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

(L)

y

b

X

. 12.1

(t),

(t), t [t1 ,t2 ],

t1 ,t2

,

(t),

z z t ,

t2

2

2

2

l

dl

xt

yt

zt

dt .

(12.2)

L

t1

b

S

ds

dxdy

[ 2 (x)

1 (x)]dx ,

(12.3)

(D )

(D )

a

y

2

x

y

1 (x),

y

x a,

x b

(

. 12.2),

D

a

b

X

,

y

. 12.2

:

(t),

t

[t1 ,t2 ],

,

x

b ,

x(t1 )

y(t)

0

t2

y(t) x'(t)dt .

S

t1

1

r 2

d .

2

Z

z

x, y ,

x, y D ,

x, y

D

( . 12.4),

0

X

q

dq

1 (

x

)2

(

y

)2

ds

(Q )

( D )

V

dxdydz .

(V )

Z

,

Q

x, y ,

OZ,

OXY

(D)

0

(D)

X

. 12.5

,

b

V

S x dx ,

a

Y

y

f x

B

Bb,

A

y f (x) ,

X

x a, x b

a

b

X

(

. 12.6),

0

b

. 12.6

V

[ f (x)]2 dx .

a

x y 2

1,

x

5 .

y

.

x

y 2

1

x

5

(

. 12.7).

B

x

y 2

1,

y

0,

A (1; 0 )

A

5

x

Ox .

0

x

y2

1,

x

5

. 12.7

C (5;

2) .

(12.3),

5

x 1

5

5

5

S

dxdy

dx

dy

x 1

dx

( x

1

x

x 1

( D )

1

x 1

1

3

1

1

2 (x

1)

5

4 (5 1) 2

(1

1) 2

10 2 .

2

3

3

3

1

3

3

2

1 2 . 2 .

x t, y

t 2 , z

t 3

2

3

.

x

t

1, y

t

t

2, z

t

t 2 ,

(12.2)

t2

1

1

t

3

1

1

l

dl

(x

')2

( y

')2

(z

')2 dt

1 2t2

t4 dt

(1 t2 )dt

t

1

.

t

t

t

3

3

(L )

t1

0

0

0

,

( )

,

a,b

,

b

b

M

(P)dx

(x)dx ,

a

a

(P)ds

(12.4)

( D )

(

)

M

(Q )

(

)

(

(P)dv

(12.5)

M y

x (P)d

xc

( )

,

M

(P)d

(

)

yc

( )

,

(P)d

(

)

M yz

x (P)d

xc

( )

,

(P)d

(

y (P)d

yc

( )

,

(P)d

(

M xy

z (P)d

zc

( )

.

( )

, I

Y)

I

0

md 2

, I

x

m y 2

z 2 , I

xy

mz 2

,

( )

( D )

( )

( D )

( )

(V )

( )

(V )

( )

(V )

,

2 2 R 2 .

P ,

d

2 2 ,

2

R

2

3

2

m

d k d k

R

d

2

k R

3

.

3

0

3

0

3

0

0

0

3

3

4g R3

2 z 2 R 2 .

I

( 2

2

z2 )dxdydz

r 4 sin

drd

d

o

v

v

2

R

4

R5

3 P

2

d

r

dr

R

0

0

0

5

5 g

R

I

z

( 2

2 )dxdydz

d sin 3 d

r 4 dr

v

v

0

0

2

R 5

2

R 5

(cos

1

cos3 )

0

2

R 2 .

5 0

5

3

5 g