,,,uebc calculus sample,
,special ,symbols
,type ,9dicators
,, "<#f1 #f"> capitaliz$ ^w
,,, "<#f1 #f1 #f"> 2g9 capitaliz$
passage
, "<#f"> 5d capitaliz$ passage
^2 bold "*
. "<#df"> ,greek lr 9dicator
,3/ruc;n ,symbols
( 2g9 frac;n
) 5d frac;n
./ frac;n l9e
< 2g9 ma? gr\p
> 5d ma? gr\p
% 2g9 radical
+ 5d radical
9 "<#ce"> sup]script
5 "<#bf"> subscript
& sup]impose previ\s symbol ) next
.5 9dex 2l
.9 9dex abv
,special ,symbols "<3t4">
^: >r[ ov] previ\s item
,pr9t ,signs
"< op5 p>5!sis
"> close p>5!sis
.< op5 bracket
.> close bracket
"6 "<#e1 #bce"> plus sign
"- "<#e1 #cf"> m9us sign
"4 "<#e1 #bef"> c5t]$ dot
"7 "<#e1 #bcef"> equal sign
@< less ?an sign
_@< less ?an or equal sign
"9 "<#e1 #ce"> a/]isk
\o "r >r[
#= 9f9;y sign
! 9tegral sign
7 prime sign
_$#d fill$ box
_\ v]tical l9e
^j degrees
@$cc transcrib]-def9$ %ape3 c.t]-
clockwise >r[
@$cl transcrib]-def9$ %ape3
,special ,symbols "<3t4">
clockwise >r[
: 5d %ape
,,,uebc calculus sample, #bgi
#e ,vector ,9tegral ,calculus
,"p ,i4 ,two-,dim5.nal ,!ory
#e4a ,,9troduc;n
,! topic ( ? *apt] is .7l9e & surface
9tegrals4. ,x w 2 se5 t ^! c bo? 2
reg>d$ z 9tegrals ( vectors & t !
pr9cipal !orems c 2 mo/ simply /at$ 9
t]ms ( vectors2 h;e ! title 8vector
9tegral calculus40
,a famili> l9e 9tegral is t ( >c l5g?3
;!.5,c ds4 ,! subscript ;,c 9dicates t
"o is m1sur+ ! l5g? ( a curve ;,c1 z 9
,fig4 #e4a4 ,if ;,c is giv5 9 p>ametric
=m ;x "7 x"<t">1 ;y "7 y"<t">1 ! l9e
9tegral reduces to ! ord9>y def9ite
9tegral3
;;;!.5,c ds
"7 !5<t5#a>9<t5#b>
%"<(dx./dt)">9#b "6 "<(dy./dt)">9#b+
dt4; #a
,if ! curve ;,c repres5ts a a#bgi
wire ^: d5s;y "<mass p] unit l5g?">
v>ies al;g ;,c1 !n ! wire has a total
mass
;;;,m "7 !.5,c f"<x1 y"> ds1;
": f"<x1 ;y"> is ! d5s;y at ! po9t
"<;x1 ;y"> ( ! wire4 ,! new 9tegral c 2
express$ 9 t]ms ( a p>amet] z previ\sly
or c 2 ?"\ ( simply z a limit ( a sum
;;;!.5,c f"<x1 y"> ds
"7 lim ,.s.5<i"7#a>.9n
f"<x5i9"91 y5i9"9"> ,.d5is4;
------------------------------------#bhj
,"h ! curve has be5 subdivid$ 9to ;n
pieces ( l5g?s
,.d;5#as1 ,.d;5#bs1 4441 ,.d;5ns1 & !
po9t ;;;"<x5i9"91 y5i9"9">; lies on !
;i? piece4 ,! limit is tak5 z ;n 2comes
9f9ite1 :ile ! maximum ,.d;5is approa*es
#j4
,a ?ird example ( a l9e 9tegral is t (
.1"w4 ,if a "picle moves f "o 5d ( ;,c
to ! o!r "u ! 9flu;e ( a =ce ^2;,f1 ! "w
d"o by ? =ce is def9$ z
#b
;!.5,c ,f;5,t ds1 a#bhj
": ,f;5,t denotes ! compon5t ( ^2;,f on
! tang5t ^2;,t 9 ! direc;n ( mo;n4 ,?
9tegral c 2 ?"\ ( z a limit ( a sum z
previ\sly4 ,h["e1 ano!r 9t]pret,n is
possible4 ,we f/ rem>k t ! "w d"o by a
3/ant =ce ^2;,f 9 mov+ a "picle f ,a to
;,b on ! l9e seg;t ;;,,ab is
;;;^2,f "4 <,,ab>^:2; = ? scal> product
is equal to
;;;_\^2,f_\ "4 cos .a "4 _\<,,ab>^:_\1;
.a 2+ ! angle 2t ^2;,f & ;;<,,ab>^:1 &
h;e to ! product ( =ce compon5t 9
direc;n ( mo;n by ! 4t.e mov$4 ,n[ !
mo;n ( ! "picle al;g ;,c c 2 ?"\ ( z !
sum ( _m small 4place;ts al;g l9e
seg;ts1 z su7e/$ 9 ,fig4 #e4b4 ,if ^!
4place;ts >e denot$ by ,.d;5#a^2r1
,.d;5#b^2r1 4441 ,.d;5n^2r1 ! "w d"o wd
2 approximat$ by a sum ( =m
;;;,.s.5<i"7#a>.9n ^2,f5i
"4 ,.d5i^2r1;
------------------------------------#bha
": ^2,f;5i is ! =ce act+ = ! ;i?
4place;t4 ,! limit+ =m ( ? is ag #c
equal to ! l9e 9tegral a#bha
;;!,f5,tds1 b 2c ( ! way ! limit is
obta9$1 we c al write x z
;!.5,c ^2;,f "4 d^2r4
,"o c ?us write
"w "7 ;;;!.5,c ,f5,tds
"7 !.5,c ^2,f "4 d^2r4;
,if ! 4place;t vector ,.d^2r & =ce
^2;,f >e express$ 9 compon5ts1
^2;,f "7 ,f;5x^2i "6 ,f;5y^2j1
,.d^2r "7 ,.dx^2i "6 ,.dy^2j1
! ele;t ( "w ^2;,f "4 ,.d^2r 2comes
^2,f "4 ,.d^2r
"7 ,f;5x,.dx "6 ,f;5y,.dy4
,! total am.t ( "w d"o is !n approximat$
by a sum ( =m
;;;,.s"<,f5x,.dx "6 ,f5y,.dy">
"7 ,.s,f5x,.dx "6 ,.s,f5y,.dy4;
,! limit+ =m ( ? is a sum ( two
9tegrals3
;;;!.5,c ,f5x dx "6 !.5,c ,f5y dy4;
,! f/ 9tegral repres5ts ! "w d"o by !
;x-compon5t ( ! =ce2 ! second 9tegral
repres5ts ! "w d"o by ! ;y-compon5t ( !
=ce4 #d
,x ?us appe>s t "o has ?ree b#bha
types ( l9e 9tegrals to 3sid]1 "nly1 !
types
;;;!.5,c f"<x1 y"> ds1
!.5,c ,p"<x1 y"> dx1
!.5,c ,q"<x1 y"> dy1;
: >e limits ( sums
,.sf"<x1 ;y"> ,.ds1
,.s,p"<x1 ;y"> ,.dx1
,.s,q"<x1 ;y"> ,.dy4
,! =ego+ gives ! basis = ! !ory ( l9e
9tegrals 9 ! plane4 ,a v sli<t ext5.n (
^! id1s l1ds to l9e 9tegrals 9 space3
;;;!.5,c f"<x1 y1 z"> ds1
!.5,c f"<x1 y1 z"> dx1 4444;
,surface 9tegrals appe> z a natural
g5]aliz,n1 ) ! surface >ea ele;t ;d.s
replac+ ! >c ele;t ds3
;;;<!!>.5,sf"<x1 y1 z"> d.s
"7 lim ,.sf"<x1 y1 z"> ,.d.s4;
,"! >e correspond+ compon5t 9tegrals
;;;<!!>.5,sf"<x1 y1 z"> dx dy1
<!!>.5,sf"<x1 y1 z"> dy dz1 444;
------------------------------------#bhb
& a vector surface 9tegral #e
;;;<!!>.5,s^2,f "4 d^2.s a#bhb
"7 <!!>.5,s"<^2,f "4 ^2n"> d.s1;
": ;d^2.s "7 ^2;n ;d.s is ! 8>ea ele;t
vector10 ^2;n 2+ a unit normal vector to
! surface4
,x w 2 se5 t ! basic !orems,-^? (
,gre51 ,gauss1 & ,/okes,-3c]n ! rel,ns
2t l9e1 surface1 & volume "<triple">
9tegrals4 ,^! correspond to funda;tal
physical rel,ns 2t s* quantities z flux1
circul,n1 div]g;e1 & curl4 ,! applic,ns
w 2 3sid]$ at ! 5d ( ! *apt]4
#e4b ,,,l9e 9tegrals 9 ! plane,
,we n[ /ate 9 precise =m ! def9i;ns
\tl9$ 9 ! prec$+ sec;n4
,by a .7smoo? curve. ;,c 9 ! xy-plane
w 2 m1nt a curve repres5table 9 ! =m3
"<#e4a"> ;;;x "7 .f"<t">1
y "7 .y"<t">1 h _@< t _@< k1;
": ;x & ;y >e 3t9u\s & h 3t9u\s
derivatives = ;h _@< ;t _@< ;k4 ,! curve
;,c c 2 assign$ a direc;n1 : w usu,y 2 t
( 9cr1s+ ;t4 ,if ,a denotes ! po9t #f
.<.f"<h">1 .y"<h">.> & ;,b b#bhb
denotes ! po9t .<.f"<k">1 .y"<k">.>1
!n ;,c c 2 ?"\ ( z ! pa? ( a po9t mov+
3t9u\sly f ,a to ;,b4 ,? pa? may cross
xf1 z = ! curve ,c;5#a ( ,fig4 #e4c4 ,if
! 9itial po9t ,a & t]m9al po9t ;,b
co9cide1 ;,c is t]m$ a .1clos$ curve2
if1 9 addi;n1 "<;x1 ;y"> moves f ,a to
;,b "7 ,a )\t retrac+ any o!r po9t1 ;,c
is call$ a .7simple clos$. curve "<curve
,c;5#b ( ,fig4 #e4c">4
,let ;,c 2 a smoo? curve z previ\sly1
) positive direc;n t ( 9cr1s+ ;t4 ,let
f"<x1 ;y"> 2 a func;n def9$ at l1/ :5
"<;x1 ;y"> is on ;,c4 ,!
------------------------------------#bhc
l9e 9tegral ;!.5,c f"<x1 ;y"> dx is
def9$ z a limit3
"<#e4b"> ;;;!.5,c f"<x1 y"> dx
"7 lim ,.s.5<i"7#a>.9n
f"<x5i9"91 y5i9"9"> ,.d5ix4;
,! limit ref]s to a subdivi.n ( ;,c z
9dicat$ 9 ,fig4 #e4d4 ,! su3essive
subdivi.n po9ts >e ,a3 "<x;5#j1 y;5#j">1
"<x;5#a1 y;5#a">1 4441 ;,b3 #g
"<x;5n1 y;5n">4 ,^! correspond a#bhc
to p>amet] values3 ;h "7 t;5#j @< t;5#a
@< 444 @< t;5n "7 ;k4 ,! po9t
;;;"<x5i9"91 y5i9"9">; is "s po9t ( ;,c
2t ;;;"<x5<i"-#a>1 y5<i"-#a>">; &
"<x;5i1 y;5i">2 t is1
;;;"<x5i9"91 y5i9"9">; corresponds to a
p>amet] value ;;t5i9"91 ":
;;;t5<i"-#a> _@< t5i9"9 _@< t5i4;
,.d;5ix denotes ! di6];e
;;;x5i "- x5<i"-#a>4; ,! limit is tak5 z
;n 2comes 9f9ite & ! l>ge/ ,.d;5it
approa*es #j1 ":
;;;,.d5it "7 t5i "- t5<i"-#a>4;
,simil>ly1
"<#e4c"> ;;;!.5,c f"<x1 y"> dy
"7 lim ,.sf"<x5i9"91 y5i9"9">
,.d5iy1;
": ;;;,.d5iy "7 y5i "- y5<i"-#a>4;
,! signific.e ( ^! def9i;ns is
gu>ante$ by ! foll[+ basic !orems3
,i ,if f"<x1 ;y"> is 3t9u\s on ;,c1 !n
;!.5,c f"<x1 ;y"> dx &
;!.5,c f"<x1 ;y"> dy exi/4
#h
,,ii ,if f"<x1 ;y"> is 3t9u\s b#bhc
on ;,c1 !n
"<#e4d"> ;;;!.5,c f"<x1 y"> dx
"7 !5h9k f.<.f"<t">1 .y"<t">.>
.y7"<t"> dt1;
"<#e4e"> ;;;!.5,c f"<x1 y"> dy
"7 !5h9k f.<.f"<t">1 .y"<t">.>
.y7"<t"> dt4;
,=mulas "<#e4d"> & "<#e4e"> reduce !
9tegrals to ord9>y def9ite 9tegrals & >e
?us ess5tial = comput,n ( "picul>
9tegrals4 ,?us let ;,c 2 !
------------------------------------#bhd
pa? ;;;x "7 #a "6 t1 y "7 t9#b1
#j _@< t _@< #a1; direct$ ) 9cr1s+ ;t4
,!n
;;;!.5,c "<x9#b "- y9#b"> dx
"7 !5#j9#a.<"<#a "6 t">9#b "- t9#d.>
dt
"7 #cb/ae1
!.5,c "<x9#b "- y9#b"> dy
"7 !5#j9#a.<"<#a "6 t">9#b "- t9#d.>
#bt dt
"7 #b#a/b4; #i
,x is logic,y easi] to prove a#bhd
,,ii f/1 = ,i is an imm 3sequ;e ( ,,ii4
,to prove ,,ii1 "o notes t ! sum
;;;,.sf"<x5i9"91 y5i9"9"> ,.d5ix; c 2
writt5 z
;;;,.s.5<i"7#a>.9n
f.<.f"<t5#a9"9">1 .y"<t5#a9"9">.>
(,.d5ix./,.d5it),.d5it4;
,n[ ;;;,.d5ix "7 x5i "- x5<i"-#a>
"7 .f7"<t5i9<"9"9>"> ,.d5it; by ! ,law (
! ,m1n4 ,h;e ! sum c 2 writt5 z
;;;,.s.5i"7#a.9n ,f"<t5i9"9">
.f7"<t5i9<"9"9>"> ,.d5it1;
": ,f"<t"> "7 f.<.f"<t">1 .y"<t">.> &
;;t5i9"9 & ;;t5i9<"9"9> >e bo? 2t
;;t5<i"-#a> & t;5i4 ,x is easily %[n
.<see ,,cla1 ,sec;n #ab-#be.> t ? sum
approa*es z limit ! 9tegral
;;;!5h9k ,f"<t">.f7"<t"> dt
"7 !5h9kf.<.f"<t">1 .y"<t">.>
.f7"<t"> dt;
z requir$4 ,=mula "<#e4e"> is prov$ 9 !
same way4
,we rem>k t ! value ( a l9e 9tegral on
;,c does n dep5d on ! "picul> #aj
p>ametriz,n ( ;,c1 b only on ! b#bhd
ord] 9 : ! po9ts ( ;,c >e trac$4 "<,see
,problem #e4">
,9 _m applic,ns ! pa? ;,c is n xf
smoo? b is compos$ ( a f9ite numb] (
>cs1 ea* ( : is smoo?4 ,?us ;,c mi<t 2 a
brok5 l9e4 ,9 ? case1 ;,c is t]m$
.1piecewise smoo?4 ,! l9e 9tegral al;g
;,c is simply1 by def9i;n1 ! sum ( !
9tegrals al;g ! pieces4 ,"o v]ifies at
once t "<#e4b">1 "<#e4c">1 & ! !orems ,i
& ,,ii 3t9ue to hold4 ,9 "<#e4d"> &
"<#e4e"> ! func;ns .f7"<t"> & .y7"<t"> w
h jump 4cont9uities1 : w n 9t]f]e ) !
exi/;e ( ! 9tegral "<cf4 ,sec;n #d4a">4
.7,"?\t ? book all pa?s ( 9tegr,n = l9e
9tegrals w 2 piecewise smoo? un.s
o!rwise specifi$4.
,if ! curve ;,c is repres5t$ 9 ! =m
;;;y "7 g"<x">1 a _@< x _@< b1;
!n "o c reg>d ;x xf z p>amet]1 replac+
;t2 t is1 ;,c is giv5 by ! equ,ns
;;;x "7 x1 y "7 g"<x">1
a _@< x _@< b;
9 t]ms ( ! p>amet] ;x4 ,if ! #aa
direc;n ( ;,c is t ( 9cr1s+ ;x1 c#bhd
"<#e4d"> &
------------------------------------#bhe
"<#e4e"> 2come
"<#e4f"> ;;;!.5,c f"<x1 y"> dx
"7 !5a9b f.<x1 g"<x">.> dx1;
"<#e4g"> ;;;!.5,c f"<x1 y"> dy
"7 !5a9bf.<x1 g"<x">.>g7"<x"> dx4;
,! ord9>y def9ite 9tegral ;;!5a9by dx1
": ;y "7 g"<x">1 is a special case (
"<#e4f">4
,simil>ly1 if ;,c is repres5t$ 9 ! =m
;;;x "7 ,f"<y">1 c _@< y _@< d1;
& ! direc;n ( ;,c is t ( 9cr1s+ ;y1 !n
"<#e4h"> ;;;!.5,c f"<x1 y"> dx
"7 !5c9df.<,f"<y">1 y.>,f7"<y"> dy1;
"<#e4i"> ;;;!.5,c f"<x1 y"> dy
"7 !5c9df.<,f"<y">1 y.> dy4;
,9 mo/ applic,ns ! l9e 9tegrals appe>
z a comb9,n1
;;;!.5,c ,p"<x1 y"> dx
"6 !.5,c ,q"<x1 y"> dy1;
: is a2reviat$ z foll[s3
;;;!.5,c .<,p"<x1 y"> dx "6 ,q"<x1 y">
dy.>; #ab
or a#bhe
;;;!.5,c ,p"<x1 y"> dx
"6 ,q"<x1 y"> dy1;
! brackets 2+ us$ only :5 nec4
,9 ! =mulas ?us f> ! direc;n ( ;,c has
be5 t ( 9cr1s+ p>amet]4 ,if ! opposite
direc;n is *os51 upp] & l[] limits >e
rev]s$ on all 9tegrals4 ,?us "<#e4d">
2comes
"<#e4d7"> ;;;!.5,c f"<x1 y"> dx
"7 !5k9hf.<.f"<t">1 .y"<t">.>
.f7"<t"> dt4;
,! l9e 9tegral is "!=e multipli$ by
"-#a4 ,(t5 x is 3v5i5t to specify ! pa?
by xs equ,ns 9 "s =m & to 9dicate !
direc;n by us+ ! 9itial & t]m9al po9ts z
l[] & upp] limits3
;;;!.5,c5,a9,b ,p dx "6 ,q dy;
or
;;;!.5,c5<"<x5#a1y5#a">>
9<"<x5#b1y5#b">> ,p dx "6 ,q dy4;
,x w 2 se5 lat] t "u c]ta9 3di;ns1 "o
ne$s only prescribe 9itial & t]m9al
po9ts3
;;;!5,a9,b ,p dx "6 ,q dy4; #ac
,,example #a ,to evaluate b#bhe
;;;!.5,c5"<#a1#j">9"<"-#a1#j">
"<x9#c "- y9#c"> dy1;
------------------------------------#bhf
": ;,c is ! semicircle
;y "7 ;%#a "- x;9#b+ %[n 9 ,fig4 #e4e1
"o c repres5t ;,c p>ametric,y3
;;;x "7 cos t1 y "7 sin t1
#j _@< t _@< .p1;
& ! 9tegral 2comes
;;;!5#j9.p"<cos9#c t "- sin9#c t">
cos t dt
"7 (#c.p./#h)4;
,"o c use ,x z p>amet]1 & ! 9tegral
2comes
;;;!5#a9<"-#a>
.<x9#c"<#a "- x9#b">9#c/b.>
("-x./%#a "- x9#b+) dx2;
? is cle>ly 9 a m awkw>d =m = 9tegr,n4
,! sub/itu;n ;x "7 cos ;t br+s "o back
to ! p>ametric =m4 ,"o c use ;y z
p>amet] b has !n to split ! 9tegral 9to
two "ps1 f "<#a1 #j"> to "<#j1 #a"> & f
"<#j1 #a"> to "<"-#a1 #j">3
#ad
;;;!5#j9#a a#bhf
.<"<#a "- y9#b">9#c/b "- y9#c.> dy
"6 !5#a9#j
.<"-"<#a "- y9#b">9#c/b "- y9#c.> dy
"7 #b!5#j9#a
"<#a "- y9#b">9#c/b dy4;
,note t ;x "7 ;%#a "- y;9#b+ on ! f/ "p
( ! pa? & ;x "7 "- ;%#a "- y;9#b+ on !
second "p4 _$#d
,,example #b ,let ;,c 2 ! p>abolic >c
;y "7 x;9#b f "<#j1 #j"> to "<"-#a1
#a">4 ,!n
;;;!.5,c xy9#b dx "6 x9#by dy
"7 !5#j9<"-#a>
"<xy9#b "6 x9#by(dy./dx)"> dx
"7 !5#j9<"-#a>"<x9#e "6 #bx9#e"> dx
"7 #a/b4 _$#d;
,if ;,c is a .1clos$ curve1 !n "! is
no ne$ to specify 9itial & t]m9al po9t1
?\< ! direc;n m/ 2 9dicat$4 ,if ;,c is a
simple clos$ curve "<trac$ j once">1 !n
"o ne$ only specify : ( ! two
possible direc;ns is *os54 ,! #ae
not,ns b#bhf
"<a"> ;;;!&@$cc ,p dx "6 ,q dy1;
"<;b"> ;;;!&@$cl ,p dx "6 ,q dy;
ref] to ! two cases ( ,figs4 #e4f"<a"> &
#e4f"<b">4 ,! c.t]clockwise >r[ ref]s to
:at is r\<ly a c.t]clockwise direc;n on
;,c2 ? w 2 t]m$ ! .1positive direc;n "<z
= angul> m1sure">2 ! clockwise direc;n w
2 call$
------------------------------------#bhg
! .1negative direc;n4 ,x %d 2 not$ t !
direc;n c 2 specifi$ by ref];e to ! unit
tang5t vector ^2;,t 9 ! direc;n (
9tegr,n & ! unit normal vector ^2;n t
po9ts to ! \tside ( ! region b.d$ by
;,c2 = ! positive direc;n1 ^2;n is #ij^j
2h ^2;,t1 z 9 ,fig4 #e4f"<a">2 = !
negative direc;n1 ^2;n is #ij^j ah1d (
^2;,t z 9 ,fig4 #e4f"<b">4
,,example #c ,to evaluate
;;;!&@$cc:.5,c y9#b dx "6 x9#b dy1;
": ;,c is ! triangle ) v]tices
"<#a1 #j">1 "<#a1 #a">1 "<#j1 #j">1 %[n
9 ,fig4 #e4g1 "o has to compute #af
?ree 9tegrals4 ,! f/ is ! a#bhg
9tegral f "<#j1 #j"> to "<#a1 #j">2 al;g
? pa?1 ;y "7 #j &1 if ;x is ! p>amet]1
dy "7 #j4 ,h;e ! f/ 9tegral is #j4 ,!
second 9tegral is t f "<#a1 #j"> to
"<#a1 #a">2 if ;y is us$ z p>amet]1 ?
reduces to
;;!5#j9#a dy "7 #a1
s9ce dx "7 #j4 ,= ! ?ird 9tegral1 f
"<#a1 #a"> to "<#j1 #j">1 ;x c 2 us$ z
p>amet]1 s t ! 9tegral is
;;!5#a9#j #bx9#b dx "7 "-#b/c1
------------------------------------#bhh
s9ce dy "7 dx4 ,?us f9,y
;;;!&@$cc:.5,c y9#b dx "6 x9#b dy
"7 #j "6 #a "- #b/c "7 #a/c4; _$#d
#e4c ,,,9tegrals ) respect to >c
l5g?,-basic prop]ties
( l9e 9tegrals,
,= a smoo? or piecewise smoo? pa? ;,c1
z 9 ! prec$+ sec;n1 >c l5g? ;s is well
def9$4 ,?us ;s c 2 def9$ z ! 4t.e
trav]s$ f ! 9itial po9t #ag
"<;t "7 ;h"> up to a g5]al ;t3 a#bhh
"<#e4aj"> ;;;s
"7 !5h9t
%"<(dx./dt)">9#b "6 "<(dy./dt)">9#b+
dt4;
,if ! curve ;,c is direct$ ) 9cr1s+ ;t1
!n ;s al 9cr1ses 9 ! direc;n ( mo;n1 go+
f #j up to ! l5g? ;,l ( ;,c4 ,let ;,c 2
subdivid$ z 9 ,fig4 #e4d & let ,.d;5is
denote ! 9cre;t 9 ;s f ;;t5<i"-#a> to
t;5i1 t is1 ! 4t.e mov$ 9 ? 9t]val4 ,"o
!n makes ! def9i;n
"<#e4aa"> ;;;!.5,c f"<x1 y"> ds
"7 lim.5<n\o#=>.5<max ,.d5is\o#j>
,.s.5<i"7#a>.9n
f"<x5i9"91 y5i9"9"> ,.d5is4;
,if ;f is 3t9u\s on ;,c1 ? 9tegral w
exi/ & c 2 evaluat$ z foll[s3
"<#e4ab"> ;;;!.5,c f"<x1 y"> ds
"7 !5h9k f.<.f"<t">1 .y"<t">.>
%.f7"<t">9#b "6 .y7"<t">9#b+ dt4;
,? is prov$ 9 ! same way z "<#e4d"> &
"<#e4e">1 ) ! aid ( ! =mula
#ah
;;;(ds./dt) b#bhh
"7 %"<(dx./dt)">9#b
"6 "<(dy./dt)">9#b+
"7 %.f7"<t">9#b "6 .y7"<t">9#b+4;
,"o c 9 pr9ciple use ;s xf z ! p>amet]
on ! curve ;,c2 if ? is d"o1 ;x & ;y
2come func;ns ( ;s3 ;x "7 x"<s">1
;y "7 y"<s">4 ,! po9t .<x"<s">1 y"<s">.>
is !n ! posi;n ( ! mov+ po9t af a 4t.e
;s has be5 trav]s$4 ,9 ? case1 "<#e4aa">
reduces to a def9ite 9tegral ) respect
to ;s3
"<#e4ac"> ;;;!.5,c f"<x1 y"> ds
"7 !5#j9,l f.<x"<s">1 y"<s">.> ds4;
,if ;x is us$ z p>amet] "o has
"<#e4ad"> ;;;!.5,c f"<x1 y"> ds
"7 !5a9b f.<x1 y"<x">.>
%#a "6 "<(dy./dx)">9#b+ dx2;
"! is an analog\s =mula = ;y4
444
,,,5d ( sample,
#ai
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