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Башкирский Государственный Аграрный Университет

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x2. Ž¯à¥¤¥«¥--ë© ¨-â¥£à «, ®á-®¢-ë¥ á¢®©áâ¢ ...

I.2. Šà¨¢ ï § ¤ - ¯ à ¬¥âà¨ç¥áª¨: x = x(t), y = y(t), t 2 [ ; ], â®£¤ ¤«¨- ªà¨¢®© ¢ëç¨á«ï¥âáï ¯® ä®à¬ã«¥:

l = Z

[x0(t)]2 + [y0(t)]2 dt:

•à¨¬¥à 7. ‚ëç¨á«¨âì ¤«¨-ã

áâà®¨¤ë x = 2cos3 t, y = 2sin3 t.

’ ª ª ª ªà¨¢ ï á¨¬¬¥âà¨ç-

®â-®á¨â¥«ì-® ®¡¥¨å ª®®à¤¨- â-ëå ®á¥©,

â® ¢ëç¨á«¨¬ á- ç «

¤«¨-ã ¥¥ ç¥â¢¥àâ®© ç áâ¨ l1, à á¯®«®¦¥--®© ¢ ¯¥à¢®¬

ª¢ ¤à -â¥, ¢ íâ®¬ á«ãç ¥ 0 6 t 6

. x0

6cos2 t

sint, y0

= 6sin2 t

cost,

®âáî¤

l1 = Z

36cos4 t sin2 t + 36sin4 t cos2 t dt =

= 6 Z

cos2 t sin2 t

dt = 6 Z

sint cost dt = 6sin2

2 = 3:

„«¨- ¢á¥© ªà¨¢®© l = 4l1 = 4 3 = 12.

•à¨¬¥à 8. ‚ëç¨á«¨âì ¤«¨-ã æ¨ª«®¨¤ë: x = (t sint), y = (1 cost),

t2 [0; 2 ].

•©¤¥¬ ¯à®¨§¢®¤-ë¥ x0t = 1 cost, yt0 = sint, â®£¤

l = Z

dt = Z

(1 cost)2

+ sin2 t

1 2cost + cos2 t + sin2 t dt =

= Z p

dt = Z

2sin

dt = 4cos

2 2cost

= 4(cos cos0) = 4( 1 1) = 8:

I.3. Šà¨¢ ï § ¤ -

¢ ¯®«ïà-ëå ª®®à¤¨- â å: r = r('), 6 ' 6 , â®£¤

¤«¨- ªà¨¢®© ¢ëç¨á«ï¥âáï ¯® ä®à¬ã«¥:

Z p

l =

(r0('))2 + (r('))2 d':

•à¨¬¥à 9. ‚ëç¨á«¨âì ¤«¨-ã ªà¨¢®© r = (1 + cos'), 0 6 ' 6 .

42	x2. Ž¯à¥¤¥«¥--ë© ¨-â¥£à «, ®á-®¢-ë¥ á¢®©áâ¢ ...

• ©¤¥¬ ¯à®¨§¢®¤-ãî r0

= (1+ cos')0 =

sin', ®âáî¤

l = Z

d' = Z

(1 + cos')2

+ sin2

2 + 2cos' d' =

= 2Z

cos

d' = 4sin

= 4 sin

sin0 = 4:

II. •«®é ¤¨

II.1. •ãáâì ªà¨¢®«¨-¥©- ï âà ¯¥æ¨ï ®£à -¨ç¥-- á¢¥àåã ¨ á-¨§ã ªà¨¢ë-

¬¨, ãà ¢-¥-¨ï ª®â®àëå y = y1(x), y = y2(x), x 2 [a; b], y1(x) > y2(x). ’®£¤ ¯«®é ¤ì ªà¨¢®«¨-¥©-®© âà ¯¥æ¨¨ ¢ëç¨á«ï¥âáï ¯® ä®à¬ã«¥:

S = Za

[y1(x) y2(x)]dx:

•à¨¬¥à 10. „ -ë í««¨¯á

= 1 ¨ ¯àï¬ë¥ x = 1, x = 1, y = 0,

- ©â¨ ¯«®é ¤ì ä¨£ãàë, ®£à -¨ç¥--®© ¯àï¬ë¬¨ ¨ í««¨¯á®¬.

3 p

ˆ§ ãà ¢-¥-¨ï í««¨¯á

, ®âáî¤

¨¬¥¥¬: y = 2

4 x

S = Z

2 4 x2 dx = 3arcsin 2

+ 4 x

4 x2

= 3arcsin

p4 1 3arcsin

p4 1 =

6 p

= 6arcsin

3 = +

•à¨¬¥à 11. Ž¯à¥¤¥«¨âì ¯«®é ¤ì ä¨£ãàë, § ª«îç¥--®© ¬¥¦¤ã ¤¢ã¬ï

¯ à ¡®« ¬¨ y2 = 6x ¨ x2 = 6y.

6x, y =

, x 2 [0; 6].

ˆ§ ãà ¢-¥-¨© ªà¨¢ëå ¨¬¥¥¬: y =

S = Z p6x

dx =

p6x3=2

0 = 12:

II.2. •«®é ¤ì ªà¨¢®«¨-¥©-®© âà ¯¥æ¨¨ ¢ á«ãç ¥,	ª®£¤ ªà¨¢ ï § ¤ - ¢
¯ à ¬¥âà¨ç¥áª®© ä®à¬¥: x = x(t), y = y(t), 6 t 6	, '( ) = a, '( ) = b,

x2. Ž¯à¥¤¥«¥--ë© ¨-â¥£à «, ®á-®¢-ë¥ á¢®©áâ¢ ...

¢ëç¨á«ï¥âáï ¯® ä®à¬ã«¥:

S = y(t) x0(t)dt:

•à¨¬¥à 12. ‚ëç¨á«¨âì ¯«®é ¤ì ®¡« áâ¨, ®£à -¨ç¥--®© í««¨¯á®¬ x = = 3cost, y = 2sint.

‚ëç¨á«¨¬ ¯«®é ¤ì ¢¥àå-¥© ¯®«®¢¨-ë ¨ ã¤¢®¨¬. ‡¤¥áì x 2 [ 3; 3], ¯®- íâ®¬ã t ¨§¬¥-ï¥âáï ®â ¤® 0,

0
S = 2 Z	2sint( 3sint)dt = 12Z0	sin2 t dt =
		1	cos2t		t	sin2t
		1	cos2t		t	sin2t
	= 12Z			dt = 12			0 = 6 :
	= 12Z		2	dt = 12	2	4	0 = 6 :
	0

•à¨¬¥à 13. ‚ëç¨á«¨âì ¯«®é ¤ì ä¨£ãàë, ®£à -¨ç¥--®© æ¨ª«®¨¤®© x = = t sint, y = 1 cost, t 2 [0; 2 ].

2			2
S = Z (1 cost)2 dt =	2 t 2sint + 4		2	= 3 :
S = Z (1 cost)2 dt =	2 t 2sint + 4		sin2t 0	= 3 :
0	3	1

II.3. •«®é ¤ì ªà¨¢®«¨-¥©-®£® á¥ªâ®à		¢ ¯®«ïà-ëå ª®®à¤¨			- â å r = r('),

6 ' 6 , ¢ëç¨á«ï¥âáï ¯® ä®à¬ã«¥:

S = 12 Z (r('))2 d':

•à¨¬¥à 14. • ©â¨ ¯«®é ¤ì ª à¤¨®¨¤ë r = cos' + 1, ' 2 [0; 2 ].

	1	2	1	3		1	2		3
	1		1	3		1	2		3
S =		Z (cos' + 1)2 d' =			' +		sin2' = 2sin' 0	=		:
S =	2	Z (cos' + 1)2 d' =	2	2	' +	4	sin2' = 2sin' 0	=	2	:
	0

•à¨¬¥à 15. • ©â¨ ¯«®é ¤ì «¥¬-¨áª âë r2 = 2cos2'.

„«ï ¢ëç¨á«¥-¨ï ®¡é¥© ¯«®é ¤¨ ¤®áâ â®ç-® ã¤¢®¨âì ¯«®é ¤ì ¯à ¢®£®

®¢ « , ª®â®à®¬ã ®â¢¥ç ¥â ¨§¬¥-¥-¨¥ ã£«				4 6 ' 6 4.
		=4		=4
S = 2 2	2	Z		=4		= 1 ( 1) = 2:
S = 2 2	2	Z	cos2' d' = sin2'		=4	= 1 ( 1) = 2:
1		=4

44	‡ ¤ ç¨ ¤«ï á ¬®áâ®ïâ¥«ì-®£® à¥è¥-¨ï

III. Ž¡ê¥¬ â¥«	¢à é¥-¨ï

Ž¡ê¥¬ â¥« , ®¡à §®¢ --®£® ¢à é¥-¨¥¬ ¢®ªàã£ ®á¨ Ox ªà¨¢®«¨-¥©-®© âà ¯¥æ¨¨, ®£à -¨ç¥--®© ªà¨¢®© y = f (x), ®áìî Ox ¨ ¯àï¬ë¬¨ x = a, x = b, ¢ëç¨á«ï¥âáï ¯® ä®à¬ã«¥:

V = [f (x)]2 dx:

•à¨¬¥à 16. • ©â¨ ®¡ê¥¬ â¥« , ®¡à §®¢ --®£® ¢à é¥-¨¥¬ í««¨¯á ¢®-

ªàã£ ®á¨ Ox			x2	+		y2		= 1.
ªàã£ ®á¨ Ox				+				= 1.
	25			9
’ ª ª ª y2 =					9		(25 x2), ¯®«ãç¨¬
’ ª ª ª y2 =					25		(25 x2), ¯®«ãç¨¬
5	9							5	9
V = Z	9	(25 x2)dx = 2 Z							9	(25 x2)dx =

	25								25
5								0					x3	5
										= 2	9	25x	x3	5
										= 2	25	25x	3	0 = 60 :

‡ ¤ ç¨ ¤«ï á ¬®áâ®ïâ¥«ì-®£® à¥è¥-¨ï

‚ëç¨á«¨âì:

287. R (x2 + 1)dx;

288.R ( x x2)dx;

289.sinx dx;

290.sin2x dx;

291.x sinx dx;

2 e

292.lnx dx;

293. R x2+2dxx+2;

294. R tg5 x dx;

‡ ¤ ç¨ ¤«ï á ¬®áâ®ïâ¥«ì-®£® à¥è¥-¨ï	45

295. R sin3 ' d';

296. R cos3 ' d';

297. R cos2 ' sin3 ' d';

298. R e x3 dx;

299. R xe x2 dx;

300. R x2e x dx;

1 p

301.arctgx dx;

	0
	R			3p
302.	R	x		3p	R			2	x	2	dx (R > 0);
302.	R	x			R				x		dx (R > 0);
	0
	e
303.	R ln2 x dx;
	1

	2		dx
304.	R		dx				.
304.	R		2+cos			x	.
	0

‚ëç¨á«¨âì ¯«®é ¤¨ ä¨£ãà, ®£à -¨ç¥--ëå «¨-¨ï¬¨:

305.y = x1, x = 1, x = e, y = 0;

306.y = x2, y = 1;

307.y = x2, y = 2 x2;

308.y = x2 1, x = 2, y = 0, £¤¥ x > 1;

309.y = sin3x, y = 0, £¤¥ 0 6 x 6 3;

310.y = sinx, y = sin3 x, £¤¥ 0 6 x 6 2;

311.y = x2, y = x;

312.y = arcsin2x, x = 0, y = 2;

313.y = sin2x, y = 1, x = 2, £¤¥ 4 6 x 6 2;

314.x2 y2 = 1, x = 2;

315.y = x3, y = 1, x = 0;

316.y = 12 (ex2 + e x2 ), x = 1, x = 1, y = 0;

317.y = x(3 x), y = x 3;

318.y = 3x x2, y = x2 x;

319.xy = 5, x + y = 6;

46	‡ ¤ ç¨ ¤«ï á ¬®áâ®ïâ¥«ì-®£® à¥è¥-¨ï

320.xy = 2, y = x 3;

321.xy = 4, x = 4, y = 4, x = 0, y = 0;

322.ª à¤¨®¨¤®© = a(1 + cos');

323.= a cos2';

324.= a sin2';

325.= 2 + sin2';

326.= ae', £¤¥ 0 6 ' 6 2 ;

327.= a sin3';

328.= a cos3';

329. ®¤-®© àª®© æ¨ª«®¨¤ë x = a(t sint), y = a(1 cost), 0 6 t 6 2 ¨ ®áìî OX;

330.= a cos4';

331.= a sin4'.

332.‚ëç¨á«¨âì ¯«®é ¤¨ ä¨£ãà, ¨§®¡à ¦¥--ëå - à¨áã-ª å 1 | 6.

‚ëç¨á«¨âì ®¡ê¥¬ë â¥«, ®¡à §®¢ --ëå ¢à é¥-¨¥¬ ä¨£ãàë, ®£à -¨ç¥--®© «¨-¨ï¬¨:

333. y = 4 x2, y = 0, x = 0, £¤¥ x > 0 ¢®ªàã£: 1) ®á¨ x, 2) ®á¨ y;

334. y = x x2, y = 0 ¢®ªàã£ ª ¦¤®© ¨§ á«¥¤ãîé¨å ¯àï¬ëå: 1) y = 0, 2) x = 0, 3) x = 2, 4) x = 2, 5) y = 1, 6) y = 2;

335.y = ex, x = 0, x = 1, y = 0 ¢®ªàã£: 1) ®á¨ x, 2) ®á¨ y;

336.y = x2, y = 4, x = 0, £¤¥ x > 0 ¢®ªàã£: 1) ®á¨ x, 2) ®á¨ y;

x3. •¥á®¡áâ¢¥--ë¥ ¨-â¥£à «ë

337.y = x2 + 1, y = 0, x = 1, x = 2 ¢®ªàã£: 1) ®á¨ x, 2) ®á¨ y;

338.y = x3, y = 1, x = 0 ¢®ªàã£: 1) ®á¨ x, 2) ®á¨ y;

339.xa22 + yb22 = 1, y = 0, £¤¥ y > 0 ¢®ªàã£ ®á¨ x;

340. y = lnx, y = 0, x = e ¢®ªàã£ ª ¦¤®© ¨§ á«¥¤ãîé¨å ¯àï¬ëå: 1) y = 0, 2) x = 0, 3) y = 1, 4) x = 1, 5) x = 1, 6) y = 1;

341.y = sinx, y = 0, £¤¥ 0 6 x 6 ¢®ªàã£ ª ¦¤®© ¨§ á«¥¤ãîé¨å ¯àï¬ëå:

1)y = 0, 2) x = 0, 3) x = 2 , 4) x = 1, 5) x = 2, 6) y = 1, 7) y = 2;

342.x2 y2 = 4, y = 2, y = 0 ¢®ªàã£ ®á¨ x;

343.y = x, y = x2 ¢®ªàã£: 1) ®á¨ x, 2) ®á¨ y;

344.y = cos2x, y = 0, x = 0, £¤¥ 0 6 x 6 4 ¢®ªàã£: 1) ®á¨ x, 2) ®á¨ y;

345.y = sinx, y = 0, £¤¥ 2 6 x 6 3 ¢®ªàã£ ª ¦¤®© ¨§ á«¥¤ãîé¨å

¯àï¬ëå: 1) y = 0, 2) x = 0, 3) x = , 4) y = 2;

346.	y = 2x x2, y = 0 ¢®ªàã£ ª ¦¤®© ¨§ á«¥¤ãîé¨å ¯àï¬ëå: 1) x = 0,
2) y = 0, 3) x4					= 1, 4) y = 1;
347.	y = x , x = 1, x = 4, y = 0 ¢®ªàã£: 1) ®á¨ x, 2) ®á¨ y;
348.	y =			1		, x = 1, x = 1, y = 0 ¢®ªàã£: 1) ®á¨ x, 2) ®á¨ y.
				1+x2
‚ëç¨á«¨âì ¤«¨-ã ¤ã£¨ ªà¨¢®©:
349.	y2		= x3, ®âá¥ç¥--®© ¯àï¬®© x = 1;
350.	y = lncosx, ®âá¥ç¥--®© ¯àï¬ë¬¨ x = 0, x = ;
							6
351.	y2		= (x + 1)3, ®âá¥ç¥--®© ¯àï¬®© x = 4;
	y	2	=		4		3
352.					9 (2x		x) x, ®âá¥ç¥--®© ¯àï¬®© x = 1;
353.	y = a				(e a		+ e a ) ¬¥¦¤ã ®áìî y ¨ ¯àï¬®© x = a;
			2

354.y = x2 1, ®âá¥ç¥--®© ®áìî x;

355.y = lnsinx ®â x = 3 ¤® x = 23 ;

356.áâà®¨¤ë x = a cos3 t, y = a sin3 t;

357. ®¤-®© àª¨ æ¨ª«®¨¤ë x = a(t sint), y = a(1 cost), 0 6 t 6 2 ;

358.ª à¤¨®¨¤ë r = 4(1 cos');

359.¯¥à¢®£® § ¢¨âª á¯¨à «¨ r = a', 0 6 ' 6 2 ;

360.y = x42 12 lnx ®â x = 1 ¤® x = e.

x3. •¥á®¡áâ¢¥--ë¥ ¨-â¥£à «ë

3.1. •¥á®¡áâ¢¥--ë¥ ¨-â¥£à «ë á ¡¥áª®-¥ç-ë¬¨ ¯à¥¤¥« ¬¨

•ãáâì äã-ªæ¨ï f (x) ®¯à¥¤¥«¥- - ¯à®¬¥¦ãâª¥ [a; +1) ¨ ¨-â¥£à¨àã¥¬

¢ «î¡®© ª®-¥ç-®© ¥£® ç áâ¨ [a; A], â ª çâ® ¨-â¥£à «	f (x)dx ¨¬¥¥â á¬ëá«
	a
¯à¨ «î¡®¬ A > a.	R

48 x3. •¥á®¡áâ¢¥--ë¥ ¨-â¥£à «ë

Ž¯à¥¤¥«¥-¨¥ 1. •¥á®¡áâ¢¥--ë¬ ¨-â¥£à «®¬ ®â äã-ªæ¨¨ f (x) ¯® ¯à®-

¬¥¦ãâªã [

;+1) - §ë¢ ¥âáï ¯à¥¤¥« (¥á«¨ ®- áãé¥áâ¢ã¥â)

lim

f (x)dx,

¥£® ¢¥«¨ç¨- ®¡®§- ç ¥âáï

1 R

lim

f (x)dx:

(1)

(

)

A!+1 Za

‚ á«ãç ¥, ¥á«¨ íâ®â ¯à¥¤¥« ª®-¥ç¥-, â® £®¢®àïâ, çâ® ¨-â¥£à « (1) áå®¤¨â- áï, ¥á«¨ ¯à¥¤¥« (1) ¡¥áª®-¥ç¥- ¨«¨ ¢®¢á¥ -¥ áãé¥áâ¢ã¥â, â® £®¢®àïâ, çâ® ¨-â¥£à « à áå®¤¨âáï.

•à¨¬¥à 1. • áá¬®âà¨¬ ¨-â¥£à «

. ”ã-ªæ¨ï f (x) =

- «î¡®¬

1+x

¯à®¬¥¦ãâª¥ [0; A] ¨-â¥£à¨àã¥¬

1+x

dx =

lim

1 + x2

A!+1 Z0

1 + x2 = A!+1 arctg

•® ®¯à¥¤¥«¥-¨î ¨-â¥£à « áå®¤¨âáï ¨ ¥£® ¢¥«¨ç¨-

à ¢-

â¥£à¨àã¥¬ -

«î¡®¬ ¯à®¬¥¦ãâª¥ [0; AR]. ’ ª ª ª

•à¨¬¥à 2.

• áá¬®âà¨¬ ¨-â¥£à «

sinx dx. ”ã-ªæ¨ï f (x) = sinx ¨--

A!+1

A!+1 Z0

(

cosA)

lim

sinx dx = lim

sinx dx à áå®¤¨âáï.

-¥ áãé¥áâ¢ã¥â, â® ¯® ®¯à¥¤¥«¥-¨î ¨-â¥£à «

•à¨¬¥à 3. • áá¬®âà¨¬ ¨-â¥£à «

dxp . ”ã-ªæ¨ï f (x) =

¨-â¥£à¨àã-

¥¬ - «î¡®¬ ¯à®¬¥¦ãâª¥ [0; A]. ’ ª ªR ª

(A1 p

1); p = 1;

lnA;

p = 1;

1 p

â® ¨-â¥£à «

dxp áå®¤¨âáï ¯à¨ p > 1 ¨ à áå®¤¨âáï ¯à¨ p 6 1.

x3. •¥á®¡áâ¢¥--ë¥ ¨-â¥£à «ë

-¯à®¬¥¦ãâª¥ [a;+1), â® ¤«ï «î¡ëå ¯®áâ®ï--ëå ¨ á¯à ¢¥¤«¨¢® à - ¢¥-áâ¢®:

+1		+1		+1
Za	( f (x)+ g(x))dx =	Za	f (x)dx +	Za	g(x)dx:


+1		c	+1
Za	f (x)dx = Za	f (x)dx + Zc		f (x)dx:
3. •ãáâì äã-ªæ¨ï f (x) -¥¯à¥àë¢- -			¯à®¬¥¦ãâª¥ [a;+1), â® ¤«ï «î-

+1
Za	f (x)dx =	Z	f ('(t))'0(t)dt:

4. …á«¨ äã-ªæ¨¨ f (x) ¨ g(x) -¥¯à¥àë¢-® ¤¨ää¥à¥-æ¨àã¥¬ë - [a;+1)

	+1			+1
¨ áãé¥áâ¢ã¥â lim f (x)g(x), â® ¨-â¥£à «ë		f (x)dx ¨		g(x)dx ®¤-®-
A!+1	a			a
¢à¥¬¥--® áå®¤ïâáï ¨«¨ à áå®¤ïâáï, ¨ ¢ á«ãç ¥R		¨å	áå®¤¨¬®áâ¨ ¨¬¥¥â ¬¥áâ®
¢à¥¬¥--® áå®¤ïâáï ¨«¨ à áå®¤ïâáï, ¨ ¢ á«ãç ¥R		¨å		R

à ¢¥-áâ¢®

+1	+	1
Z		1

f (x)g(x)dx = f (x)g(x)

f 0(x)g(x)dx:

‡ ¬¥ç -¨¥. €- «®£¨ç-® (1) ®¯à¥¤¥«ï¥âáï ¨-â¥£à « ®â äã-ªæ¨¨ f (x) - ( 1;a]

a					a
Z	(	x	)	dx	= A! 1 Z
	f	x		dx	lim f (x)dx;
1					A

à ¢-® ª ª ¨ ¨-â¥£à « äã-ªæ¨¨ f (x) - ¯à®¬¥¦ãâª¥ ( 1;+1):

+1			A
Z	f (x)dx =	lim	Z	f (x)dx:

1	B! 1
	A!+1 B

x3.

•¥á®¡áâ¢¥--ë¥ ¨-â¥£à «ë

•à¨¬¥à 4.

= lim

lim

(

arctgA) =

1 + x2

A! 1 Z

1 + x2 = A! 1

•à¨¬¥à 5.

= :

1 + x2

•ãáâì äã-ªæ¨ï f (x) § ¤ -

ª®-¥ç-®¬ ¯à®¬¥¦ãâª¥ [a;b], -® -¥®£à -

-¨ç¥- ¢ íâ®¬ ¯à®¬¥¦ãâª¥. •®«®¦¨¬, çâ® ¢ «î¡®¬ ¯à®¬¥¦ãâª¥ [a;b "]

(0 < " < b a) f (x) ®£à -¨ç¥-

¨ ¨-â¥£à¨àã¥¬ , -® ®ª §ë¢ ¥âáï -¥®£à -¨-

b ¢ íâ®¬ á«ãç ¥ - §ë¢ ¥âáï

®á®¡®© â®çª®©.

Ž¯à¥¤¥«¥-¨¥ 2. •¥á®¡áâ¢¥--ë¬ ¨-â¥£à «®¬ ®â äã-ªæ¨¨ f (x) -

¯à®-

¬¥¦ãâª¥ [a; b] - §ë¢ ¥âáï ¯à¥¤¥« lim

f (x)dx, ¥£® ¢¥«¨ç¨- ®¡®§- ç ¥âáï

" 0 a

"!0

b "

(

)

(

)

dx:

(2)

lim

‚ á«ãç ¥, ¥á«¨ íâ®â ¯à¥¤¥« ª®-¥ç¥-, £®¢®àïâ, çâ® ¨-â¥£à « (2) áå®¤¨âáï. …á«¨ ¦¥ ¯à¥¤¥« (2) ¡¥áª®-¥ç¥- ¨«¨ ¢®¢á¥ -¥ áãé¥áâ¢ã¥â, â® £®¢®àïâ, çâ® ¨-â¥£à « (2) à áå®¤¨âáï.

‡ ¬¥ç -¨¥. ‚ á«ãç ¥, ¥á«¨ f (x) ®£à -¨ç¥- ¨ ¨-â¥£à¨àã¥¬

¢ «î¡®¬

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¢ ª ¦¤®¬ ¯à®¬¥¦ãâª¥ [a;a +"] á¯à ¢

®â â®çª¨ a (®á®¡ ï â®çª ), â®£¤

¯à®¬¥¦ãâª¥ [a; b] ®¯à¥¤¥«ï¥âáï à ¢¥-áâ¢®¬

(

)

"!0aZ+"

(

)

dx:

(3)

lim

‡ ¬¥ç -¨¥. •ãáâì c 2 [a; b] ¨ äã-ªæ¨ï f (x) -¥®£à -¨ç¥-

¢ â®çª¥ c,

¯à¨ç¥¬ - ¯à®¬¥¦ãâª å [a;c "1] (0 < "1 < c a) ¨ [c +"2; b] (0 < "2 < b c)

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