. (3.4)
f(x,y) .
(dl ).
, m
, (3.5)
d (x,y) .
I :
1. ; = const
2 2.
3 3. ( ). f(x,y) , ,
(.., , ).
I , , ( , ).
4. I :
.
, (.. ),
( Dlk -1 , , , 0, , , ò = ò.
.
( ), , .
I
.
.
.
3 :
I. = j ()
j () j1 ()
.(1;1), . (2; 2).
, dl dx, :
(3.6)
1 2 .
3.1.1
, Z - :
(0;0), (4;0), (4;2), (0;2).
.
1) ():
2) ():
3) (): .
4) ():
, .
2. :
= j (t), y = y (t) (t1 £ t £ t2);
j (t) y (t) j1(t) y1(t).
, t = t11, t = t2, t t1 t2 .
, (3.7)
t1 t2 t .
, ò f(x,y) dl, , 1) = j(t), = y(t), 2) t.
|
|
3.1.2
,
> 0 = cos3t, = sin3t , .
.
3. r = r(q); q1 £ q £ q2; r(q) r1(q).
.
3.1.3
Z - 2 + 2 = 2.
I =? : 2 + 2 = 2 Û r2 = 2 Þ r= .
3.1.4
= , 0 £ £ 2, .
.
3.2 II ( )
, , (,) .
n 1, 2,...,n-1, . ..
º 0, º n.
( = 0,1,2,.., n).
È-1 Nk (x, h). (x, h) (,).
D = -1 -1 s :
.
-1 .
s , (,) (, , ) .
n , , .
.
3.2.1
(,) () , s, , , 1) $ 2) , (x, h) .
( ).
,
. (3.8)
:
. (3.9)
D = -1 -1.
(3.1) (3.2) , (,) .
.
, , , , , .
, 1) P(x,y) Q(x,y) 2) ò P(x,y) d ò Q(x,y) d,
( )
|
|
.
.
,
.
.
( I ). . , L. .
, :
,
(,) Q(x,y) - .
.
3.2.1
, .
.
1. , = cons t.
2.
3.( ). 2 (,) , ,
+ .
, ( ).
4. ( ), :
( ).
, , , -1 ( = 1,2,..., n) , , D = -1, s, -1 - , s
.
, lim , ; , .
.
, , .
5. (,) L, , L ( ) .
, . .
AmB BnA
.
.
. , .
, L , , , .
. L , .
3.2.2
II .
.
I .
y = f(x), f() ; . (1;1), . (2;2).
3.2.2.1
, L , = 1, = 1, = 3, = 5. .
|
|
1) ; : = 1 Þ dy = 0
2) .
3) , .. CD: y = 5 Þ dy = 0.
4)
2 .
= j (t), y = y(t), j(t) y(t) .
. t = t1; . t = t2.
t, dx dy - t dt :
. (3.10)
3.2.2.2
, = R × cos t, y = R × sin t ( 3.9).
.
3.2.3
L, D, D .
, , .
D, L.
3.2.1
D, L, (,) Q(x,y), . :
, (3.11)
( ).
1. :
.
[ , b]; :
2 = 2 () ( AmB)
1 = 1 () - ( AnB),
1() 2() [ a,b]; 2() ³ 1():
.:
. (3.12)
.: = ( AnB BmA) =
. (3.13)
(3.1) (3.2) :
. (3.14)
2.: .
D .
D [ c,d ].
NnM: = 1(); NmM: 2 = 2(). Î [c,d].
1() 2() [c,d], 2() ³ 1().
a)
(3.15)
)
(3.16)
(3.15) (3.16) :
(3.17)
(3.17) (3.14) , .
, D L, D. 1) , 2) .
.
, .
3.2.3
, , L
.
:
Q D.
= ( S = p×a×b) = .
3.3
.
D L. D.
(3.18)
1) (3.1) (,) º ,
Q(x,y) = 0.
: ; (3.19)
2) (1) (,) = 0, Q(,) º x.
; (3.20)
3) (3.19) (3.20) 2,
. (3.21)
(3.19),(3.20),(3.21) , .
, .
|
|
3.3.1
, = a×cos t, y = b×sint
.
3.3.2.
, .
, ; = 1.
, .. = Û Þ t1 = 0, t2 = 1
(.).
3.4 II
,
.
L, (0;- 2) (1;0), L :
1) = 2 2 (L1);
2) 2 = 4 4 (L2);
3) = 1 (L3)
L3 t Î[ -p/2; 0].
.
) L1: y = 2x 2 Þ dy = 2dx.