Определенный интеграл. Часть 2


    

  


  





  





    
       
  
     
   

 

   
    !"  # $ ! "

!"# $" $%"& !'& (& ')* +,- !  . $+ /"%"# $0 /$')  1  ( 21+ ! ' -* " 0+'34 5+"'6 "&7 #) 4 8 59 4 4 - $%"& 1"-   ' $:"*7 #" (-;  /"%"#  $( /$')  4 < 1%'" ! ( (0"  + "# $" +% ! '- $+ 2( ("# $"* '"%37 &=" & $ !*  ( 7 "%#) (& ')4 5 # & 1 + > "? $/ !+- !"// + :"')-; " " 0+')-; + "*  @ (&  04 A1+$' B C4

D 
 E4  ( +"# $" 1+"'6 "& 1+ ! ' 0 " 0+' 4444444444444 F -#"' " 1'=! * 444444444444444444444444444444444444444444 F -#"' " !'"- !0" 4444444444444444444444444444444444444444 G -#"' " ,H (  1(=)? 1+ ! ' 0 " 0+' 444 .. 5'=!) 1 +;" += "& 444444444444444444444444444444 . I4 ;"# $" 1+"'6 "& 1+ ! ' 0 " 0+' 4444444444444 .I   !+!0  +6& 444444444444444444444444444444444 .I +!"- : + &6 " 4444444444444444444444444444444444 .I , 1 + ( * "'- 444444444444444444444444444444444444444 F 9" ++ 44444444444444444444444444444444444444444444444444444444 J




8◦ 

   
   
 


% &'
 
( )


      

- % (7 # $ 1&"? 1+ ! ' 0 " 0+' 1+"!" %! #  ;6! "" 1'=!" $+"'" ** +1 :"" A(4 %!# .C4
$7 1+ !'" *  + ,
 y = f (x)
0  f (x)    [a, b]   

    


b S=

A.FC

f (x) dx. a


   
 D = {(x, y)

y

6

a  x  b, y = f (x) < 0}

 D∗
   D   
b SD = SD ∗ =

y = −f (x)

D∗


b (−f (x)) dx = −

a

f (x) dx.

-

O a

b x D

a

y = f (x)


    
    
 D = {(x, y) : a  x  b,

f (x)  y  g(x)}

 
 0  f (x)  y  g(x)           !
     "  

 "
"



     g(x)  f (x)

y

6

y = g(x) D y = f (x) a

O

b

-

x



#  
b SD =


b g(x) dx −

a


b f (x) dx =

a

A.JC

[g(x) − f (x)] dx. a

$ 
 f (x)  %!      &  ! % '  
     y = g(x)+c  y = f (x)+c c (         %
 
  f (x) + c
0 % x ∈ [a, b] y

y

6

6

y = g(x) D∗ D O y = f (x) b

a

-

x

a

b

O

-

x

) D∗   
%    
  D *  
b SD = SD ∗ =


b [g(x) + c − f (x) − c] dx =

a

[g(x) − f (x)] dx. a


   -
 f (x) % !   [a, b]    
 

       !  [a, b]
 Ox      y = f (x) 
%
%   
b S=

|f (x)| dx. a

#    


b f (x) dx

   
+
    

a


  &"     
+ Ox
b

a

6

S3

S1 f (x) dx = S1 − S2 + S3 − S4 .

a

y

S2

-

S4 b x




       

(+"(  $+? /$:"? r = r(ϕ)7 1+ ! ' ?  [α, β]4 ! ( #")7 # r " ϕ K 1'&+- $+!"- #$"4 0! '?,( ϕ ∈ [α, β]    r = r(ϕ ) "7 %#"7 #$ M (ϕ , r )7 0! ϕ 7 r K 1'&+- $+!"- #$"4 '" ϕ ,!  ( &)&7 21+, 0&3  ) [α, β]7  1 + ( & #$ M 1">   $+? $+"? AB 7 %!? + " ( r = r(ϕ)4 0

0

0

0

0

0

0

0

.  
 +/ , 
    ! 
%   

 %  %   ϕ = α ϕ = β    AB  !     %" 

 "   r = r(ϕ) α  ϕ  β  B

r = r(ϕ) A β α

-

x

O

1+ !'" ' !?=& *  +


 % r(ϕ)
0     [α, β]   

  
  
%
%    S


  

1 = 2


β

A.@C

r2 (ϕ) dϕ.

α

%,) (  ) [α, β] 1+"%')-( ,+%(  n # * α = ϕ < ϕ < ... < ϕ < ϕ < . . . < ϕ = β 4 5+ ! ( '#" ϕ = ϕ 7 k = 1, . . . , n − 14 0!  ) $+"'" *-*  $+ +%,) &  n L' ( +-; $+"'" *-;  $+7 1'=!) $+-; 1+",'" 6  + 1'=!" $+00  $+7 0+"# 0 '#(" ϕ = ϕ 7 -  ! 
 0

1

k−1

k

n

k

k−1



" !0* $+6" r = r(θ )7 θ K 1+"%')& #$  0(  , ϕ ]4 ϕ 5'=!) L' ( +0 $+" '" *0  $+ ,!  1+" 1 r (θ )∆ϕ 7 ,'"6  +&)& 2 θ ∆ϕ = ϕ − ϕ 4 $"( ,+%(7 r(θ C " + ?=&  1'=!)

ϕ = ϕk [ϕk−1

k

k

k

k

2

O

   k        ∆ϕk 

k

k

k

k

k

k−1

1 2 S≈ r (θk ) · ∆ϕk . 2 n

-

ϕk−1

< +   ( # 7 # ( ( )> ∆ϕ 4 ,%#"( λ = max ∆ϕ " +(+"(

k=1

k

k

1kn

1 2 σ= r (θk ) · ∆ϕk . 2 n

k=1

# "!7 # σ ) " 0+')& ((7 ' & !'& /$:"" 12 r (ϕ) "  "'  1+ +-" r(ϕ)  [α, β] L " 0+')& (( "(  $  #-* 1+ ! ' 1+" λ → 07  %"&="* "  +%," "&  0(  [α, β]
1  #"7 "  -,+ # $ θ M  +  2 r (ϕ) dϕ4 ' ! ')7 $# ') 2

β

2

k

α

S


  

# " + ,') !$%)4

1 = 2


β

r2 (ϕ) dϕ,

α

%% &'

' 

5) 1'$& $+"& AB %! + " ( y = f (x)7 x ∈ [a, b] " f (x) /$:"&  1+ +-&  [a, b]4 %,) ( [a, b] 1+"%')  n # * #$(" x 8 a = x < x < < x < . . . < x = b4 5+ ! ( 1+&(- x = x 7 k = 1, . . . , n − 14 ... < x k

k−1

k

n

k

0

1



5+" L( !0 AB +%,) &  n # * #$(" A = M , M , . . . , M 7 0

y6

Mrn = B

Mk−1Mk

 1+' ""  A $ B4  !""( L" #$" + %$(" 1+&(-; " 1'#"( '(? '""? M M . . . M 4 ,%#"( !'" L* '(* # + % L 7 !'" !0 %   # + % ∆ 7 λ = max ∆ 4 0

A = M0 r

xk−1 xk

b = xk

k−1

Mk , . . . , Mn = B

M1

O a = x0

1

-

x

1

n

n

k

1kn

k

.  
 +0 , % AB  ! 
%
%%  




       " 
"  + AB  
   λ → 0 .    !         !

  LAB  


A.GC

LAB = lim Ln . λ→0

*  + 
  % AB !     y = f (x)  f (x)  f  (x)

   [a, b] AB % %
%
%%   LAB


b  = 1 + [f  (x)]2 dx.

A.EC

a

%,) ( + %$ [a, b] 1+"%')  n # *7 0! AB +%,) &  n # * #$(" M (x , f (x ))7 k = 1, 2, . . . , n4 ( +"( '(? '""? M M . . . M M . . . M 4  !'" -  ! 


k

0

Ln =

n  k=1

∆k =

5  + ( 90+6

1

k−1

k

k

k

n

n  

(xk − xk−1 )2 + [f (xk ) − f (xk−1 )]2 .

k=1

f (xk ) − f (xk−1 ) = f  (ξk )∆xk ,

∆xk = xk − xk−1 ,

ξk ∈ [xk−1 , xk ],

n  n    2  2 (∆xk ) + (f (ξk )∆xk ) = 1 + [f  (ξk )]2 · ∆xk . Ln = k=1

k=1



5' ! -+6 " &'& & " 0+')* ((* !'&  1+ +-  *7 7 ' ! ')7 " " 0+"+ (*  [a, b] /$:"" 1 + [f (x)] 4 ?! - $ 7 # +(+" (& $+"& &'& & 1+&('& (* " !'" -#"'& & 1 /+(' 

LAB

2


b  = lim Ln = 1 + [f  (x)]2 dx, λ→0

a

# " + ,') !$%)4


    
 AB !      
 ⎧ ⎨ x = x(t)

,

⎩ y = y(t)

α  t  β;

A(x(α), y(α));

B(x(β), y(β)).


  x(t) y(t)    

  /01  & ! 
  LAB

[α, β]

' 


b  = 1 + [y  (x)]2 dx. a

#  !  "      y (t) dy =  2 dx = x (t) dt 
   dx

x (t)


β LAB =




α





y  (t) 1+  x (t)

LAB

2

x = x(t)



y  (x) =


β  · x (t) dt = [x (t)]2 + [y  (t)]2 dt, α


β  [x (t)]2 + [y  (t)]2 dt. =

A.IC

α


    
  % AB !  
%  %" 

 "

r = r(ϕ) ϕ ∈ [α, β]


  !  +

3 &

  !         

⎧ ⎪ ⎨ x = r(ϕ) cos ϕ ⎪ ⎩ y = r(ϕ) sin ϕ

,

ϕ ∈ [α, β].



'  !   /41   LAB


β  = r2 (ϕ) + [r (ϕ)]2 dϕ.

AC

α

   

⎧ ⎨ x = ϕ(t)

, t ∈ [α, β]4 (+"( !0 AB7 %!? + "&(" ⎩ y = ψ(t) 5) /$:"" ϕ(t)7 ψ(t)  1+ +-!"// + :"+ (-  [α, β]4 -,  + (  $+? 1 + ( ? #$ M ∈ AB4 *    %# " 1+( + t4 '" 1 + ( * !0" AM ,%#"( L(t) " 0' /+ (' A.IC
t  L(t) = ϕ 2 (τ ) + ψ  2 (τ ) dτ. α

$ $$ 1!" 0+')& /$:"&  1+ +-7  L(t)7 $$ " 0+'  1 + ( -(  +;"( 1+ ! '(7 ) /$:"& !"// + :"+ (& " L (t) =



ϕ 2 (t) + ψ  2 (t).


'" L2 (t) = ϕ2 (t) + ψ 2 (t) =⇒ [L (t) dt]2 = [ϕ (t) dt]2 + [ψ  (t) dt]2 =⇒ =⇒ dL2 = dϕ2 + dψ 2

"'" 1   



dϕ dL

2

 +

dψ dL

2

A.C

= 1.

$% + !'& 1'$"; !0 +1++& & "  1++ - !0"8 ⎧ ⎪ ⎪ ⎪ ⎨ x = ϕ(t)

y = ψ(t) ⎪ ⎪ ⎪ ⎩ z = g(t),

t ∈ [α, β].



'" /$:"" ϕ(t)7 ψ(t)7 g(t)  1+ +-!"// + :"+ (-  [α, β]7 
β  L= ϕ2 (t) + ψ 2 (t) + g 2 (t) dt. α

%%% &'
 23  (4   
 



     !

     "  #

(+"(  $+  ' T 4 %)( ( 1+"%') x " 1+ ! ( 1'$)7 1 +1 !"$'&+ " Ox4   # "" 1'#"( 1'$? /"0+4 5+ !1'6"(7 # 1'=!) L* /"0+- S(x) ( "%  " /$:"& S(x)  1+ +-   0(  [a, b]4 *! ( ,H ( L0  ' T 4 %,) (  0(  [a, b] 1+"%')  n # * a = x < x < . . . < < x < . . . < x 4 N + % $6!? #$ ! ' "& x 1+ ! ( 1'$) x 1 +1 !"$'&+? " Ox4 5+" L(   ' T +%,) &  1+ !  ' - L' ( +- '"7 -#"'"( 1+",'"6  ,H ( V !0 '&4 '& L0 -, + ( 1+"%')? #$ ξ ∈ [x , x ] " 1+ ! ( # + % L #$ 1'$)7 1 +1 !"$'&+? " Ox4 5'=!) 1'$* /" 0+-7 1'# *   # ""7 + S(ξ )4 ( "(  1 +) L' ( +-* '* :"'"!+(  " (7 1'=!) $+0 ) S(ξ )7 " -* ∆x = x − x 4 0! V ≈ S(ξ )∆x "7 ' ! ')7 0

k−1

k

n

1

k

k

k

k−1

k

k

k

k

k

k−1

k

k

V ≈

k

n 

S(ξk )∆xk .

k=1

5'# & (( )7 # "!7 " 0+')& ((7 ' & !'&  1+ +-* /$:"" S(x)7 x ∈ [a, b]4 ' ! ') L " 0+')& (( "(  $ #-* 1+ ! ' 1+" λ = max ∆x → 07  %"&="* "  1, +%," "& [a, b] "  -,+ # $ ξ 4
$7 $# ') k

k


b V =

S(x) dx. a

AC



$
%   #

5) /$:"& y = f (x)  1+ +- "  +": ')  [a, b]4 - #"'"( ,H (  '7 1'# 0  + %') += "& $+"*7 %!* + " ( y = f (x) $+0 " Ox4 # "!7 # L %!# &'& & #-( '# ( 6 +(+ *  14. %!#"4 y  *" ')7 1+ !& # + % y = f (x) 1+"%')? #$ x ∈ [a, b]  # " 1'$)? 1 +1 !" $'&+* " Ox7 (- 1'#"( b O a x $+07 +!" $+0 +  f (x) "7 %#"7 1'=!) L0  # "& S(x) = πf (x)4 $"( ,+%(7 "1')%& /+(' AC7 1'#"(
V = π f (x) dx. AC 6

-

U

2

b

2

Ox

a

1   '0"# (6 1'#") /+(' !'& -#"'

(  ' += "& $+0 " Ox8


d

VOy = π

x2 (y) dy.

"& ,H AFC

c

%5 
(    6 (

(+"( 1 +;)7 ,+%? += " ( $+"* y = f (x) A/$:"& y = f (x)
0  1+ +-!"// + :"+ (&  [a, b]C $+0 " Ox4 *! ( 1'=!) 1 +;" += "&4 #' -&"(7 # ,! ( 1"() 1! 1'=!)? 1 +;" += "&  '4 %,) (  0(  [a, b] 1+"%')-( ,+%(  n # * a = x0 < x1 < . . . < xk−1 < xk < . . . < xn = b.



y6

Mn

Mk

6!* #$ x    M #$ M (x , f (x ))  $+"*4  !"" #$" M 7 1'#"( ' (?7 1"?  !? $+" x x O a b x ?4 (+"( !'& 1+- ! %  '(* M M 4 5+" +=  y "" 0 $+0 " Ox 1'#"(   M ∆ M ∆y # -* $7 1'=!) 1 +;" ∆x $+0 + 2π y 2+ y ∆ 7 0! ∆ K !'" + %$ M M 4 N  x x x O + % P ,%#"( (( 1'=! * 1 +; *  ; $ Mk−1

k

0

k

k

k

k

-

k−1

k

k−1

k

6

k

k

k−1

k

k−1

k

k

-

k−1

k

k−1

k

k

k

n

Pn =

n 

2π

k=1

yk−1 + yk · ∆k . 2

.  
 +7


    Pn 

λ = max ∆xk → 0   !    +  "
   % 1kn

  !  P 

AJC

P = lim Pn . λ→0

*  + -
 % y = f (x)    

[a, b]     "
   % P
b P = 2π


%
%   

 f (x) 1 + [f  (x)]2 dx.

A@C

a

-  ! 


"( P

%,) ( [a, b] 1+"%')  n # *4 0' AJC

n

Pn = 2π

n  yk−1 + yk k=1

2

∆k = 2π

n  f (xk−1 ) + f (xk ) k=1

2

∆k .




# "!  ∆k = (∆xk )2 + (∆yk )2 =



 1+

1')% (&  + (* 90+68

∆yk ∆xk

2

· ∆xk .

∆yk = f (xk ) − f (xk−1 ) = f  (ξk )(xk − xk−1 ) = f  (ξk ) · ∆xk ,

0! ∆k =

$"( ,+%(



1 + (f  (ξk ))2 · ∆xk .

n  f (xk−1 ) + f (xk )  Pn = 2π · 1 + [f  (ξk )]2 · ∆xk 2

"'"

k=1

Pn = 2π

n  f (xk−1 ) − f (ξk )  2

k=1

+

ξk ∈ [xk−1 , xk ].

n  f (xk ) − f (ξk ) 

2

k=1

1 + [f  (ξk )]2 · ∆xk +

1 + [f  (ξk )]2 · ∆xk +

n 

 f (ξk ) 1 + [f  (ξk )]2 · ∆xk

 .

k=1

# "!7 # 1' ! '0 ( ) " 0+')& ((7 ' & !'& /$:"" 2πf (x) · 1 + [f (x)] 4 5$6 (7 # 1 +- ! '0 (-; + (&& $ '? 1+" λ → 04 $ $$f (x)  1+ +-  [a, b]7   0+"#    ( "7 ' ! ')7 1 + [f (ξ )] < M 4 $ $$ f (x)  1+ +-  [a, b]7   " +( +  1+ +-   (7 " 1L( !'& ∀ ε > 0 =   δ = δ(ε) $ 7 # |f (x ) − f (x )| < ε $$ ')$ |x − x | < δ, ∀ x , x ∈ [a, b]. $ $$ λ = max ∆x → 07  (6 #")7 # |x − x | < δ7  1( " 1! |x − ξ | < δ4 ' ! ')7 |f (x ) − f (ξ )| < ε " %#" 

2









k

2



k

k

k



k

k



k−1

k

n   ε f (xk ) − f (ξk )     2 · 1 + [f (ξk )] ∆xk   · M.  2 2 k=1

'0"# : " & " 1 + '0 ( 4
$7 ;!& 1+ ! ' !'& - +6 "& P 1+" λ → 07 (- 1'#"( /+(' A@C4  + ( !$%4 n



1   .C '" 1

+;) 1'#  += " ( $+0 " Ox $+" * AB7 %!* 1+( +"# $" + "&(" x = ϕ(t)7 y = ψ(t)7 α  t  β 7 1+"# ( ψ(t)
0 " a  ϕ(t)  b 1+" α  t  β 7 ϕ(α) = a7 ϕ(β) = b7  ! '& %(  1 + ( -;  A@C x = ϕ(t)7 "( (
β POx = 2π

 ψ(t) ϕ2 (t) + ψ 2 (t) dt.

α

C '" $+"& AB %!  1'&+-; $+!"; r = r(ϕ)7 ϕ ∈ [α, β]7 0! r(ϕ)  1+ +-!"// + :"+ (&  [α, β]7  L !"& $ 1+( +"# $( %!"? $+"* x = r(ϕ) cos ϕ7 y = r(ϕ) sin ϕ7 α  ϕ  β " %#"7
β

POx = 2π

r(ϕ) sin ϕ

 r2 (ϕ) + r2 (ϕ) dϕ.

α

(+"(  $')$ 1+"( +7 "''?+"+?="; 1&" 1+ !  ' 0 " 0+'4    *" 1'=!) /"0+-7 0+"# * '"" * 3x +2y −4 = 0 " )? Ox4 y 56 & $+"& ) 1+,'7 2 + " ' 0$ 1+" " $ "! −3x = 2(y − 2)7  )  +>"  #$ (0, 2)4 #$" 1 +   # "&  )? Ox√ *! ( "% + "& 3x − 4 = √ √ 2/ 3 x −2/ 3 0 0 −→ x = ±2/ 34 2

6

2

2

-

√ 2/
3

S=

√ 2/
3

y(x) dx = √ −2/ 3

√ −2/ 3

√ 2/√3 x3 2/√3 16 4 − 3x2 3   dx = 2x √ −  √ = . −2/ 3 2 2 −2/ 3 9

   *" 1'=!)7 0+"#

? 1 +* +$* +$* :"$'"!-



⎧ ⎨ x = a(t − sin t), ⎩ y = a(t − cos t)

y

" )? Ox4

6

2a 0

56
% +

2πa

-

x

"& $+"* dx = a(1 − cos t) dt4 5 +& +$ :"$'"!   "%(  "? 1+( + t  0 ! 2π4 ' ! ')7
2π
2π S = a2 (1 − cos t)2 dt = a2 (1 − 2 cos t + cos2 t)dt = 0

= a2


2π

0

dt − 2a2

0


2π 0

a2 cos t + 2


2π (1 + cos 2t)dt = 0

2π 2π 2π a2 2π a2 a2     2 2 = a t −2a sin t + t + sin 2t = a 2π + 2π = 2πa2 . 0 0 0 2 0 4 2 2

  - *" 1'=!)7 0+"#

? $+"* r = 4(1 + cos ϕ)4 56  "' "(( +"#" /"0 +- " ') " Ox7 ,! ( -#" '&) 1'=!)  +; * #"4 5'=!) %$'?#  ( 6! !(& '#(" ϕ = 0 " ϕ = π4 1 S = 2 2


π 0

r2 (ϕ)dϕ =


π

16(1 + cos ϕ)2 dϕ = 16

0


π

(1 + 2 cos ϕ + cos2 ϕ)dϕ =

0

  π 1
π π   (1 + cos 2ϕ)dϕ = = 16 ϕ +2 sin ϕ + 0 0 2 0

 π 1 π 1  = 16 π + ϕ + sin 2ϕ = 24π. 0 2 0 4

  / *" !'" +"!- x2/3 + y2/3 = a2/34



"// + :"+& + " +"!-7 1'# ( y = −y /x 4 5L( !'" !0" !* #  +" +"!- -#"'& & 1 /+(' 56



1 L = 4

1/3

y

1/3

ra


a 

r

y 2/3 1 + 2/3 dx = x

0


a  =

+ x2/3

x2/3

0

y 2/3


a dx = 0

L=




2π 

a2 (1

−

cos t)2

+

0

a2 sin2 t



=

0

  7 *" !'" 

⎧ ⎨ x = a(t − sin t), ⎩ y = a(1 − cos t)

4

7 1L(

dy = a sin t dt


2π = 2a

a r

3 a dx = a. 2 x1/3

= a(1 − cos t)7 y ( x = dx dt

r

0

1/3

  0 *" !'" !* +$" :"$'"!-

56
(

6

t 2π t sin dt = 4 cos  = 8a. 2 2 0

* $+"* r = a sin ϕ3 4 56 & $+"& 1"- & # $* (ϕ, r) 1+" "%(  "" ϕ  0 ! 3π4
( ( r = a sin ϕ3 cos ϕ3 7 1L( !'"  * !0" $+"* 

3

2


3π 
3π 3πa γ ϕ ϕ ϕ . L= a2 sin6 + a2 sin4 cos2 dϕ = a sin2 dϕ = 3 3 3 3 2 0

  + *" ,H x2 + (y − b)2  a2

0

( +7 ,+%0 += " ( $+0 Ab
aC $+0 " Ox4

-

x



y

56 %+

>"( +√ " $+0  "√ ') y8 y = b − a − x 7 y = b + a − x 4 5L( 2

1

2

2

r

6

b

2

2


a 

y2

6

y1 −a O

VOx = π

6

a

?

?

w

-

x


   2  2  2 2 2 2 a2 − x2 dx. b+ a −x − b− a −x dx = 4πb a



−a

−a

! ' ( %(  1 + ( -; x = a sin t4 0! dx = a cos t dt4 5 +  #" ( 1+ ! '- " 0+"+"& x = −a −→ t = −π/2M x = a −→ t = π/24 ?! "( ( VOx


π/2 
π/2 = 4πb a2 (1 − sin2 t) a cos tdx = 4πa2 b cos2 tdt = −π/2

−π/2


π/2

2

= 2πa b

π/2 π/2   2 (1 + cos 2t)dt = 2πa bt +πa b sin 2t = 2π 2 a2 b. 2

−π/2

−π/2

−π/2

  8 *" 1'=!) 1

+;"7 ,+%* += " ( $+0 " Ox 1 '" $+"* 9y = x(3 − x )4 y 56 '&  +; * #" $+"* 1+" √ 1 0  x  3 "( ( y = (3−x) x4 ?! 3 x O !"//+ :"' !0" 3 x+1 d = 1 + y (x) dx = √ dx4 0! x 2

2

6

r

2


3 POx = 2π 0

π = 3


3 0

√ x+1 π 1 (3 − x) x · √ dx = 3 3 2 x

(2x − x2 + 3) dx = 3π.


3 (3 − x)(x + 1) dx = 0

-



9◦ 

6   
   
 


%        

5)   0(  [a, b] +1'6   !+!-*  +6 )7 '" * & 1') $+0 ρ(x) ) /$:"&  1+ +-&  [a, b]4 1( "(7 # ρ(x) = lim ∆m/∆x7 0! ∆m K ( #"  +6&   0(  [x, x + ∆x]4 %,) ( [a, b] #$(" a = x < x < . . . < x = b7  $6!( #"#(  0(  [x , x ] -, + ( 1+"%') #$ ξ " 1+ !1' 6"(7 #  L(  0(  1') )  '"#" 1&&7 +& ρ(ξ )4 0! ( m L0  0(  ,!  +&)& 1+",'"6  1+"%  ! "? ρ(ξ )∆x 7 0! ∆x = x − x 7  ( m  0  +6& 1+",'" 6  %1"> & $8 m ≈  ρ(ξ )∆x 4 '& ;6! "& #0 %# "& m ! 1 + *" $ 1+ ! '  1' ! ( +  1+" λ = max ∆x → 04 $# ') 1'#"(
m = ρ(x) dx. AGC ∆x→0

0

k−1

k

1

k

n

k

k

k

k

k

k

n

k−1

k

k

k=1

1kn

k

b

a

( "(7 # L " 0+' =  7 $ $$ ρ(x)  1+ +-&  [a, b]4

%% !  ' " 

.C (+"( #'  1'$" Oxy " ( ( +"')-;  # $ A (x , y )7 A (x , y )74447 A (x , y )7 (- $+-; +-     m , m , . . . , m 4 ,%#"( # + % (x , y ) $+!"- : + &6 " L* " (-4 1')% (& "% -( /$( "% $+ ( ;"$" " % 1"> ( ' !?=" +  1

1

1

2

1

2

2

n

2

n

n

n

c

c

n 

mk x k m1 x 1 + m2 x 2 + . . . + mn x n My k=1 , xc = =  = n m1 + m2 + . . . + mn m mk k=1 n 

mk yk m 1 y 1 + m 2 y 2 + . . . + mn y n Mx k=1 yc = =  = , n m1 + m2 + . . . + mn m mk k=1

AEC



0!

My =

Mx =

n 

n 

) "# $"* ((  " (- " ') " Oy7 K "# $"* ((  " (- " ') " Ox7

mk x k

k=1

mk yk

k=1 n 

4 "6 (- "1')% ( L" /+('- !'& ;6! "& $+!" :  + &6 " +%'"#-; /"0+4 C  + &6 " 1'$* $+"*4 5) $+"& AB %! + " ( y = f (x)7 0! f (x)  1+ +- !"// + :"+ (   0(  [a, b]M " 1) L $+"& ) ( +"')& '""&7 '" *& 1') ρ )  '"#" 1&& A$+"& ! +!C4 %,) ( $+"? 1+"%')  n # * #$(" A = A , A 74447 A = B 7 !'"- !0 A A 7 A A 7 y A A 4447 A B +-    B = A P ∆s 7 ∆s 7 4447 ∆s 4 - m L"; A A  C(x , y ) !0 ,! +&)& 1+"% ! "? y !'"  1') ρ8 m = ρ∆s 4 A =A x x  $6!* !0 A A -, + ( m = ρ∆s 4  $6!* !0 A A -, + ( 1+"%') #$ P [ξ , f (ξ )] " ,! ( #")7 # ( L* !0" ρ∆s + !#   #$ P 4 5+" L( $+"? AB (6 1+",'"6  %( ") " (* ( +"')-; # $ P , P , . . . , P  +1'6 -("  "; ((" ρ∆s 7 ρ∆s 7 4447 ρ∆s 4 ?!7  "' +  AEC7 !'& $+!" : + &6 " 1' $* $+"* (x , y ) 1'# ( ' !?=" 1+",'"6 - + 8 m=

mk

k=1

0

0

n

6

r

k−1 r k

2

r

1

c

1

2

n

k

k−1

k

k−1

k

k

k

k

k

1

2

2

k

-

c

k

1

c

c

0

1

n−1

n

k

r

1

k

k

k

1

n

2

n

c

n 

c

ξk ρ∆sk

xc ≈ k=1 n  k=1

ρ∆sk

n 

n 

ξk ∆sk

= k=1 n  k=1

, ∆sk

yc ≈

n 

f (ξk )ρ∆sk

k=1

n 

k=1

= ρ∆sk

f (ξk )∆sk

k=1

n 

k=1

. ∆sk

$ $$ /$:"" f (x) " f (x)  1+ +--  [a, b]7   ((-7 &="  #"'" ' " %(  ' , "; !+, * "( ? 1+ ! '- 1+" λ → 07 +



1+ ! '(  ?="; " 0+')-; ((4 $"( ,+%( 1'# ( $# ') b xc =

yc =

x ds

a

b

a b  a

b  x 1 + f 2 (x) dx =

a

b

ds

a

f (x) ds b

= ds

a

 1 + f 2 (x) dx b a

b  x 1 + f 2 (x) dx =

a

 f (x) 1 + f 2 (x) dx b

 1 + f 2 (x) dx

;

S b =

AIC

 f (x) 1 + f 2 (x) dx

a

S

,

a

0! S K !'" $+"* AB4

% /+('- !'& y 1'# (7 # S · y = (6& , #" 1' ! 0 +   2π7 c

c

b

 f (x) 1 + f 2 (x) dx

"'"7

a


b 2πyc · S = 2π

 f (x) 1 + f 2 (x) dx,

a

0! 1+& #) ) 1'=!) 1 +;"7 1'# * 1+" += "" $+"* AB $+0 " Ox7  2πy ) !'" $+6" +!" y 4 $"( ,+%(7 "(  (  c

c

*  + /     
       "
 

        
         
  + 
 
  &   & 

    & 
 
     %&
    &           *" $+!"- :

7

4

x2 + y 2 = a2 y > 0

+ &6 " 1'$+6"



56
(

( y = √a

7

− x2 y  = − √

2

a xc = a

a −a a −a

√ x dx a2 −x2 √ dx a2 −x2

a √ yc =

−a

7

x a dS = √ dx a2 − x2 a2 − x2

4 0!

a a2 − x2 −a 0  = 0; =− = a πa arcsin xa −a √

a2 − x2 √aa2dx −x2 πa

=

2a2 2a = . πa πa

C  + &6 " $+"'" ** +1 :""4 $ ,-#7 +%,) ( $+" y '" *? +1 :"?  L' (  y = f (x) +- +1 :""  " ( ∆x = x − x " %( "( $6!? P $? +1 :"? 1+&(0')"$(  x ξ x b=x x a=x  ( 6 " ( " -* f (ξ )7 0! ξ + !&& #$ [x , x ]4  1+&(0')"$ + ρ · f (ξ )∆x Aρ ) 1 +;& 1')7 4 4 (7 1+";!&=&&  !"": 1' =!"C4
% ( ;"$" "% 7 # : + &6 " 1+&(0')"$ ' 6"  #$ 1 +  # "& 0 !"0' * "7 ' ! ')7 $+!"- :  + &6 " k0 1+&(0')"$ +- ξ 7 f (ξ )4 ! ( #")7 # ( L0 1+&(0')"$ ;!"&  #$ P (ξ , f (ξ ))4 $"( , +%(7 +1 :"? (6 1+",'"6  %( ") " (* ( +"')-; # $ P , P , . . . , P  ((" ρf (ξ )∆x 7 ρf (ξ )∆x 74447 ρf (ξ )∆x 7 "7 '  ! ')7 "1')%& +  AEC7 1'#"( 1+",'"6 - +  6

k

k

r k

k−1

0

k

k

k−1

k−1 k

k

k

1 2

k

k

k

1

2

1

n

n 

xc ≈

n

k

1

2

n 

ξk ρf (ξk )∆xk

k=1 n  k=1

ρf (ξk )∆xk

=

1 k 2

k

2

n

ξk f (ξk )∆xk

k=1 n  k=1

, f (ξk )∆xk

n

k



yc ≈

1 2

n 

1 2

f (ξk )ρf (ξk )∆xk

k=1 n 

n 

k=1 =  n

ρf (ξk )∆xk

k=1

f 2 (ξk )∆xk . f (ξk )∆xk

k=1

5 + ;!& $ 1+ ! '  1' !"; + ; 1+" λ → 07 1'# ( $#  ') b

xc =

b

xf (x) dx

a

b

=

xf (x) dx

a

f (x) dx

S

;

yc =

a

1 2

b

a b

f 2 (x) dx = f (x) dx

1 2

b

f 2 (x) dx

a

S

,

AC

a

0! S K 1'=!)  * +1 :""4 $ "  1+ !-!= ( '# 7 "% +0 +  AC "( ( +  
b

f 2 (x) dx,

2πyc · S = π

$+ /+('"+ & $

a

*  + 0 &    
   78   % 

      
 + 
 
  &   & 

           &     &
 
     %&
      



FC  + &6 "  !+!0  +6&4 %,"&7 $$  0!7  0(  [a, b]  #" A(4 OC7 ( m #"  +6& [x , x ] *! ( 1 /+(' AGC k

k−1

k


xk mk =

ρ(x) dx = ρ(ξk )(xk − xk−1 ) = ρ(ξk )∆xk xk−1

A%! ) (- 1')%'")  + (*  + ! (7 ξ  $+& #$  0 (  [x , x ]C4 ! ( #")7 # L ( + !#  "(   #$ P (ξ )4 $"( ,+%( (- ) 1'# ( " ( # $   +6 k

k−1

k

k

k




 ((" ρ(ξ )∆x , ρ(ξ )∆x , . . . , ρ(ξ )∆x 4 6!& $$  1+ !-!="; 1$;7 (6 ( %1")

P1 , P2 , . . . , Pn

1

1

2

n 

2

n

n

ρ(ξk )ξk ∆xk

xc ≈ k=1 n 

ρ(ξk )∆xk

k=1

"'"7 $# ')7 1 + ;!& $ 1+ ! ' 1+" λ → 07 1'#"( b xc =

xρ(x) dx

a

b

ρ(x) dx

a

Ax K $+!"- : + &6 "  !+!0  +6&C4 c

%%%  2  ) 
'

5) ( +"')& #$ M !"6 & 1  $+* 1+&(* OS 1! ! *" ( "'- F 7 1+' " $+* 1!   1+' " ( !"6  "&4 1+ ! '") +,7 1+"% ! ? "'*7 1+" 1 + ( = "" #$" "% 1'6 "& s = a  1'6 " s = b4 '" F 1&7 7 $$ "% 7 +, A ;!"& 1 /+(' A = F (b − a).

5+ !1'6"(7 # F %""  1'6 "& #$"7 4 4 F = F (s)7 0! F (s)  1+ +-  [a, b]4 %,) ( [a, b]  n 1+"%')-; # * [s , s ]7 -, + ( 1+"%') #$ ξ ∈ [s , s ] " ,! ( #")7 #  [s , s ] F (s) 1& " + F (ξ )4 0! +, A ,!  1+",'"6  + k−1

k

k−1

k

k−1

k

k

k

A≈

?! 1+" λ = ;!"(

n 

F (ξk )∆sk ,

k=1

max ∆sk → 0

1kn

∆sk = sk − sk−1 .

A#"-&7 # F (s)  1+ +-  [a, b]C
b

A=

F (s) ds. a




 P.Q
        - ( ("# $0 '"%4 K 48 $7 .I@G4 PQ    ->& ( ("$4 K 48 ->& >$'7 .II4 PQ     !#" " 1+6 "& 1 ( ("# $( '"% A!'& %C4 K 48 + ') 7 4 PFQ 
        !" #  $!%     & ' 5+"'6 "& 1+ ! ' 0 " 0+'4 !"# $" $%"& " "!""!')- %!"& !'& !  . $+ /"%"#  $0 /$') 4 K  R8 59 7 .IIF4