математика

b

ɍ

l ³ 1 (yc)2 dx .

a

ȿɫɥɢ ɠɟ ɤɪɢɜɚɹ ɡɚɞɚɧɚ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢɦɢ ɭɪɚɜɧɟɧɢɹɦɢ x=x(t), y=y(t) (DdtdE),

Ɍ

ɬɨ l

E

(x c)2

(yc)2 dt .

ɇ

³

t

t

D

Ⱥɧɚɥɨɝɢɱɧɨ ɧɚɯɨɞɢɬɫɹ ɞɥɢɧɚ ɞɭɝɢ

ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɪɢɜɨɣ, ɨɩɢɫɚɧɧɨɣ

Ȼ

ɩɚɪɚɦɟɬɪɢɱɟɫɤɢɦɢ ɭɪɚɜɧɟɧɢɹɦɢ: x = x(t), y = y(t), z = z(t), D d t d E

l

E

ɣ

³

(x c)2 (yc)2

(zc)2 dt .

t

t

t

D

ɢ

ȿɫɥɢ ɡɚɞɚɧɨ ɩɨɥɹɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤɪɢɜɨɣ U = U(M), D d M d E, ɬɨ

ɪ

l

E

U

2

c

2

dM .

(U )

ɨ³

ɬ

D

ȿɫɥɢ ɩɥɨɳɚɞɶ S(x) ɫɟɱɟɧɢɹ ɬɟɥɚ ɩɥɨɫɤɨɫɬɶɸ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɨɫɢ Ox,

ɢ

ɹɜɥɹɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɮɭɧɤɰɢɟɣ ɧɚ ɨɬɪɟɡɤɟ [a, b], ɬɨ ɨɛɴɟɦ ɬɟɥɚ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ

ɮɨɪɦɭɥɟ

ɡ

ɨ

b

V

³S (x )dx .

ɩ

a

Ɉɛɴɟɦ V ɬɟɥɚ, ɨɛɪɚɡɨɜɚɧɧɨɝɨ ɜɪɚɳɟɧɢɟɦ ɜɨɤɪɭɝ ɨɫɢ Ox ɤɪɢɜɨɥɢɧɟɣɧɨɣ

ɟ

ɬɪɚɩɟɰɢɢ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɤɪɢɜɨɣ y=f(x), (f(x)t0), ɨɫɶɸ ɚɛɫɰɢɫɫ ɢ ɩɪɹɦɵɦɢ x=a ɢ x=b

(a<b), ɜɵɪɚɠɚɟɬɫɹ ɢɧɬɟɝɪɚɥɨɦ

Ɋ

b

V

S³ f 2 (x)dx .

ɉɪɢɦɟɪ 2.8. ȼɵɱɢɫɥɢɬɶ ɞɥɢɧɭ ɞɭɝɢ ɤɪɢɜɨɣ

y2

x 3 ,

ɨɬɫɟɱɟɧɧɨɣ ɩɪɹɦɨɣ

x

4

(ɪɢɫ. 2.5).

3

y

x

4

ɍ

Ɍ

y2

x3

Ⱥ

3

0

ɯ

ɇ

4

Ȼ

3

ɣ

ȼ

ɪ

Ɋɢɫ. 2.5

ɢ

Ɋɟɲɟɧɢɟ. Ⱦɥɢɧɚ ɞɭɝɢ ȺɈȼ ɪɚɜɧɚ ɭɞɜɨɟɧɧɨɣ ɞɥɢɧɟ ɞɭɝɢ ɈȺ.

3

1

ɨ

y

x 2 , yc

3 x 2 .

Ɍɨɝɞɚ

ɬ

1

2

ɢ

4

4

·2

4

1

3

§

3

3

9

4 3 §

9

·

§

9

·

2

³

ɡ

³

³

2

l

lOA

1 ¨

2

x

¸

dx

1

4

xdx

¨1

4

x

¸ d¨1

4

x¸

0

©

¹

0

9 0 ©

¹

©

¹

§

9

·

3

2

ɩ

4

¨1

4

x¸

4 / 3

8

56

56

112

©

¹

ɨ

43 / 2 1

; l 2

.

9

3

0

27

27

27

27

ɟ

Ɋ

2

­

(t

2

2)sin t 2t cost;

ɉɪɢɦɟɪ 2.9. ȼɵɱɢɫɥɢɬɶ ɞɥɢɧɭ

ɞɭɝɢ

ɤɪɢɜɨɣ

°x

®

2

°

(2 t

)cost 2t sin t,

¯y

xtc

2t sin t (t 2 2)cost 2cost 2t sin t

t 2 cost,

ytc

2t cost (2 t 2 )sin t 2sin t 2t cost t 2 sin t,

ɨɬɤɭɞɚ

(x c)2

(yc)2

t4 cos2 t t4 sin 2 t

t4 (cos2 t sin 2 t)

t2 .

t

t

ɍ

S

t

3

S

S

3

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, l

³t2dt

.

Ɍ

0

3

0

3

ɉɪɢɦɟɪ 2.10. ɇɚɣɬɢ ɞɥɢɧɭ ɞɭɝɢ ɤɚɪɞɢɨɢɞɵ U=a(1+cosM ), (a>0, 0dMd2S)

(ɪɢɫ. 2.6).

Ȼ

2

2

(Uc )2 U2

Ɋɟɲɟɧɢɟ. ɁɞɟɫɶUc

asin M,

2a2

(1 cosɇM)

M

M

4a cos

M

2a

cos

M

ɣ

S

M

2a³cos

dM 8a.

2

2

. ȼ ɫɢɥɭ ɫɢɦɦɟɬɪɢɢ l

2

2

0

M

S

ɢS

2

M

4

ɪ

M S

ɨ

Ɉ

ɬ

2

O

M 0

ɢ

ɡ

M

3S

ɨ

2

Ɋɢɫ. 2.6

Ɂɚɦɟɱɚɧɢɟ. ɉɨɫɬɪɨɟɧɢɟ ɥɢɧɢɢ ɜɟɞɟɬɫɹ ɜ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɩɨ

ɟ

ɬɨɱɤɚɦ,ɩɤɨɬɨɪɵɟ ɜ ɞɨɫɬɚɬɨɱɧɨɦ ɤɨɥɢɱɟɫɬɜɟ ɡɚɩɢɫɵɜɚɸɬɫɹ ɜ ɜɢɞɟ ɬɚɛɥɢɰɵ ɢɯ

Ɋ

ɤɨɨɪɞɢɧɚɬ.

ɉɪɢɦɟɪ 2.11. ɇɚɣɬɢ ɨɛɴɟɦ ɬɟɥɚ, ɨɛɪɚɡɨɜɚɧɧɨɝɨ ɜɪɚɳɟɧɢɟɦ ɜɨɤɪɭɝ ɨɫɢ Ox

ɮɢɝɭɪɵ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɥɢɧɢɹɦɢ 2y

x2 ɢ 2x 2y 3 0 (ɪɢɫ. 2.7).

y

x2

ȹ y

3 2x 3 x :

x2

3 x; x2 2x 3 0; x 3, x

2

1.

2

2

2

2

2

1

y

ɍ

2 y

x2

Ɍ

Ⱥ

ɇ

Ȼ

ȼ

2x 2 y 3

0

-3

O

1

x

Ɋɢɫ. 2.7

ɢ

ɂɫɤɨɦɵɣ ɨɛɴɟɦ ɟɫɬɶ ɪɚɡɧɨɫɬɶ ɞɜɭɯ ɨɛɴɟɦɨɜɣ: ɨɛɴɟɦɚ V1

ɬɟɥɚ,

ɩɨɥɭɱɟɧɧɨɝɨ

ɜɪɚɳɟɧɢɟɦ ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɬɪɚɩɟɰɢɢ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɩɪɹɦɨɣ y

3

x

( 3 d x d1),

2

ɢ ɨɛɴɟɦɚ

V2

ɬɟɥɚ,

x

ɨ

ɩɨɥɭɱɟɧɧɨɝɨɪɜɪɚɳɟɧɢɟɦ ɤɪɢɜɨɥɢɧɟɣɧɨɣ ɬɪɚɩɟɰɢɢ,

ɬ

a

2