Конспект лекций по математике для студентов инженерно-технических специальностей. В 3 ч. Ч. 2

n

T

S lim

v( pk ) tk

v(t)dt .

0

k 1

t 0

b

9.1.

f (x)dx

,

f x

,

,

a;b , . .

,

.

9 . 1 .

,

1,

x -

;

f (x)

x -

0;1 .

0,

n

n

k 1

k 1

n

n

k 1

k 1

,

n

0;1

,

0,

,

1.

,

, . .

9.2.

f x

a;b ,

f (x)dx

f (t)dt

f (s)ds ...

(9.3)

x,

(x) :

x

x0

const

(x)

f (x)dx

,

(9.3):

x0

x

[ ; ]

x

x0

const

(x)

f (t)dt

.

(9.4)

x0

x

[ ; ]

.

(x) ,

(x)

x,

y

f (x)

. .

y

.

B

A

(x)

9.4.

f

x

x

0

x

. 9.5

,

y f x

f

,

x

x

f ,

x x

f

0

x0

x

x x

. 9.6.

x

x

x .

f

f (x

) ,

x

x

0 ,

x

x

(x) lim

lim f (x ) f (x) ,

x

x 0

x 0

x0

x

a;b .

,

9.2,

f (t)dt

x0

x.

,

9.4

x

f (x)dx

f (t)dt C ,

(9.6)

x0

9.2.6.

,

(x)

f (x)

a;b ,

f (x)

f (x)dx F ( ) F ( ) ,

(9.7)

:

[ ; ]

f (x)

,

x

x

.

f (x)dx F (x)

F ( ) F ( ) ,

(9.8)

f ( x)d x

( F( x) C)

( F( )

C) ( F( ) C)

F ( ) F ( ) .

,

.

1

1

3

1

3

3

x

1

0

1

9 . 2 . x2 dx

x2 dx

.

3

0

3

3

0

0

3

(9.8)

,

,

.

f (x) F (x) ,

(9.7)

:

F (x)dx F ( )

F ( ) ,

1.

(

) ,

:

f (x)dx 0 .

2.

f (x) 1 ,

dx

.

def

n

n

dx lim

1 xk

xk

.

0

k 1

k 1

,

3

.

3

:

f (x)dx

f (x)dx

,

.

,

xk

xk xk 1

n

x0 x1 ... xn

def

n

n

c f ( x) dx

lim

cf (

k ) xk

c lim

f ( k ) xk c f (x)dx .

0

k 1

0

k 1

5.

[ ;

]

f1(x), f 2 (x),..., fn (x)

.

.

4 5

:

f1 (x)

f2 ( x)

[ ; ] ,

c1 f1(x) c2

f2 (x),

c1 , c2 R,

[ ; ]

f (x)dx

f (x)dx ,

f (x)dx

, , :

.

xm , . .

[ ; ]

[ ; ] [ ;

] ([ ; x1 ]

[x1 ; x2 ] ...

[xm 1 ; xm ]) ([xm ; xm 1 ] ... [xn 1 ; ]).

n

m

n

f ( k ) xk

f ( k ) xk

f ( k ) xk .

k

1

k 1

k

m

max

xk

0,

6

,

[

;

]

[

; ] [

;

].

7.

f (x) 0 , x [ ; ] ,

f ( x)dx 0 ,

.

.

f ( k )

0

xk

0,

n

def

lim

f ( k ) xk

f (x)dx 0.

0 k

1

f (x)

(x),

x [ ; ],

f (x)dx

(x),

.

.

f (x)

(x) 0,

x [ ; ],

9.

f (x)dx

f (x)

dx.

8.

f (x)

f (x)

f (x)

, x [ ; ],

f (x)

f (x)dx

f (x)

f (x)dx

f (x)

dx.

dx

dx

m( ) f (x)dx M ( ),

.

(9.9)

.

m f (x)

M , x [ ; ].

8

,

m dx

f (x)dx M dx.

2, dx

,

, m(

)

f (x)dx M (

).

10

,

f (x) 0,