,,,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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