Área e volume: a transição da noção de medida à de área e de volume

!"#$

% & ' & ( & ) * + ', - . /

0

+ ' 123 4 5'6', 7 86' 9 : ;< ( 2=>,8& ?

1 0@?%?==

!" #$

%&' () * +

,* -( .!(.!/ * 0 () * %&+!" #$'1234

56+- +

7(- &(

8 .!9 ( :;(* <"= > ? @'( A( ' A( B

3+>+1+(A( 9%:5+ C( + 4+ ++ &('++(* D"= > ? D+( A( ' A( B ++EC +

% & ' & ( & ) * + ', - . /

0

+ ' 123 4 5'6', 7 86' 9 : ;< ( 2=>,8& ?

89

0@?%?==

:A

1

0B?%B?8)

: 3488

0@?%?007C'=5

:A

%' )= &, ,& '=, '& '&=? & ( ,= , = ? &' & ( =- ' ' %, & + & & 0= ?

. & . % ( '= -? . &, , ( 0 &' '= 0 '., & 0' & '&' C . ' ' ? & & ', &', 9&'=,0 ' '+ ?

&0 =(& =9&6',& & & ' '=' ? .' ) &0,( '=' 9&6'?

'. ' 12, & & ' , & 9 &'& & '. '= 3 0+' .+ ?

0, . ( . 0 & & = ' )*&' (=-?

1)- &'& )= '' '.6'&' &*. &?& 53E(0( 6' . + '.6' & ' &*.? ' & & %=( ('.6'&' . ,.,( ?0& . & 0' '.6' & ' . &*.'+?

3'= ??8.6'&' ?

! "

#

#$

% &' (

! " # ! $ " $# %" $% &" $ '()**) * + ,')*- *./ $. $ $0)$'$*$# %12! %, %*)3*-$- $ 4*%5 %%6*78 9$%+ %%%*9$$* %: %&'7 - /' $ ;'/ < $ 4*& &1 &, 6 =<&#

D

1&)- )=0'' + 9 6' ' 0F + '- &*. & &' ?

4 C'' '.6'&' &*.&, ''.6'&' &*.) (.+ &*. ( '. & ' , &' & 53E&*.&+, '. & ' ? . . &. & & , '. & ' G & 9 &53E&&?&. '&)& & + 'H) '. ' + ' #I"", ( & 6 &) ) ? #I"! '= 9 %= & . &. ,0 (') . '. & ' ? 1 & '= %= ' ( & '. & ' & .? +& '&* ! )=?

2 & . & 0'

'.6' & ' . &*. ' +? . 0/ & '' + &*. & 0'&?

I

!

"

#

$

#

; => 0. &' J""" , &' .' CF.*',&' ) =&&'&+3 ?

. .,' C.*'0')'?.

;H /> ' & . = .. H/ , . . ) .' ? 8 F &' && .*'? , , '+ )==&'',' )& '' ),,' ,. '?

, . . & = &. & ' ) ( '= ,(, C,& = (0 ' ?% && C= ( .& K'?& ( , ) , C= ' && &3 - & +'? ,&' & ' .' 6 9 0, '' ( = ) ? 1 0C, , ' ' 0 ( &. ) .?

#"

, &F '& '= .,( '& K.?

& ( 600 , F..0 ',&)'&&' 30*' ?

8 00 ,' =,'

0.#?#

2.#?#

&= ( .' ) ( = && ? && + , ' K. K.)?,-) K.'& K.*+. K.,) 0.#?!?

2.#?!

( '=' 0/ & '' + K. K. (+.,&

N

! ?

N

! ?

N)?

,'&' '&& +('=' 0/(& =G'' ' '63?

'& 0. & ., ' 0. #?O, ( '& '&& )?

##

2.#?O

' '&('='0/,( = 6 '?&F '0 '&'& '& & ('& 0. = =0(& '& ?

P & '& &*. &*. , '. , .0' ( &*. & '& . , & '=Q&*.'. &' Q?

",- ?% 0. R# R! '= '. K.

'& 6 ', ''R#R!?

?&*.R&'& &*.R#,R!,RO,???,

RL0.#?$M, ' RNR#SR!SROSSR

2.#?$

",- ?%&*../'. &' ' 7T, &'& <0 &*. 0( & ' &*.?&' ,

7N7#S7!S7OSS7,N#S!SOSS

#!

1&/9 0'9&&*.'. &' ? 00? K.'. &' K.?

% % K.58,' K.').

K.L&9&,58M%2&)&&& H L58M?'. &&'H,) 0. #?J? 1 K. H '. K. 5H% 82 &' ?

2.#?J

K.58 K.58%'. &' ?

$%

#

( 0 , '' '.6' & ' & C&'& =&*.? ) ( & &+ ' & & 0C '&F = &*.? 0C &' ?.9&''' <'&*.,-,00'- &*.? -1 '- &*. & '? : 0 01'=,- 2 0C. &&

4? LRMV"L M?

44? R#R!'. &' , LR#MNLR!ML4 K'M?

444? RNR#SR!SSR, LRMNLR#MSLR!MSSLRML M?

4? (&R#,LRMN#?

#O

9 6' ' 0 +, ' &*. R '& &*. R#, R!, ???, R? && 444 L M

,LRMNLR#MSLR!MSSLRM?

: C('&*.&&'& K. K. + &*. + & + K. K.?

8# ! 0F +? &*. '&

K.RNR#SR!SSR,

#LRMN#LR#MS#LR!MSS#LRMN!LR#MS!LR!MSS!LRMN!LRM

,#!.?

8 RLM K. . '& ' 0 + RLM? 4 , ( LRL MM N ' &?

-LRLMMNLM0 + ,

3RLMNRLM L,MNL,M?

3RL,M'& RL#,MRL!,M,(#S!N,&'&&44

L K'M) L,MNLRL,MMNLRL#,MMSLRL!,MMNL#,MSL!,M?

4 ( 0 LM 0C . 'F 4XL& MLMV"LV"M

44XL K'MLMNLM

444XL ML03+MNL0MSL+ML0+V"M

4XL#,#MN#

0+0 0C'F4X,44X,444X4X, ,L,MN & ?

% % ( &?

#B? (LMN&< ? < , ,&&& ,

LMNL,)M &, LMN

, ???

L MNLMNLMNLMNLM : C(L#,#MN#, LMN?

!B? (<', LMN -4

# ₄

#$ LMN # , N # # , # , L#,#MN #

##, # N#

# # , #

8( , LMN

#

_L#,#MN

OB? (Y5Y5 LMYL55M 3NXN6, /) ?

3ZXN6, L6MNLSL6M,MNLMSL6MVLM 3N6Z6, L6MNL6MVLMNLM

3Z6,Z6, L66MVL6MVLM

$B?% (<& , L,MN -<& ,&'<& [,9 <' #,!,#,!, (

[Z#YY!ZS[[Z#YY!ZS[

7. L##MYLMYL!!ML[ML[MZ##YLMY!!ZL3[ML3[M

,3[L3A[MZLMZS[L3S[M

8[&((<& &(, [",) LMN?

,0+9 , + K.

.& '& .?

8 K.'. &' K.,). K. K., .

07?0+9 , + K.& L8*+

% ) (+ K.&'=),& ' K. 58 L) 0. #?\M H#, 5H! 8HO

K. 58, 0' ( 5 9 8HON 8 9 5H!N 58 9 H#? (

K.5H!8HO=

O

! (9

( 9

,& 598HON895H!?1

#J

2.#?\

,& ?0F+#!, #LRMN!LRM

& K.R?8'&*.&'& K. K. 0+9 & 09' ?,' 00& ?

% ) ) ( + &*. & . '=?4 0 &(+ ' 9?

8(+ 58%?W (58%&'& K.&.5%&.8?8& '.,(+ +'& ( K.,' 0.#?]?

2.#?]

(#N^#S^O,!N^!S^$,ON^#S^!$N^OS^$?

% (L#MSL!MNLOMSL$M,&'

..

L^#MSL^OMSL^!MSL^$MNL^#MSL^!MSL^OMSL^$M?

+ (+ & '&? 0 &&*.0 = ?

#\

& K' & 0?8& ( R, ( . ' ( &, ( + LRM N # ( & && ? ( &*. R '& R#, R!,???,R? RN R# S R!SS R? ' R L N #,???,M

'& ' K.9&&RNR#SR!SSR,

'- K.R-'N#,!,???,N#,!,???,' .R

& ,

LRMNLR##MSL_#!MSSL_##M SLR!#MSLR!!MSSL_!!MS

SLR#MSL_!MSSL_MNLR#MSLR!MSLROMSSLRM

% &&?

, 0 + :%8 & R &*. 0? 8 ,=+= 0+?, 9 6' '0+?

$&

#

!'

'

K' 0+, &*. '. & ' &+?+( '*&'G& & ?%0 #DO! 2_`0.. 5 , #DOO, ? E & .

0;L53EM&*.&+, '. &' ?

, &' & . ?.00?-R7&*.,RNR#SR!SROSSR7N7#S7!SS7

'&FR7,&' ?& 40R7'. &

' R7'. &' ?

)0'.6'&' ?

.0+'.6'&' ( 6'?

% ?&& 09 '?% & ?

#]

87#,7!,???7'&7,(7N7#S7!SS7

&*.R(&' *R#,R!,???R('. '

& 7#, 7!,???, 7 '& 7? R N R# S R! SS R, ' R7

LN#,!,O????,M

8 7a#, 7a!, ???, 7a ' '& 7 (

7N7a#S7a!SS7a&*.(&' C&#,!,???,

( '. ' & 7a#, 7a!, ???, 7a '& 7?

N#S!SS'7a--LN#,!,???M?

8'&7) &')'&F7 '',-,7+0&& 7-N77a-LN#,!,???,b-N#,!,???,M?

8*'7'' *' ,)

'& 7? '& 7 =+ '&

'& R?% ,) 0 cRd'&cRd

R?

,'*'7'' *',

) 3 '& 7X? '& 7X '&

'&?% ,) 30 c-d'&cd

?

R--R'. &' ?

.07:&.'. &' K.'&

=C K..'& )&.'& '. &.?

% 8 &. 58% K. a5a8a%a )%a8a.)%8 a%a. %HL) 0.#?DM?

2.#?D

#D

'. a%a? &. 58% '. & ' K. a5a8a%a?

2.#?I

1&/90 53E0 '& ' ?

.0;?% K.'+.'. &' ?

& ', (( K. '. & ' K., '-'& =C .#?

? - 58% 2H K. ' + . 5 ? 8 . H '= & 7 ( '& 7.5L 0.#?#"M?

#?O,&.2R7'. &' '2H?: C ( + K. 58% . &. 2R7 '& =C 58%.'& )2R7, 58%2R7 .? ,&#?!58%2H'. &' ?

2.#?#"

. 5E?

- R# R! &*. (LR#M NLR!M? 8 R# &

'& e#, e!, ???, e, -, R# N e#S e!SS e? #?#, '

K.eLN#,???,M'. &' ' K., R#N#S!S

#I

2.#?##

. 0;''. &' "

"

# '

'& '.#?#?#R#R#"'. &' ? ( '& =C R"

#L#M?1) 0.#?#!? LR#MNLR_#"M

?

2.#?#!

&' &' & R!, ' * K. R"_! (

'. &' R!&'& '.#? LR!MNLR

" !M?

8LR#M N LR!M '& =C R#" . '&

=C R"!, ''* ( R#"f R"! & #?#, R# R! '. &

' ?

$(

'

") *

+

*

,'

.,'F 6 +&*.? ,

'&0+ 0.&G

!"

8=, ( '*', 0 '&C '' + <, 0. . ' & ' '' ? 1) 0. #?#O(0' * . &

#?8 /3 K. K.58

!?8' 1L& 5M 3'*'1N15? O?8' %L& 8M 3'*'%N%8? $?8' L& 58M 3'*'8N5?

2.#?#O

0. <+7#7!?

+<7#S7!,. K. K.58?%0 ,) (

+<&) 3+'*' K 885 K.58) 3 +'*' K 5?

7#S7!NL'*'8MSL'*'85MSLe58MAL'*'5M

+ '*' K 8 58 . + '*' K 5 L +., & ' K. 58 K.,885' 5=& M?7.

7#S7!NL'*'5MSLe58MAL'*'5MNLe58M

1) LM& +?

H&/' & + < 0 =+LC +&*.= .( C=M?

!#

+0. &= ' )& &&( .. ' ( &* ' ( '- + . '*', &.''&? ' , &* ' +'*' C ?' ( &)0 < 00 ) '#I?:CF&( & & ' &) &( '+ & ' C<gL<'M?

' ' & '' + '*', ' 0.#?#$?1'*' ''?8' '*') 0.&/9&.?h( ' C,)&.&9 '& ''06' &. &9 '*'?

, (+'*' i?

2.#?#$

9 &''+'*',&9&,' &..'*'L0.#?#JM?

2.#?#J

!!

'. ) ,''+L&M&*..0

L-,M,)

! ? ? ?

?&* LM&*.,&

! ? 1

% '*', , ' &*. . ' '*',''' <+&*.L&M&9

+ '*' L'M, ' &* &9 '&

''06' K.&9'*'?

1

!

'N

N !

? 1

N !

!

N_?!

'N?!

(''+'*'' '*'&*. .&''' '*''&*.,) 0.#?#\?

!O

C =9+., &'& &', 0 , '=. &*. I\ ? ., '' + &*. ( )' 0 + '*' + &*. 9 ( )' &? &', '=. &9&+'*'( ,. , ?

P ' + 9 '*' 0C '+?1 + '..90& + '(&'&' & ' + '*'? 1 9 '= 9 ? ' /+'*', ) '&K?

!$

%

-)

( +) ')? :') ' . '&' ')' #?/&'& ')&C, ( / '4'C &0/'+' )' ., & 9&, ' )' . '& , . L, < M + '& ') &, & )' .+?

'0C&'' /(&'& ')&G

' C.*&'-+'' ,=+=, ./?1 '.*&' = 0''' ,'' ''0C=-'& +)& ?

.(,%/' 0&'( '

#jO'') .,( &K#jO &') ? 0F0 )/9,(& 9 ?

!J 2.!?# +)&) )&K0.= C= N

_{,( ) ) &}

&K )&?+)(C +0.= (C=? , ( '&+

! '

'. , &

# ! ! ! ! '

O # !

P LM, (&9

O # M O # L' ?

!\

1'' '.6'&' &&00= '.6' & ' & &*.? 0 & & ) 0 ' && = 0 +, -, & , K' ,,, ')&0'# ? & K' ''*(&'. &' & ? '&* 0''*&' ,-, 9 &5E&&?

&. . & & C ' l4l & #I""0& &H) 8.4 ' + ''= ' &) H) ? : , '= 9 %=, & . &. ?&' ,%=& . +0L%=M : . # ') & '. &' ?

2.!?!

& %= ) C ' ? 0 & 0F ? 4' 0 )0F && ?

%$%$$./ 0

%0-'- <?:0,'=

0 & 9, L9SMNL9MSLM?

1 ,- +0-0 ? L<=8N<L=8,& <

'<=?

!]

, L 9 S L=MM N L 9 M SL3=M, , L"M N L 9 M SL3=M, & L3=MN3L=M? ) && &<V "? 0

&< ?2& &0 ?1&&/) L=8NL=8=? LL3#M=8NL=3=84L9MSL=MNL=83L=MNL3#ML=M?

8 (* ? L=84 = # 4 = = 4 # _L =8

0C,-<4

,< ,

L<=8N = 4 = # 4 # L=84 L=8

( 2 +09 (LMN& '?

% 5 C&& &=N#?8'(L

L0M? ) 4L0M?

1 .+ C %=?

++?-m#,m!,0+ (

! #

<'? 9 0 0(Lm#MN#L>!MN!?

0 ' ' ' ? 9&, -m#N#m!N !' ! <', 9 0

(L#MN"L ! MN#? 0,&&L=MN=

NL#MN" !NL !MN#?

P' ?

9 9&*' & ' 0 ( 0 &&'? ,0,0 ,9 ?

%$%$%$0

0 C

%=? 0 0(& '.6'&' &,-,&'. &' , & .?

0 '' K. ? - k# k! & .

!D

k# k!

'= K. L 0. !?OM? m K. ,

K.'& )K.?

2.!?O

%&,9 K. 0'( '& ?1K.''&K. &? -R&,m#,m!,???,mK.#,!,???,

'&

'& K.?80 0

n

+7:98 0 0C LgM N ", n

, , & R '& RN R#S R!SSR ,

nLRMNnLR#MSSnLRM&((0 (LgMN"?

% && ? - R" N R?

8 & R",R#,???,R?-.-LM,N#,!,O,??,,

&R,N#,!,???,,< R?'. ,&

= ' &R & ' ' &0 R?4'&.LM&& '.

.-LMN.#LMS.!LMSS.LM

1 < ' . LM, < &

'' ) &R?8= &(

. .LMNLN#,!,O,???,?N#,!,???,b_N#,!,???M'?'=

'.LM ) & R? - -_ LM '& .-_ m-_LM

K.'& .LM& ' &R?LMmLM

'& .LMK. .LM,&' , LMmLM& '.

!I

LMN#LMS!LMSSLM?

.,' ' &R ''?

&,) ,.nLRM&. '+'

L N

L MN

L NnLRM

., & (

L NnLR"M? LoM

: C( .& ,

NnLRAM,

(+'& & ?

& && LgMN ", + ( ' &' )& ?

'' '.?-'& .?

LoM, ' ('&., ,

(&LM, 0(.N.LML#YY@M L#,#,#M,

L!,!,!M, ???, L,,M& ' L,M? (, '. + '

& R#, R!, RO, ???, R? &' LoM ( C &

' '&

##L#M?Lm##L#MMS!!L!MLm!!L!MMSSLM?LmLMM NcLm##L#MMSLm!!L!MMSSLmLMMd

&0' , 9

m##L#MNk#,m!!L!MNk!,U,mLMNk

., 9(' ' '.,&,

' 6'

L4M&'('. + R"L& 9 M,

,=+''

LM'(9 &(R#,R!,_O,???,R,#Y#,!,???,Y@,

(' 6. ?&R"' && .'.,

O"

2.!?$

,k#Sk!SSkN!gLk#MSLk!MSSLkMN!LgMN"

L)M9 &0 R#,R!,RO,???,R,#Y#,!,???,Y@,' .

9 L 0.!?JM' +' 0'&( K.&k#Sk!SSkNg?

2.!?J

,Lk#MSLk!MSSLkMNLgMN"

L44M.',''. +)0'&R",

9 L 0.!?\M? ',( L)ML4M?, Lk#MSLk!MSSLkMNLgMN"

2.!?\

L444M '','. +) .-L"MR"? ',9 R (

O#

LM,R",=+&0 R#,R!,RO,???,R,#Y#,!,???,Y@,(

' 6.9 L) 0.!?]M?

2.!?]

,k#Sk!SSkNm-L"MNm-_L"M?8( Lk#MSLk!MSSLkMNLm-_L"MM

L)M,_",9 &(R#,R!,RO,???,R,#Y#,!,???,Y@

(' 6.) L) 0.!?DM?

2.!?D

k#Sk!SSkNm-L"MNm-_L"M?8( ,Lk#MSLk!MSSLkMNLm-_L"MM

, LoM ( &' & ' L444M, , ' +L"M?Lm-_L"MM?

: C( L#M.", M?

L MM " L L ?M "

L "

# #

" "

%

=

,&& 0 ?

%$&

"

!

+

)

1

+

2

)

3

: C( .58%.,

O!

K. (+ 58, & ' = ) 58 & 58. %2 K.58%?1'& . 2#jO2LN%2ML 0.!?IM?

2.!?I

,'mN

O #

.+0> K. ' . < ,

'>N #

O ,#< * &O?

? : & & 0 ? N #, 0 L#N#M?. &(0 & < .? ,'>N

O 'L3#M>N_O#

, * &O?

P ( 0 & 30? : C .

LS#M>N L>S>MN > >A>>

L3#M>N L>>MN > >S>>

)

LS#M>S L3#M>N! > >,

,

LS#M>N! > >3 L3#M>

:=&/ ,

LS#M>N!?

O ?O #

3 _#

O

OO

LS#M>N _#

O I !

=&/ & O, & !3I ) ? , &

0 ,0 & ?

.++?>K.' .,gjm<'?

?&=(

N

,< ? >NB

8( '>N'BNp#,()&.+0?

.&& %=? !?!!?!,9 0 (

LBMN"L>MN#?

!?O,

n

.C, C( K.'.B*+

,& .,

nLMN\L>MN\q",

'& ? ') . &

0 ,& & !?O&'. &' , .?4 0C %=?

O$

&

0

1)- . 0' 53E&*.&+, '. &

' ? , & &, ' , ' .C 0.? (& 00 ?

). C . '' K., (, K.,&., &C,& , &&' &?1=. ''.'&+9 & K',& &'0.0 .?

). )= F ( . ( & '9 &0 '& 0.? C && K. &, '.6' 0. & = K.,'& +?

0 20'( K.'. &' ' K.?

1)- 0' ( K. '. & ' ' K. ) 0 0/ & + K.?

%

8 K.58((b

8 K.')'

K.58b

K. %(&&& 58 . 2&&' %b

OJ

' K.58 0'(&'9&0 '& K.b

2.O?!

(.F .#! K.'.

K.%22('&F K.58b

2.O?O

%00/&+ K.& + K.L)9 M?

0 %20'( &.'. &' ' K.

1)- 0'( &.'. &' ' K. ) 00/&+&.?

%

8 &.58%((b

8 K.')

&.58%b

O\

2.O?$

' &. 0'(&'9&0 '& K.?

2.O?J

(. . K.'. K.%(

'&F&.58%?

%00/&+&.& + K.?

0 &20'( &C'. &' ' K.

1)- 0 &C K.00&+ &C& + K.?

%

8 &C58%((b

2(&&& %58. %8H&&' 2b

O]

' &C' K.'+b

2.O?]

) &C58%' K.').) &C&&&& %58b

2.O?D

( K.%8H2 &C'. K. #! K. ((+ 8%H &C'.

K.#?

%00/&+ &C& + K.?

0 (20'( .'. &' ' K.

1)- 0. K.00&+ .& + K.?

%

8 .58%((b

.58%,.+ .(

OD

2.O?I

(.F .5a58>('& K.) & .'. K.%8%.?

2.O?#"

%00/&+.& + K.?

OI

>? 6'$*@'<

6?%AA,$*(9 ' B4<CCDDD/ *$/0/$/C!50E- <5F'%A&

;66!?GH1?!*(9 '

<B4<CCDDD%$0)/C $ -C$9$ I9 $C$9%+C$/$9$C $9%+4J$9$ E- %:$)%A&

KHK?6L,A0$/$'4 *-'

M2- *-$$ !$ .-$NO*9 $ 6 $' $*$$$*$ 6'$*@'?%AA

2!? 2GPH ? G!H? ?

1' # $* <>!?%A

2!? 2GPH ? G!H? ?

1' %# $* <>!?%A

2?!"#$ 9$0

!$ .-$?$'?*:%?$%AA*(9 ' <

B4<CCDDD)/C-* C:%C-$9$' 0E- <$)%A&

1>2 ;Q! !"!

%$'< ?9&?*5?,O5#?%AA&

;H?KG?&'''<4 * '$R> D *')R?

6*-*?H R?$*') /$!?##+

!S?!$*(9 ' <B