) ( , ) ( , ) .D D D

f x y g x y dxdy f x y dxdy g x y dxdy = (5)

3. D f(x, y) g(x, y)

f(x, y) g(x, y) ,

( , ) ( , ) .D D

f x y dxdy g x y dxdy (6)

:

4. D D1 D2

f(x, y) D,

1 2

( , ) ( , ) ( , ) .D D D

f x y dxdy f x y dxdy f x y dxdy= + (7)

.

D :

1 2

( ) ( ) ( ) ,i i i i i iD D D

f P S f P S f P SD = D + D

D , D1 D2

. max 0id ,

(7).

5. D f(x, y)

| f(x, y) |,

• 7

( , ) | ( , ) | .D D

f x y dxdy f x y dxdy (8)

.

( ) | ( ) | ,i i i iD D

f P S f P SD D

max 0id (8).

6. ,DD

dxdy S= SD D.

, f(x, y) 1.

7. D f(x, y)

m f(x, y) M,

( , ) .D DD

mS f x y dxdy MS (9)

( ) .D i i DD

mS f P S MS D

8 ( ). f (,)

D, (0 , 0),

0 01 ( , ) ( , )D D

f x y dxdy f x yS

= , (10)

, ,

( , ) , .DD

f x y dxdy S m Mm m= (10)

, (9) SD.

( -)

.

• 8

V,

S.

f(x, y, z). V

vi , vi ,

( )i iV

f P vD , (11)

Pi vi .

V.

, vi

, .. .

3. 0r (11),

V Pi

,

f(x, y, z) V:

0

( , , ) limV

f x y z dxdydzr

= ( )i iV

f P vD (12)

1.

(,

..) , ,

, .

2.

.

3.

, ,

- .

• 9

V, ,

z = f(x, y), D

,

, .

. 2

, Si D,

f(Pi), Pi Si.

max 0iSD , ,

( , ) ,D

V f x y dxdy= (13)

, z = f(x, y),

D.

2.

D,

1 2 1 2( ), ( ) ( ( ) ( )),y x y x x xj j j j= = x = a, x = b ( a < b ), 1() 2()

• 10

[a, b]. ,

D,

: N1 N2 (.1),

. ,

. ,

, . ,

.3.

f(x, y) D.

2

1

( )

( )

( , )b x

Da x

I f x y dy dxj

j

= , (14)

f(x, y) D.

( )

, .

:

.3

2

1

( )

( )

( ) ( , ) .x

x

x f x y dyj

j

F =

b.

( ) .b

Da

I x dx= F

• 11

.

1. D, ,

D1 D2 , ,

D

D1 D2:

1 2D D DI I I= + . (15)

.

) = D D1 D2 ,

.

2

1

( )

( )

( , ) ( ) ( ) ( )b x b c b

Da x a a c

I f x y dy dx x dx x dx x dxj

j

= = F = F + F =

2

1

( )

( )

( , )c x

a x

f x y dy dxj

j

+

2

1 2

1

( )

( )

( , ) .b x

D Dc x

f x y dy dx I Ij

j

= +

) y = h D

D1 D2 (.2). M1 (a1, h) M2 (b1, h)

y = h L D.

.4.

D1

1) y = 1(x);

• 12

2) 112, y = 1*(x),

1*() = 2() 1 b1 x b, 1*() = h 1 b1;

3) x = a, x = b.

D2 y = 1*(x), = 2(), 1 b1.

: *

2 1 2

*1 1 1

( ) ( ) ( )

( ) ( ) ( )

( , ) ( , ) ( , )b x b x x

Da x a x x

I f x y dy dx f x y dy f x y dy dxj j j

j j j

= = + =

*1 2

*1 1

( ) ( )

( ) ( )

( , ) ( , ) .b x b x

a x a x

f x y dy dx f x y dy dxj j

j j

= +

:

2

*1

( )

( )

( , )b x

a x

f x y dy dxj

j

=

1 2

*1

( )

( )

( , )a x

a x

f x y dy dxj

j

+

1 2

*1 1

( )

( )

( , )b x

a x

f x y dy dxj

j

+

+2

*1 1

( )

( )

( , )b x

b x

f x y dy dxj

j

.

1*() = 2() 1 b1 x b,

. ,

ID =

*1

1

( )

( )

( , )b x

a x

f x y dy dxj

j

+

1 2

*1 1

( )

( )

( , )b x

a x

f x y dy dxj

j

,

1 2D D DI I I= + .

. D

. D

.

• 13

1. 1

, ,

:

mS 2

1

( )

( )

( , )b x

a x

f x y dy dxj

j

,MS (16)

f(x, y) D, S ,

ID = f(P)S, (17)

, D .

2.

2

1

( )

( )

( , )b x

a x

f x y dy dxj

j

=

2

1

( )

( )

( , ) .b x

a x

dx f x y dyj

j (18)

2. f(x, y)

D

,

( , )D

f x y dxdy = 2

1

( )

( )

( , )b x

a x

f x y dy dxj

j

. (19)

.

D , ,

( ) S1, S2,, Sn.

1

1 2

1...

n i

n

D S S S Si

I I I I ID D D D=

= + + + = .

(16) : 1

( ) , ( )i

n

S i i D i ii

I f P S I f P SD=

= D = D ,

, f

• 14

D, ID .

max 0iSD , (19).

1.

z = x + y ,

(0,0), (0,1)

(1,0) (.5).

. 5

= 0, b = 1, 1(x) = 0, 2(x) = 1 x.

1 1

0 0

( ) ( )x

D

x y dxdy dx x y dy-

+ = + =

1 1 22

0 0

1 (1 )( (1 )02 2

x xydx xy x x dx- - = + = - + =

1 32

0

11 1 1(1 ) .02 2 3 3

xx dx x= - = - =

3.

,

. ()

( ).

• 15

. 6 . 7

(. 6)

: (,).

, , , > 0,

.

. [0,2] [-,

],

(