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2 ; 0 5:12. <, p sin t 14 2+ t 2 R.4;33 1 ) 5e ; e 1 ) (e + 1)=2:15. )64p;72 9,& ': .5 , 5 ) * M7. 1, $, $ $ y = f (x) x = a, x = b, y = 0 (. . 7.1).0 $ f (x) : a b ],, (. M 7), * S $ZbS = f (x) dx:(9:1)aP $ D, x = a,x = b $ : a b ] $ y = f1(x) y = f2(x), f1(x) f2(x) (. 9.1).
> * SD $D * , * $ f2(x) f1(x), $Zb;; SD =. 9.1af2 (x) ; f1(x) dx:(9:2). 9.2 9.1. 1 * S $, <73x2 + y2 = 1:a2 b2' * S 2 S1, S1 { * p , Ox $ $ y = ab a2 ; x2,;a x a (. 9.2). ( $ (9.1) Za pbS = 2 S1 = 2 aa2 ; x2 dx:a2 $ x = a sin t, dx = a cos t dt. ( t ;a = a sin t,a = a sin t. %) , ; 2 t 2 . 9 ,==Z2 2Z2qb2a2 ; a2 sin t a cos t dt = 2 a b cos t dt =S=2 a;=2;;=2!=2 !sin2t= a b (1 + cos 2 t) dt = a b t + 2 = a b 2 + 2 = a b:=2=2=Z2;;', ", * a b S = a b.
, a = b = R, ae * S = R2 R.;. 9.3. 9.4 9.2. 1 * SD $ D, ! y = x2 + 1 x + y = 3 (. 9.3).P# 748< y = x2 + 1: y = 3 ; x ! ! : x1 = ;2,x2 = 1.( f1(x) = x2 +1 f2(x) = 3 ; x, $ (9.2)Z1 Z12SD = (3 ; x) ; (x + 1) dx = (2 ; x ; x2) dx =;2;20123 1xx= @2 x ; 2 ; 3 A = 2 (1 + 2) ; 1 ;2 4 ; 1 +3 8 = 6 + 32 ; 3 = 29 :2;5 , ) $ . , ) * , < . 1 , , . O, &. ( O Ox, & ( 9.4) ! #! .( M { .
9 M O OM . 9.1. 2 OM = M , x OM = ' ) ;!, , ;OM M . $ M '.0 M ', #M ( '), , '. ( < # , .. 0 < + 1, ' , * !: , 0 ' < 2 .675 , !Y # 2 , ) .!) 7 . P Oxy O , ) Ox(. 9.5). > M x y ' OA = x AM = y OM = x OM = ' ,OA = OM cos ' AM = OM sin ' () x = cos ' y = sin ':', ) $:8< x = cos '(9:3): y = sin '( 0 0 ' < 2 ):5 x y M , ) ' * $:q = x2 + y2- cos ' = px2x+ y2 - sin ' = px2y+ y2 :6;; . 9.5. 9.6 9.3. 5 ) x2 + y2 = R2 .' $ (9.3), x2 + y2 = R2 () 2 cos2 ' + 2 sin2 ' = R2 () 2 = R2 () = R:> !, ) R = R (0 ' < 2 ).76 9.4. ( , = a ', a { ) (< & 8().' , ! ' :' 0141232252 01 a41 a2a3 a22 a5 a2G O , !* , ! .
1 ) , , ! . 9 < , ! (. 9.6).) $" ( (. ( ; = f (') ' :> $ G, ; , * (. 9.7), .;; . 9.7. 9.8 9.1. f (') { : ] , " S G, = f (') ' = , ' = , < , 77 Z1S = 2 f 2 (') d':(9:4). P! : ] n '0 = < '1 < : : : < 'k 1 < 'k < : : : < 'n 1 < 'n = :1 ) : 'k 1 'k ] ! k (k = 1 : : : n).= ' = '1, ' = '2, : : : , ' = 'n 1 ! n G1, G2, : : : , Gn, ) ( 9.7). ( < Gk , ' = 'k 1, ' = 'k k = f (k ) T 'k = 'k ; 'k 1 (k = 1 : : : n).> * R ! $S = 12 R2 ! * Sn ! $, , ) $nnXXSn = 12 2k T 'k = 12 f 2(k ) T 'k :(9:5)k=1k=16 , * S * (9.5) = 1maxT 'k ! 0, ..k nnX(9:6)S = lim0 Sn = lim0 21 f 2(k ) T 'k :k=15, * # (9.6) $ 12 f 2(') : ].
> f (') : ] $ 1 f 2(') , 7.2 M 7 2 (9.6) * 21 f 2(') : ]. 9 ,Z 1 2S = 2 f (') d'78;;;;;;!!.. $ (9.4) .. ( $ , = f1('), = f2('),0 f1(') f2('), ' = , ' = , < (. 9.8). > * S < $ * ) ! $Z122S = 2 f2 (') ; f1 (') d':(9:7) 9.5. 1 * $, 4 = a ', a { ) , ' = 32 ,0 ' 23 (. 9.9).' $ (9.4), (3Z=2)(3Z=2)2 (3Z=2)23 (3=2)2 31aa'9a1222 d' = 2(a ') d' = 2' d' = 2 3 = 16 :S=20000;; .
9.9. 9.10 9.6. 1 * $, ) = 3 = 2 (1 ; cos ') ( ).E = 3 { < ) 3 O(. 9.3).2 = 2 (1 ; cos ') ! ' :79'0 16 02; 3p 13 12 23 65 1232+ 3p4 ) , ', ! , < (. 9.10).2 ) A C ) # * :8< = 3: = 2 (1 ; cos ')88< = 3< = 3() : 3 = 2 (1 ; cos ') () : cos ' = ;1=2 =)8<=) : ' == 3 (2=3) :9 , A (3- ; 32 )- C (3- 23 ).>! * $ ABCOA, * $ OBCO. > $ OBCO = 3, = 2 (1 ; cos ') ' = 0, ' = 23 , , $ (9.7), S = 2 SOBCO =(2Z=3)(2Z=3)!1+cos2'1223 ;4 (1;cos ') d' =5 + 8 cos ' ; 4d' == 2 2200=(2=3)p(2=3)93+8 cos ';2 cos 2' d' = 3' + 8 sin ' ; sin 2' 0 = 2+ 2 3 :Z 0 !" #$"1.
* " !! : ? (c. (9.1)).2. < : , & & y = ln x " x = e, y = 0.3. < :, & + 4! 25 x2 + 9 y2 = 225 (c. 9.1).804.* : &" D, & " x = a, x = b & "") A a b ] y = f1(x) y = f2(x), & f1(x) f2(x)?(c. (9.2)).5. < : &", & & y = ex,y = e x x = 1.6. !!" ") (c. 9.1).7. * !" ! &" " "? C p&" " M (31 3 ), " "1 " " M (41 23 ), &" ".8. C8 ! x2 + y2 = R2 ") ) (c. 9.3).9.
* : & ! ") )? (c. 9.1).10. < : ! !!, & = 2 (1 ;cos ') (c. 9.6 (9.4)).11. * : &", & ", " !! = f1('), = f2('), 0 f1(') f2('), ' = , ' = , < ? (c. (9.7)).12. < : &", & ",p "p !+: ") ): = 2 sin 2 '1 = 3 ( 3 ).;2. 1. 5. e + ep1 ; 2:p7. @ M (31 3) " " = 2 3 ' = =6 pM (41 32 ) { &" " x = ;21 y = 2 3.p33 ; .10. 6: 12.6;81 10,& ': * / <@% 5 < # , ) ) . 2 ) ! .( ;AB (. 10.1). P!O n M0 = A M1 : : : Mn 1 Mn = B .P M0 M1 : : : Mn 1 Mn, # , ! [ n <, \ { ! # :;;[n =nXk=1Mk 1 Mk - \ = max fM0M1 - M1M2 - : : : - Mn 1Mn g :;;0 \ ! 0, ) #, n !.. 10.1.
10.2 10.1. 0 * [ n - ;AB \ ! 0 < ! ! , ;AB ! s. E, *, .', :s = lim0 [ n = lim0!! 10.1.nXk=1Mk 1Mk :(10:1); ;AB ( ( y = f (x), a x b, f (x) 082 : a b ], ;AB Zb qs=as1 + :f (x) ]2 dx:(10:2)0P! : a b ] n x0 = a < x1 < : : : < xk 1 < xk < : : : < xn 1 < xn = b (. 10.2) * :M0 (x0- y0) M1 (x1- y1) : : : Mk 1 (xk 1- yk 1) Mk (xk - yk ) : : : Mn 1 (xn 1- yn 1) Mn (xn- yn) yk = f (xk ) k = 1 2 : : : n. ( , # , :qMk 1Mk = (xk ; xk 1)2 + (yk ; yk 1)2 k = 1 2 : : : n: (10:3)( =)yk ; yk 1 = f (xk ) ; f (xk 1 ) = f (k ) (xk ; xk 1)(10:4) k : xk 1 xk ].( T xk = xk ; xk 1, (10.3) (10.4) qqMk 1Mk = (T xk )2 + : f (k ) (T xk ) ]2 = 1 + : f (k ) ]2 T xk (10:5)(k = 1 2 : : : n):( = max fT x0- T x1- : : : - T xn g \ = max fM0M1 - M1M2 - : : : - Mn 1Mn g :> (.
(10.5)), \ ! 0, ! 0, < (. (10.1)) # (10.5) .;;;;;;;;;;;0;;;;;0;0;s = lim0!nXk=1Mk 1Mk = lim0;!n qXk=11 + : f (k ) ]2 T xk :(10:6)05, , * $(10.6), q $ g (x) = 1 + :f (x) ]2 : a b ]. ' f (x) q , $ g (x) = 1 + :f (x) ]2 00083: a b ]. 9 , (10.6) *q 1 + :f (x) ]2 : a b ],..Zb qs = 1 + :f (x) ]2 dx:00a', ;AB s ( ) $ (10.2) . > 10.1 .p32 10.1. 1 y = 3 x p (3- 2 3 ).p> f (x) = 23 x3=2, 0 x 3, f (x) = 23 32 x1=2 = x, , $ (10.2), 0s=Z3 q01 + :f (x) ] dx =02Z3 p03 223=2 1 + x dx = 3 (1 + x) 0 = 3 (23 ; 1) = 143: 10.2.
$ ;AB x = ' (t), y = (t), t0 t T , ' (t) (t) & ' (t), (t) : t0 T ]. ' ;AB s 0s=ZT qt00:' (t) ]2 + : (t) ]2 dt00, ( (,s=ZT qt0:x (t) ]2 + :y (t) ]2 dt:0(10:7)0( ) , ' (t) 6= 0, t 2 : t0 T ]. 6 , ' (t) > 0 (' (t) < 0, ). > $ x = ' (t) * : t0 T ].9 , : a b ], a = ' (t0),b = ' (T ), * ! $ t = K (x). (< x = ' (t), y = (t) $ .00084y = f (x) = (K(x)), $dy = (t) . 2$ $ , : dx' (t) $ (10.2) x = ' (t):00vv012u dy !2TuuZb uZ(t)uuA ' (t) dt =s = t1 +dx = t 1 + @0dxa=' (t)t0ZT q :' (t) ]2 + : (t) ]200' (t)0t000' (t)dt =0ZT qt0:' (t) ]2 + : (t) ]2 dt:00> 10.2 . 10.2.
1 , x = R (t ; sin t), y = R (1 ; cos t), 0 t 2 (< - ) , ) R, * ! ) ).2 $ ! x y t : 0 2 ].t0x0 Ry01212;1R32RR2R232+1R2 R01 * (x y) Oxy , (.10.3). 2 ) s $ (10.7). >x (t) = R (t ; sin t) = R (1 ; cos t)- y (t) = R (1 ; cos t) = R sin t0000hi:x (t) ]2 +:y (t) ]2 = R2 (1 ; cos t)2 + sin2 t = R2 (2 ; 2 cos t) = 4 R2 sin2 2t 0085 2 Z q:x (t) ]2 + :y (t) ]2 dt =s=00u2 Z2 vZ2 utt2 tt2= 4 R sin dt = 2 R sin dt = ;4 R cos = 8 R:020202. 10.30.
10.4 10.3. $ ;AB ( ( = f ('), ' , f (') & & f (') : ]. ' ;AB s 0s=Z qf 2(') + :f (') ]2 d'0, ( (,s=Z q 2 + 2 d':(10:8)0' ) (. $ (9.3)), ;AB 8 $ ( ' ):< x = cos ' = f (') cos ': y = sin ' = f (') sin ' ' :1 x y 'x (') = f (') cos ' ; f (') sin '- y (') = f (') sin ' + f (') cos ':x (') ]2 + :y (') ]2 == :f (') ]2 (cos2 ' + sin2 ') + f 2(') (cos2 ' + sin2 ') = :f (') ]2 + f 2 (').(<, $ (10.7), .00000000s=Z q:x (') ]2 + :y (') ]2 dt =00> 10.3 .86Z qf 2 (') + :f (') ]2 d':0 10.3.
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