,,bana
,,uebc ,,sampl] #b
,prep>$ "u ! auspices (!
,brl ,au?or;y ( ,nor? ,am]ica
,june #bjja
,volume #a ( #b
,pages p#a-p#c & #a-ade p#a
,3t5ts
,volume #a
,sample ,page
,9troduc;n """""""""""""""""""""""""" #a
,"qnaire """""""""""""""""""""""""""" #e
#a4 ,spati,y ,>rang$ ,ma!matics """"" #i
#a;a4 ,>i?metic ,pro#ms """""""""" #aa
#a;b4 ,sy/em ( ,equ,ns """"""""""" #ag
#a;c4 ,matrix ,multiplic,n """"""" #ai
#a;d4 ,l;g ,divi.n """"""""""""""" #bc
#a;e4 ,l;g ,multiplic,n """""""""" #bg
#a;f4 ,c.ell$ ,digits """""""""""" #bi
#a;g4 ,c.ell$ ,frac;ns """"""""""" #ca
#a;h4 ,numb] ,l9e """""""""""""""" #cc
#a;i4 ,a4i;n ,puzzle """"""""""""" #cg
#b4 ,algebra3 s"eal pages f a
textbook """"""""""""""""""""""" #ci
#c4 ,calculus3 s"eal pages f a
textbook """"""""""""""""""""""" #ii
,volume #b
#d4 ,t1*+ ,mat]ial """""""""""""""" #adg
p#b
#d;a4 ,ma? ,"w%op """"""""""""""" #aeg
#d;b4 ,*apt] #f ,review """"""""" #afg
#d;c4 ,algebra ,i ,*apt] #g ,te/ #aga
#d;d4 ,algebra ,,ii ,*apt] #c
,te/ """""""""""""""""""""""" #age
#d;e4 ,a ,sample ,>ticle """""""" #agi
#e4 ,-put]3 s"eal pages f a manual #ahc
#f4 ,*emi/ry3 s"eal pages f a
textbook """""""""""""""""""""" #bec
,9troduc;n
,! ,brl ,au?or;y ( ,nor? ,am]ica
7,,bana7 is pl1s$ 6s5d y ,,uebc ,sampl]
#b4 ,x illu/rates ! draft brl code 2+
develop$ 0! ,unifi$ ,5gli% ,brl ,code
,rese>* ,project "u ! direc;n (!
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7,,iceb74 ,if y h n se5 ,,uebc ,sampl]
#a1 pl1se 3tact ! appropriate p]son f !
li/ ( a4resses at ! 5d ( ? ,9troduc;n4
,,uebc ,sampl] #b 9cludes examples (
publi%$ te*nical mat]ials4 ,fr ( !m1
ea* #aj 3secutive pr9t pages1 >e brld 9
,,uebc & 9 ! ,,bana code 9 use 9 ,nor?
,am]ica = t subject1 e4g4 ,neme? ,code
=! algebra & calculus1 ,-put] ,code =!
-put] not,n1 & ,*emi/ry ,code =!
*emi/ry4 ,"! >e al3 "s ele;t>y >i?metic
samples2 a li/ ( examples us$ 9 h] t1*+
0,susan ,o/]haus1 a ma!matics t1*] at !
,texas ,s*ool =! ,bl & ,visu,y ,impair$2
& an example provid$ 0,jane ,corcoran1 a
,cali=nia transcrib]4
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symbols requir$ 6r1d ! mat]ials
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examples foll[ ! same =mat us$ 0! pres5t
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! spac+ (! orig9al text4 ,= 9/.e1
ma!matical signs ( op],n >e spac$ or
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,9 ! n>rative por;n ( ^! examples y w
notice t ei<t 3trac;ns f.d 9 ,5gli% ,brl
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3trac;n3 9to & a;n4 ,all (! o!r #aha
3trac;ns1 ^wsigns & %ort=ms >e un*ang$4
,ea* ,,uebc symbol is unambigus--a
pr9t symbol is repres5t$ 0! same brl
symbol reg>d.s (! subject4 ,2c "! >e
only #fc possi# s+le cell dot -b9,ns _m
symbols h 6be made up ( two or ?ree
cells4 ,if ! brl symbol uses ! same dot
3figur,n z a 3trac;n x m/ 2 prec$$ 0a
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,? design f1ture w make x easi] =a brl
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9dicators >e al omitt$ :5 numb]s >e
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,a punctu,n 9dicator is us$ ": nec
64t+ui% signs ( punctu,n f digits4
,,uebc ,r1d+ ,notes
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sec;ns ( text 3ta9+ symbols t wd o!rwise
2 r1d z lit]>y 3trac;ns4 ,! numb] #c
9dicator al sets grade "o mode = unspac$
symbols or lrs t foll[ a numb]4
,pr9t ,copies
,6obta9 a pr9t copy (! examples 9
,sampl] #b or a pr9t copy ( ,sampl] #a
3tact
,! ,am]ican ,f.d,n =! ,bl
,n,nal ,lit]acy ,c5t]
,3tact3 ,fr.es ,m>y ,d',&rea
#djd-ebe-bcjc or ,,afb's ,9=m,n ,c5t]
#hjj-bcb-edfc
;,e-mail3 _+literacy@afb.net_:
,6obta9 a brl copy ( ,sampl] #a 3tact
,kim ,*>lson1 ,brl ,au?or;y ( ,nor?
,am]ica1
#fag-igb-gbdi or ;e-mail
_+charlsonk@perkins.pvt.k12.ma.us_:
,= ,canadians1 pr9t & brl copies (
,sampl] #a may 2 obta9$ 03tact+
,d>le5 ,bog>t
,! ,canadian ,n,nal ,9/itute =! ,bl
#daf-dhj-gecj or #hjj-bfh-hhah
;,e-mail3 _+bogartd@lib.cnib.ca_:
,"qnaire
,?ank y = r1d+ ,,uebc ,sampl] #b4
,,bana wants yr -;ts & has prep>$ a %ort
li/ ( "qs z a guide 76get y />t$74 ,y
may respond 9 brl1 pr9t1 on audio tape
or 0;e-mail4 ,up-to-date 9=m,n on ,,uebc
is availa# 0visit+ ! ,,bana web site at
_+http://www.brailleauthority.org_:
.,direc;ns3 ,pl1se -plete ! foll[+
"qnaire af r1d+ "? ,sampl] #b4 ,write yr
answ]s 2l1 on a sep>ate %eet ( pap]1 or
9 an ;e-mail message4 ,s5d yr -plet$
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#abcj ,gov]n;t ,/reet
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_>uebc_<input@aol.com_:
,yr 9put w 2 use;l 6! ,,bana ,bo>d 9
/udy+ ! ,,uebc4 ,?ank y6
,"qs ab ,,uebc & ,o!r ,issues
#a4 ,2f y r1d ,sampl] #b1 :at 7 yr
feel+s t[>d unify+ ! brl codes8
#b4 ,hav+ r1d "? ,sampl] #b1 h[ h yr
id1s/feel+s *ang$ t[>d a unifi$ brl
code8
#c4 ,:at d y re,y l ab ! ,unifi$ ,5gli%
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#d4 ,:at d y re,y 4like ab ! ,,uebc8
#e4 ,:at issues d y feel ! ,,bana ,bo>d
%d 3sid] 9 mak+ a deci.n on adop;n (!
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#f4 ,>e yr -;ts bas$ on all six samples8
,yes ,no
,if n1 pl1se *eck ^? : >e 9clud$4
#a #b #c #d #e #f
,op;nal ,backgr.d ,9=m,n
#g4 ,:at k9d ( mat]ials d y typic,y r1d
9 brl8 ,*eck all t apply4
a4 magaz9es
b4 fic;n
c4 nonfic;n
d4 textbooks
e4 o!r 7li/73
#h4 ,:at ma? or te*nical mat]ials d y
r1d8 ,*eck all t apply4
a4 textbooks
b4 te*nical mat]ials = my job
c4 -put] brl
d4 *emi/ry or o!r sci5tific not,n
e4 o!r 7li/73
#g
#i4 ,:at ma? code did y le>n 9 s*ool8
a4 upp] numb]s 7,taylor ,code7
b4 l[] numb]s 7,neme?7
c4 upp] numb]s 7,,bauk--,brl ,au?or;y
(! ,unit$ ,k+dom7
d4 no ma? code us$
#aj4 ,:5 did y le>n brl8
a4 2f age #ah
b4 #ah-ee ye>s ( age
c4 #ef & abv
,op;nal ,p]sonal ,9=m,n
,"n3
,age3
,g5d]3
,o3up,n3
,pref]r$ ,lit]acy ,m$ium3
;,e-mail a4ress3
,a4ress3
,teleph"o3
,sample #a4
,spati,y ,>rang$ ,ma!matics
,? sample is transcrib$ us+ ..,!
,neme? ,brl ,code = ,ma!matics & ,sci;e
,not,n #aigb .,revi.n 7on left-h& pages7
&! ,unifi$ ,5gli% ,brl ,code z ( ,june
#bjja 7on "r-h& pages74
#i
#a;a4 ,>i?metic ,pro#ms
9 ,neme? ,code
7777777777777777777777777777777777777777
,neme? ,symbols
_ 7#def7 punctu,n 9dicator
+ plus sign
- 7#cf7 m9us sign
@* "ts sign 7cross7
.k equal sign
( op5+ p>5!sis
) clos+ p>5!sis
, 7#f7 ma!matical -ma
gggggggggggggggggggggggggggggggggggggggg
#a;a4 ,>i?metic ,pro#ms 9 ,,uebc
7777777777777777777777777777777777777777
,,uebc ,symbols
.="6 plus sign
.="- m9us sign
.="8 "ts sign 7cross7
.="7 equal sign
.="< op5+ p>5!sis
.="> clos+ p>5!sis
gggggggggggggggggggggggggggggggggggggggg
#aa
,add4
#28_4 8 #29_4 5 #30_4 2
+5 +8 +7
3333 3333 3333
#31_4 7 #32_4 3 #33_4 8
+2 +0 +0
3333 3333 3333
#34_4 (2+6)+5
#35_4 #2+(6+5)
#36_4 (7+3)+0
#37_4 #4+5+8
,f9d ! sums & di6];es4 ,use a
calculator if nec4
#1_4 2,964 #2_4 6,587
+5,682 +2,744
33333333 33333333
#3_4 4,532 #4_4 4,430
+1,607 - 726
33333333 33333333
,add4
#bh4 #h #bi4 #e #cj4 #b
"6#e "6#h "6#g
"33333 "33333 "33333
#ca4 #g #cb4 #c #cc4 #h
"6#b "6#j "6#j
"33333 "33333 "33333
#cd4 "<#b "6 #f"> "6 #e
#ce4 #b "6 "<#f "6 #e">
#cf4 "<#g "6 #c"> "6 #j
#cg4 #d "6 #e "6 #h
,f9d ! sums & di6];es4 ,use a
calculator if nec4
#a4 #b1ifd #b4 #f1ehg
"6#e1fhb "6#b1gdd
"333333333 "333333333
#c4 #d1ecb #d4 #d1dcj
"6#a1fjg "- #gbf
"333333333 "333333333 #ac
#5_4 6,429 #6_4 7,000
-5,161 -2,674
33333333 33333333
,f9d ! products4
#1_4 4 #2_4 1 #3_4 2
@*1 @*3 @*0
33333 33333 33333
#4_4 0 #5_4 2
@*3 @*1
33333 33333
#6_4 #5@*4 .k #20
#4@*5 .k n
#e4 #f1dbi #f4 #g1jjj
"-#e1afa "-#b1fgd
"333333333 "333333333
,f9d ! products4
#a4 #d #b4 #a #c4 #b
"8#a "8#c "8#j
"33333 "33333 "33333
#d4 #j #e4 #b
"8#c "8#a
"33333 "33333
#f4 #e "8 #d "7 #bj
#d "8 #e "7 ;n
#ae
#a;b4 ,sy/em ( ,equ,ns
9 ,neme? ,code
7777777777777777777777777777777777777777
,neme? ,symbols
+ plus sign
- 7#cf7 m9us sign
.k equal sign
gggggggggggggggggggggggggggggggggggggggg
,equat+ coe6ici5ts l1ds 6! sy/em
c1+2c2+ c3 .k #0
-#2c1+ c2+8c3 .k #0
#3c1+8c2+7c3 .k #0
#a;b4 ,sy/em ( ,equ,ns 9 ,,uebc
7777777777777777777777777777777777777777
,,uebc ,symbols
.=""= dot locator 72f l[] symbol on a
l9e 0xf7
.=;;; 2g9 grade "o passage
.=; 5d grade "o passage
.=5 subscript next item
.="6 plus sign
.="- m9us sign
.="7 equal sign
gggggggggggggggggggggggggggggggggggggggg
,equat+ coe6ici5ts l1ds to ! sy/em
""=;;;
c5#a "6 #b;c5#b "6 c5#c "7 #j
"-#b;c5#a "6 c5#b "6 #h;c5#c "7 #j
#c;c5#a "6 #h;c5#b "6 #g;c5#c "7 #j
""=;
#ag
#a;c4 ,matrix ,multiplic,n
9 ,neme? ,code
7777777777777777777777777777777777777777
,neme? ,symbols
( op5+ p>5!sis
) clos+ p>5!sis
,( multil9e op5+ p>5!sis
,) multil9e clos+ p>5!sis
@( op5+ bracket
@) clos+ bracket
, 7#f7 ma!matical -ma
; 7#ef7 subscript 9dicator
^ 7#de7 sup]script 9dicator
" 7#e7 return 6basel9e af subscript or
sup]script
' 7#c7 prime sign
- 7#cf7 m9us sign
.k equal sign
.a ,greek alpha
gggggggggggggggggggggggggggggggggggggggg
#a;c4 ,matrix
,multiplic,n 9 ,,uebc
7777777777777777777777777777777777777777
,,uebc ,symbols
.=""= dot locator 72f l[] symbol on a
l9e 0xf7
.=;;; 2g9 grade "o passage
.=; 5d grade "o passage
.="< op5+ p>5!sis
.="> clos+ p>5!sis
.=.< op5+ bracket
.=.> clos+ bracket
.=,"< multil9e op5+ p>5!sis
.=,"> multil9e clos+ p>5!sis
.=5 subscript next item
.=9 sup]script next item
.=444 ellipsis
.=4]4]4 v]tical ellipsis
.=7 prime sign
.=< 2g9 -p.d item
.=> 5d -p.d item
.="- m9us sign
.="7 equal sign
.=.a ,greek alpha
gggggggggggggggggggggggggggggggggg #ai
,= ea* fix$ ;x 9 @(a, b@), ! matrix
equ,n
(11)
,(f1(x) f2(x) ''' f;n"(x) ,)
,(f'1(x) f'2(x) ''' f';n"(x) ,)
,(''''''''''''''''''''''''''''''',)
,(f1^(n-1) f2^(n-1) ''' f;n^(n-1),)
,( "(x) "(x) "(x) ,)
,(.a1 ,)
,(.a2 ,)
,(''''',)
,(.a;n",)
.k ,(#0 ,)
,(#0 ,)
,(''',)
,(#0 ,)
,= ea* fix$ ;x 9 .<a1 ;b.>1 ! matrix
equa;n
""=;;;
"<#aa">
,"<f5#a f5#b 444 f5n ,">
,"< "<x"> "<x"> "<x">,">
,"<f75#a f75#b 444 f75n ,">
,"< "<x"> "<x"> "<x">,">
,"<4]4]4 ,">
,"<f5#a9< f5#b9< 444 f5n9< ,">
,"< "<n"- "<n"- "<n"-,">
,"< #a">> #a">> #a">>,">
,"< "<x"> "<x"> "<x">,">
,"<.a5#a,">
,"<.a5#b,">
,"<4]4]4,">
,"<.a5n ,">
"7 ,"<#j ,">
,"<#j ,">
,"<4]4]4,">
,"<#j ,">
""=; #ba
#a;d4 ,l;g ,divi.n 9 ,neme? ,code
7777777777777777777777777777777777777777
,neme? ,symbols
_ 7#def7 punctu,n 9dicator
" 7#e7 basel9e 9dicator
o curv$ divi.n sign
, 7#f7 ma!matical -ma
- 7#cf7 m9us sign
gggggggggggggggggggggggggggggggggggggggg
#a;d4 ,l;g ,divi.n 9 ,,uebc
7777777777777777777777777777777777777777
,,uebc ,symbols
.="> curv$ divi.n sign
.="- m9us sign
.=" num]ic space
gggggggggggggggggggggggggggggggggggggggg
#bc
#4_4 ,br+ d[n ! next digit 9 ! divid5d4
,rep1t ^! /eps until "! >e no digits
left 6br+ d[n4
620 ,r"19
333333333333333
32o 19,859
-19 2
333333333333333
65
-64
333333333333333
19
- 0
333333333333333
19 ,rememb] 6*eck
! answ]4
#d4 ,br+ d[n ! next digit 9 ! divid5d4
,rep1t ^! /eps until "! >e no digits
left to br+ d[n4
#fbj ,r#ai
"3333333333333333
#cb"> #ai1hei
"-#ai"b
"3333333333333333
#fe
"-#fd
"3333333333333333
#ai
"- #j
"3333333333333333
#ai ,rememb] to
*eck ! answ]4
#be
#a;e4 ,l;g ,multiplic,n
9 ,neme? ,code
7777777777777777777777777777777777777777
,neme? ,symbols
_ 7#def7 punctu,n 9dicator
777 7#bcef1 #bcef1 #bcef1 '''7 c>ry l9e
@* "ts sign 7cross7
gggggggggggggggggggggggggggggggggggggggg
#2_4 ,multiply 0! t5s digit4
1
7777777
643 ,rememb] 6regrp4
@*35
3333333
3215
19290 #30@*643
,y d n h 6write ! z]o4 ,/>t ! answ]
9 ! t5s place4
#a;e4 ,l;g ,multiplic,n 9 ,,uebc
7777777777777777777777777777777777777777
,,uebc ,symbols
.="8 "ts sign 7cross7
gggggggggggggggggggggggggggggggggggggggg
#b4 ,multiply by ! t5s digit4
#a
#fdc ,rememb] to regrp4
"8#ce
"3333333
#cbae
#aibij #cj "8 #fdc
,y d n h to write ! z]o4 ,/>t !
answ] 9 ! t5s place4
#bg
#a;f4 ,c.ell$ ,digits
9 ,neme? ,code
7777777777777777777777777777777777777777
,neme? ,symbols
[ op5+ c.ell,n 9dicator
] clos+ c.ell,n 9dicator
- 7#cf7 m9us sign
. 7#df7 decimal po9t
gggggggggggggggggggggggggggggggggggggggg
,subtract3 #16-3.98
9
5 [10] 10
1[6].[ 0][ 0]
- 3 . 9 8
3333333333333333
1 2 . 0 2
#a;f4 ,c.ell$ ,digits 9 ,,uebc
7777777777777777777777777777777777777777
,,uebc ,symbols
.="- m9us sign
.=@: l9e "? previs item
.=4 decimal po9t
gggggggggggggggggggggggggggggggggggggggg
,subtract3 #af "- #c4ih
#i
#e #aj@:#aj
#a#f@:4 #j@: #j@:
"- #c 4 #i #h
"33333333333333333333
#a#b 4 #j #b
#bi
#a;g4 ,c.ell$ ,frac;ns
9 ,neme? ,code
7777777777777777777777777777777777777777
,neme? ,symbols
[ op5+ c.ell,n 9dicator
] clos+ c.ell,n 9dicator
@* "ts sign 7cross7
.k equal sign
? op5+ frac;n 9dicator
# clos+ frac;n 9dicator
_? op5+ mix$ frac;n 9dicator
_# clos+ mix$ frac;n 9dicator
/ horizontal frac;n l9e
gggggggggggggggggggggggggggggggggggggggg
?3/8#@*1_?5/9_#
#1 #7
[3] [14]
.k ?333#@*?3333#
[8] [9]
#4 #3
.k ?7/12#
#a;g4 ,c.ell$ ,frac;ns 9 ,,uebc
7777777777777777777777777777777777777777
,,uebc ,symbols
.="8 "ts sign 7cross7
.="7 equal sign
.=@: l9e "? previs item
gggggggggggggggggggggggggggggggggggggggg
#c/h "8 #a#e/i
#a #g
#c@: #ad@:
"7 "33333 "8 "333333
#h@: #i@:
#d #c
"7 #g/ab
#ca
#a;h4 ,numb] ,l9e 9 ,neme? ,code
7777777777777777777777777777777777777777
,neme? ,symbols
_ 7#def7 punctu,n 9dicator
[ left >r[h1d
333 7#be1 #be1 #be1 '''7 horizontal %aft
( numb] l9e
r v]tical m>k
o "r >r[h1d
_? op5+ mix$ frac;n 9dicator
_# clos+ mix$ frac;n 9dicator
/ horizontal frac;n l9e
= miss+ numb] 7blank "ul9e7
gggggggggggggggggggggggggggggggggggggggg
#a;h4 ,numb] ,l9e 9 ,,uebc
7777777777777777777777777777777777777777
,,uebc ,symbols
.=; grade "o symbol
.=[ left >r[h1d
.=333 horizontal %aft ( numb] l9e
.=w v]tical m>k
.=" 7at 5d ( l9e7 3t9u,n 9dicator
.=o "r >r[h1d
.=,- da% 7us$ = blank "ul9e7
.=4 decimal po9t
gggggggggggggggggggggggggggggggggggggggg
#cc
,use frac;ns1 mix$ numb]s1 &
decimals 6"n po9ts on ! numb] l9e4
,write ea* miss+ numb]4
2 2_?1/10_# 2_?2/10_#
#1_4 [3333r333333333r333333333r333333333
2.0 2.1 =
2_?3/10_# = 2_?5/10_#
r333333333r333333333r333333333
2.3 2.4 2.5
2_?6/10_# = 2_?8/10_#
r333333333r333333333r333333333
2.6 = 2.8
2_?9/10_# 3 3_?1/10_#
r333333333r333333333r333333333
2.9 3.0 =
3_?2/10_#
r333333333o
3.2
,use frac;ns1 mix$ numb]s1 &
decimals to "n po9ts on ! numb] l9e4
,write ea* miss+ numb]4
#b #b#a/aj #b#b/aj
#a4 ;[333w3333333w3333333w3333333"
#b4j #b4a ,-
#b#c/aj ,- #b#e/aj #b#f/aj
w3333333w3333333w3333333w3333333"
#b4c #b4d #b4e #b4f
,- #b#h/aj #b#i/aj #c
w3333333w3333333w3333333w3333333"
,- #b4h #b4i #c4j
#c#a/aj #c#b/aj
w3333333w333333o
,- #c4b
#ce
#a;i4 ,a4i;n ,puzzle
9 ,neme? ,code
7777777777777777777777777777777777777777
,neme? ,symbols
_ 7#def7 punctu,n 9dicator
,' 7#f1 #c7 transcrib]'s note 9dicator
+ plus sign
- 7#cf7 m9us sign
gggggggggggggggggggggggggggggggggggggggg
,ea* lr 9 ^! pro#ms repres5ts a di6]5t
digit4
#1_4 ,:at is ! value ( ;,c_8
#2_4 ,:at is ! value ( ;,d_8
,',lrs 9 ! pro#ms 2l >e capitaliz$
9 pr9t4,'
8789 deff
3ba7 -e2f6
482a 3333333
+7ab5 1997
3333333
2c287
#a;i4 ,a4i;n ,puzzle 9 ,,uebc
7777777777777777777777777777777777777777
,,uebc ,symbols
.="6 plus sign
.="- m9us sign
gggggggggggggggggggggggggggggggggggggggg
,ea* lr 9 ^! problems repres5ts a di6]5t
digit4
#a4 ,:at is ! value ( ;,c8
#b4 ,:at is ! value ( ;,d8
#h#g#h#i ,d,e,f,f
#c,b,a#g "-,e#b,f#f
#d#h#b,a "33333333333
"6#g,a,b#e #a#i#i#g
"33333333333
#b,c#b#h#g
#cg
,sample #b4
,algebra
,? sample is transcrib$ us+ ..,!
,neme? ,brl ,code = ,ma!matics & ,sci;e
,not,n #aigb .,revi.n 7on left-h& pages7
&! ,unifi$ ,5gli% ,brl ,code z ( ,june
#bjja 7on "r-h& pages74
#ci
,algebra ,sample 9 ,neme? ,code
7777777777777777777777777777777777777777
,neme? ,symbols
_ 7#def7 punctu,n 9dicator
^ 7#de7 2g9 sup]script
" 7#e7 return 6basel9e af sup]script
? 2g9 frac;n
/ horizontal frac;n l9e
# 5d frac;n
( op5+ p>5!sis
) clos+ p>5!sis
@( op5+ bracket
@) clos+ bracket
+ plus sign
- 7#cf7 m9us sign
+- plus or m9us sign
-+ m9us or plus sign
* "ts sign 7dot7
_/ sla%
_l id5t;y sign 7#c horizontal b>s7
.k equal sign
/.k n equal sign
@# a/]isk
, 7#f7 ma!matical -ma
gggggggggggggggggggggggggggggggggggggggg
,algebra ,sample 9 ,,uebc
7777777777777777777777777777777777777777
,,uebc ,symbols
.=; grade "o symbol
.=;;; 2g9 grade "o passage
.=; 72f a space7 5d grade "o passage
.=""= dot locator 72f l[] symbol on a
l9e 0xf7
.=,,, 2g9 capitaliz$ passage
.=, 72f a space7 5d capitaliz$ passage
.=.1 italic ^w
.=.7 2g9 italic passage
.=. 72f a space7 5d italic passage
.=^1 bold ^w
.=^7 2g9 bold passage
.=^ 72f a space7 5d bold passage
.=( 2g9 frac;n
.=./ frac;n l9e
.=) 5d frac;n
.=9 sup]script next item
.=< 2g9 -p.d item
.=> 5d -p.d item
#da
,,uebc ,symbols 73t47
.="< op5+ p>5!sis
.="> clos+ p>5!sis
.=.< op5+ bracket
.=.> clos+ bracket
.=4 decimal po9t
.="6 plus sign
.="- m9us sign
.=_6 plus or m9us sign
.=_- m9us or plus sign
.="4 "ts sign 7dot7
.="7 equal sign
.="7@: n equal sign
.=_= equival5t sign 7#c horizontal l9es7
.=_/ sla%
.="9 a/]isk
gggggggggggggggggggggggggggggggggggggggg
#dc
#3-4 ,,special ,,products #dg
,"! >e c]ta9 special products : o3ur s
frequ5tly 9 algebra t !y h be5
classifi$4 ,^! >e giv5 2l4 ..,! lrs 9 !
=mulas may /& = any algebraic .expres.n4
,ea* is a direct result (! axioms 9
,*apt] #2_4 ,! r1d] %d n only v]ify ea*
0actu,y c>ry+ ! /eps & giv+ ! r1sons1
b al memorize !m1 s t he c recognize bo?
! product f ! factors &! factors f !
product4
(3-11) a(x+y) _l ax+ay_4
(3-12) (x+y)(x-y) _l x^2"-y^2_4
(3-13) @# (x+-y)^2 _l x^2"+-2xy+y^2_4
@# ,! sign +- is r1d 8plus or
m9us40 ,if ! upp] (l[]) sign is us$
9 ! left memb]1 x is al us$ 9 ! "r1
s t (x+-y)^2 _l x^2"+-2xy+y^2 m1ns
#c-#d ,,,special products, #dg
,"! >e c]ta9 special products : o3ur s
frequ5tly 9 algebra t !y h be5
classifi$4 ,^! >e giv5 2l4 .7,! lrs 9 !
=mulas may /& = any algebraic expres.n4.
,ea* is a direct result ( ! axioms 9
,*apt] #b4 ,! r1d] %d n only v]ify ea*
by actually c>ry+ ! /eps & giv+ !
r1sons1 b al memorize !m1 s t he c
recognize bo? ! product f ! factors & !
factors f ! product4
""=;;;
"<#c-#aa"> a"<x "6 y"> _= ax "6 ay4
"<#c-#ab"> "<x "6 y">"<x "- y">
_= x9#b "- y9#b4
"<#c-#ac">"9 "<x _6 y">9#b
_= x9#b _6 #bxy "6 y9#b4
""=;
"9 ,! sign _6 is r1d 8plus or
m9us40 ,if ! upp] "<l[]"> sign is
us$ 9 ! left memb]1 x is al us$ 9 !
"r1 s t ;;;"<x _6 y">9#b
_= x9#b _6 #bxy "6 y9#b; m1ns
#de
(x+y)^2 _l x^2"+2xy+y^2 & a#dg
(x-y)^2 _l x^2"-2xy+y^2_4
(3-14) (x+a)(x+b) _l x^2"+(a+b)x+ab_4
(3-15)
(ax+b)(cx+d) _l acx^2"+(ad+bc)x+bd_4
(3-16)
(x+-y)^3
_l x^3"+-3x^2"y+3xy^2"+-y^3_4
(3-17)
(x+-y)(x^2"-+xy+y^2") _l x^3"+-y^3_4
,! r1d] %d det]m9e : (! abv =mulas is
us$ 9 ! foll[+ illu/r,ns4
-------------------------------------#dh
,,illu/r,n #1_4
(2x^2"-3y)(2x^2"+3y)
_l (2x^2")^2"-(3y)^2
_l #4x^4"-9y^2_4
;;;"<x "6 y">9#b a#dg
_= x9#b "6 #bxy "6 y9#b; &
;;;"<x "- y">9#b
_= x9#b "- #bxy "6 y9#b4;
""=;;;
"<#c-#ad"> "<x "6 a">"<x "6 b">
_= x9#b "6 "<a "6 b">x "6 ab4
"<#c-#ae"> "<ax "6 b">"<cx "6 d">
_= acx9#b "6 "<ad "6 bc">x "6 bd4
"<#c-#af"> "<x _6 y">9#c
_= x9#c _6 #cx9#by "6 #cxy9#b
_6 y9#c4
"<#c-#ag"> "<x _6 y">
"<x9#b _- xy "6 y9#b">
_= x9#c _6 y9#c4
""=;
,! r1d] %d det]m9e : ( ! abv =mulas is
us$ 9 ! foll[+ illu/ra;ns4
-------------------------------------#dh
^7,illu/ra;n #a4^
;;;"<#bx9#b "- #cy">"<#bx9#b "6 #cy">
_= "<#bx9#b">9#b "- "<#cy">9#b
_= #dx9#d "- #iy9#b4;
#dg
,,illu/r,n #2_4 a#dh
(x+2)(x+5)
_l x^2"+(2+5)x+10
_l x^2"+7x+10_4
,,illu/r,n #3_4
(3x+4y)(2x-3y)
_l #6x^2"+(-9+8)xy-12y^2
_l #6x^2"-xy-12y^2_4
,,illu/r,n #4
(x+y-1)^3
_l @((x+y)-1@)^3
_l (x+y)^3"-3(x+y)^2"+3(x+y)-1
_l x^3"+3x^2"y+3xy^2"+y^3"-3x^2"-6xy
-#3y^2"+3x+3y-1_4
,"h (x+y) is 3sid]$ f/ z "o t]m4
^7,illu/ra;n #b4^ a#dh
;;;"<x "6 #b">"<x "6 #e">
_= x9#b "6 "<#b "6 #e">x "6 #aj
_= x9#b "6 #gx "6 #aj4;
^7,illu/ra;n #c4^
;;;"<#cx "6 #dy">"<#bx "- #cy">
_= #fx9#b "6 "<"-#i "6 #h">xy
"- #aby9#b
_= #fx9#b "- xy "- #aby9#b4;
^7,illu/ra;n #d^
;;;"<x "6 y "- #a">9#c
_= .<"<x "6 y"> "- #a.>9#c
_= "<x "6 y">9#c "- #c"<x "6 y">9#b
"6 #c"<x "6 y"> "- #a
_= x9#c "6 #cx9#by "6 #cxy9#b
"6 y9#c "- #cx9#b "- #fxy
"- #cy9#b "6 #cx "6 #cy "- #a4;
,"h "<;x "6 ;y"> is 3sid]$ f/ z "o t]m4
#di
,,illu/r,n #5 b#dh
(3x+2y)(9x^2"-6xy+4y^2")
_l (3x+2y)
@((3x)^2"-(3x)(2y)+(2y)^2"@)
_l (3x)^3"+(2y)^3
_l #27x^3"+8y^3_4
,,pro#ms
,f9d ! foll[+ products4
#1_4 #2a(3x-4y)
#2_4 -#3x(2x+7y)
#3_4 -#7xy(3x^2"+4y)
#4_4 #4x^2"yz(z^2"+xy+yz)
#5_4 (2x-3y)(2x+3y)
#6_4 (7x+5y^2")(7x-5y^2")
#7_4 (x+2y)(x-2y)(x^2"+4y^2")
#8_4 (x-3)^2
#9_4 (2x+7y)^2
#10_4 (3x^2"y-5z^2")^2
#11_4 (x-2)(x-5)
^7,illu/ra;n #e^ b#dh
;;;"<#cx "6 #by">
"<#ix9#b "- #fxy "6 #dy9#b">
_= "<#cx "6 #by">
.<"<#cx">9#b "- "<#cx">"<#by">
"6 "<#by">9#b.>
_= "<#cx">9#c "6 "<#by">9#c
_= #bgx9#c "6 #hy9#c4;
,,problems
,f9d ! foll[+ products4
""=;;;
#a4 #b;a"<#cx "- #dy">
#b4 "-#cx"<#bx "6 #gy">
#c4 "-#gxy"<#cx9#b "6 #dy">
#d4 #dx9#byz"<z9#b "6 xy "6 yz">
#e4 "<#bx "- #cy">"<#bx "6 #cy">
#f4 "<#gx "6 #ey9#b">"<#gx "- #ey9#b">
#g4 "<x "6 #by">"<x "- #by">
"<x9#b "6 #dy9#b">
#h4 "<x "- #c">9#b
#i4 "<#bx "6 #gy">9#b
#aj4 "<#cx9#by "- #ez9#b">9#b
#aa4 "<x "- #b">"<x "- #e"> #ea
#12_4 (2x+3)(x-5) c#dh
#13_4 (xy^2"-z^2"w)^2
#14_4 (?1/2#x+?2/3#y)^2
#15_4 (4x-3y)(7x+3y)
#16_4 @((x+1)-z@)@((x+1)+z@)
#17_4 (2x+3y+3)(2x+3y-3)
#18_4 (2x+3y+4z)^2
#19_4 (x-2y-z)^2
#20_4 (2a+b)^3
#21_4 (x+2)(x^2"-2x+4)
#22_4 (x-3)(x^2"+3x+9)
#23_4 (x+3y+2z-4w)(x+3y-2z+4w)
#24_4 (4x-2y-3z+3w)(4x+2y+3z+3w)
#25_4 (a-b+c-d)^2
#26_4 (2a+3b-c-4d)^2
#27_4 @(2(x+2y)-3@)@(2(x+2y)+4@)
#28_4 @(2(x-3y)+5@)@(3(x-3y)-2@)
#29_4 (2x+3y)^3
#30_4 (5x-3y)^3
#ab4 "<#bx "6 #c">"<x "- #e"> c#dh
#ac4 "<xy9#b "- z9#bw">9#b
#ad4 "<#a/bx "6 #b/cy">9#b
#ae4 "<#dx "- #cy">"<#gx "6 #cy">
#af4 .<"<x "6 #a"> "- z.>
.<"<x "6 #a"> "6 z.>
#ag4 "<#bx "6 #cy "6 #c">
"<#bx "6 #cy "- #c">
#ah4 "<#bx "6 #cy "6 #dz">9#b
#ai4 "<x "- #by "- z">9#b
#bj4 "<#b;a "6 b">9#c
#ba4 "<x "6 #b">"<x9#b "- #bx "6 #d">
#bb4 "<x "- #c">"<x9#b "6 #cx "6 #i">
#bc4 "<x "6 #cy "6 #bz "- #dw">
"<x "6 #cy "- #bz "6 #dw">
#bd4 "<#dx "- #by "- #cz "6 #cw">
"<#dx "6 #by "6 #cz "6 #cw">
#be4 "<a "- b "6 c "- d">9#b
#bf4 "<#b;a "6 #c;b "- c "- #d;d">9#b
#bg4 .<#b"<x "6 #by"> "- #c.>
.<#b"<x "6 #by"> "6 #d.>
#bh4 .<#b"<x "- #cy"> "6 #e.>
.<#c"<x "- #cy"> "- #b.>
#bi4 "<#bx "6 #cy">9#c
#cj4 "<#ex "- #cy">9#c #ec
#3-5 ,,factors ,,& ,,factor+ #di
,! process ( factor+ an algebraic
expres.n is simil> 6t ( f9d+ ! factors (
a -posite numb]4 ,recall ! 4cus.n (
prime & -posite 9teg]s 9 ,>ticle #1-4_4
,? process1 : is usu,y re/rict$ at ?
ele;t>y /age 6factor+ polynomials )
r,nal coe6ici5ts & 6factors -pletely
free f irr,nal numb]s1 is frequ5tly
p]=m$ 0rev]s+ ! processes 3sid]$ 9
,>ticle #3-4_4 ,s* a factoriz,n is
3sid]$ -plete :5 ea* algebraic factor is
a .prime .factor2 t is1 an algebraic
expres.n t _c 2 factor$ )t violat+ !
abv re/ric;ns4
,! m -mon types ( factor+ >e illu/rat$
2l4 ,note ! import.e & applic,n (!
4tributive axioms 9 ? 4cus.n4
,,example #1_4 ,factor
#2ax^2"-4ay^2"+8a^2"x_4
""=; d#dh
-------------------------------------#di
#c-#e ,,,factors & factor+,
,! process ( factor+ an algebraic
expres.n is simil> to t ( f9d+ ! factors
( a composite numb]4 ,recall ! 4cus.n (
prime & composite 9teg]s 9 ,>ticle
#a-#d4 ,? process1 : is usually re/rict$
at ? ele;t>y /age to factor+ polynomials
) ra;nal coe6ici5ts & to factors
completely free f irra;nal numb]s1 is
frequ5tly p]=m$ by rev]s+ ! processes
3sid]$ 9 ,>ticle #c-#d4 ,s* a
factoriza;n is 3sid]$ complete :5 ea*
algebraic factor is a .7prime factor2. t
is1 an algebraic expres.n t _c 2 factor$
)t violat+ ! abv re/ric;ns4
,! m common types ( factor+ >e
illu/rat$ 2l4 ,note ! import.e &
applica;n ( ! 4tributive axioms 9 ?
4cus.n4
^7,example #a4^ ,factor
#b;ax9#b "- #d;ay9#b "6 #h;a9#bx4 #ee
.,solu;n4 ,! polynomial 9 ? a#di
pro#m has #2a z a -mon factor4
#2ax^2"-4ay^2"+8a^2"x
_l #2a(x^2"-2y^2"+4ax)_4
,,example #2_4 ,factor
x(a+2b)-3y(a+2b)_4
.,solu;n4 ,ea* (! two expres.ns has !
-mon t]m (a+2b)_4 ,"!=e1
x(a+2b)-3y(a+2b) _l (x-3y)(a+2b)_4
,,example #3_4 ,factor
(4x^2"_/y^2")-(9a-b)^2_4
.,solu;n4 ,? expres.n is ! di6];e 2t
two p]fect squ>es4
?4x^2"/y^2"#-(9a-b)^2
_l (?2x/y#)^2"-(9a-b)^2
_l @(?2x/y#+(9a-b)@)
@(?2x/y#-(9a-b)@)
_l (?2x/y#+9a-b)(?2x/y#-9a+b)_4
,,example #4_4 ,factor
.1,solu;n4 ,! polynomial 9 ? a#di
problem has #b;a z a common factor4
;;;#b;ax9#b "- #d;ay9#b "6 #h;a9#bx
_= #b;a"<x9#b "- #by9#b "6 #d;ax">4;
^7,example #b4^ ,factor
x"<a "6 #b;b"> "- #cy"<a "6 #b;b">4
.1,solu;n4 ,ea* ( ! two expres.ns has
! common t]m "<a "6 #b;b">4 ,"!=e1
;;;x"<a "6 #b;b"> "- #cy"<a "6 #b;b">
_= "<x "- #cy">"<a "6 #b;b">4;
^7,example #c4^ ,factor
"<#dx9#b_/y9#b"> "- "<#i;a "- b">;9#b4
.1,solu;n4 ,? expres.n is ! di6];e 2t
two p]fect squ>es4
;;;(#dx9#b./y9#b) "- "<#i;a "- b">9#b
_= "<(#bx./y)">9#b
"- "<#i;a "- b">9#b
_= .<(#bx./y) "6 "<#i;a "- b">.>
.<(#bx./y) "- "<#i;a "- b">.>
_= "<(#bx./y) "6 #i;a "- b">
"<(#bx./y) "- #i;a "6 b">4;
^7,example #d4^ ,factor #eg
#9x^2"-30xy+25y^2_4 b#di
.,solu;n4 ,? algebraic expres.n is a
p]fect squ>e4
#9x^2"-30xy+25y^2 _l (3x-5y)^2_4
,,example #5_4 ,factor
#27x^3"+(8_/y^3")_4
.,solu;n4 ,! algebraic expres.n is !
sum ( two cubes4 ,acly1
#27x^3"+?8/y^3"#
_l (3x+?2/y#)
(9x^2"-?6x/y#+?4/y^2"#)_4
-------------------------------------#ej
,,example #6_4 ,factor
#12x^2"+7xy-10y^2_4
.,solu;n4 ,? trinomial 9 ! =m (
,eq4 (3-15) is factor$ 0trial & ]ror4 ,!
result w 2 9 ! =m (ax+by)(cx+dy), ":
ac .k #12, bd .k -#10, & ad+bc .k #7_4
,"h ;a & ;c >e bo? plus1 & ;b & ;d >e
di6]5t 9 sign4 ,! correct -b9,n1 we f9d1
is
#ix9#b "- #cjxy "6 #bey9#b4 b#di
.1,solu;n4 ,? algebraic expres.n is a
p]fect squ>e4
;;;#ix9#b "- #cjxy "6 #bey9#b
_= "<#cx "- #ey">9#b4;
^7,example #e4^ ,factor
#bgx9#c "6 "<#h_/y9#c">4
.1,solu;n4 ,! algebraic expres.n is !
sum ( two cubes4 ,acly1
;;;#bgx9#c "6 (#h./y9#c)
_= "<#cx "6 (#b./y)">
"<#ix9#b "- (#fx./y)
"6 (#d./y9#b)">4;
-------------------------------------#ej
^7,example #f4^ ,factor
#abx9#b "6 #gxy "- #ajy9#b4
.1,solu;n4 ,? trinomial 9 ! =m (
,eq4 "<#c-#ae"> is factor$ by trial &
]ror4 ,! result w 2 9 ! =m
"<ax "6 by">"<cx "6 dy">1 ": ;ac "7 #ab1
bd "7 "-#aj1 & ad "6 bc "7 #g4 ,"h a &
;c >e bo? plus1 & ;b & ;d >e di6]5t 9
sign4 ,! correct comb9a;n1 we f9d1 is
#ei
#12x^2"+7xy-10y^2 a#ej
_l (4x+5y)(3x-2y)_4
,,example #7_4 ,factor
#6x^4"+7x^2"y^2"-3y^4_4
.,solu;n4 ,? is ! same type z ,example
#6_4
#6x^4"+7x^2"y^2"-3y^4
_l (3x^2"-y^2")(2x^2"+3y^2")_4
,al? ! f/ factor on ! "r is ! di6];e (
two squ>es1 x _c 2 factor$ fur!r1 = s*
factoriz,n wd 9troduce irr,nal
quantities4
,,pro#ms
,factor ! foll[+ -pletely4
#1_4 #4x-20
#2_4 #10x+15yz
#3_4 #3y^2"-9y
#4_4 #4x^3"y^2"+6x^2"y^3
#5_4 xy^2"z^3"-3x^2"yz^2"+5xy^3"z^2
#abx9#b "6 #gxy "- #ajy9#b a#ej
_= "<#dx "6 #ey">"<#cx "- #by">4
^7,example #g4^ ,factor
#fx9#d "6 #gx9#by9#b "- #cy9#d4
.1,solu;n4 ,? is ! same type z
,example #f4
;;;#fx9#d "6 #gx9#by9#b "- #cy9#d
_= "<#cx9#b "- y9#b">
"<#bx9#b "6 #cy9#b">4;
,al? ! f/ factor on ! "r is ! di6];e (
two squ>es1 x _c 2 factor$ fur!r1 = s*
factoriza;n wd 9troduce irra;nal
quantities4
,,problems
,factor ! foll[+ completely4
""=;;;
#a4 #dx "- #bj
#b4 #ajx "6 #aeyz
#c4 #cy9#b "- #iy
#d4 #dx9#cy9#b "6 #fx9#by9#c
#e4 xy9#bz9#c "- #cx9#byz9#b
"6 #exy9#cz9#b #fa
#6_4 a^2"b^3"c^4"-a^3"b^4"c^5 b#ej
"+2a^2"b^4"c^4
#7_4 #3y(2x+5)-4x(2x+5)
#8_4 #3y(4-y)+6x^2"(4-y)
#9_4 #2z^2"(x+3y)-6xz(x+3y)
#10_4 #3x(3-2y)-2xy(3-2y)
#11_4 #9-a^2
#12_4 #16x^2"-9y^2
#13_4 #225a^8"-64b^2
#14_4 (c^6"_/d^8")-121
#15_4 x^3"y^4"-25xd^6
#16_4 #0.01x^4"-196y^8
#17_4 (x+2y)^2"-z^2
#18_4 (3x-2y)^2"-25z^2
#19_4 (a+b)^2"-(c+d)^2
#20_4 #9(2x-y)^2"-4(2a+b)^2
#21_4 #81(4x-3y)^2"-25(3z+w)^2
#22_4 x^2"+6x+9-(y^2"+4y+4)
#23_4 x^2"-8x+16
#f4 a9#b;b9#c;c9#d b#ej
"- a9#c;b9#d;c9#e "6 #b;a9#b;b9#d;c9#d
#g4
#cy"<#bx "6 #e"> "- #dx"<#bx "6 #e">
#h4 #cy"<#d "- y"> "6 #fx9#b"<#d "- y">
#i4
#bz9#b"<x "6 #cy"> "- #fxz"<x "6 #cy">
#aj4
#cx"<#c "- #by"> "- #bxy"<#c "- #by">
#aa4 #i "- a9#b
#ab4 #afx9#b "- #iy9#b
#ac4 #bbe;a9#h "- #fd;b9#b
#ad4 "<c9#f_/d9#h"> "- #aba
#ae4 x9#cy9#d "- #bexd9#f
#af4 #j4jax9#d "- #aify9#h
#ag4 "<x "6 #by">9#b "- z9#b
#ah4 "<#cx "- #by">9#b "- #bez9#b
#ai4 "<a "6 b">9#b "- "<c "6 d">9#b
#bj4 #i"<#bx "- y">9#b
"- #d"<#b;a "6 b">9#b
#ba4 #ha"<#dx "- #cy">9#b
"- #be"<#cz "6 w">9#b
#bb4 x9#b "6 #fx "6 #i
"- "<y9#b "6 #dy "6 #d">
#bc4 x9#b "- #hx "6 #af #fc
#24_4 #4a^2"-12ab+9b^2 c#ej
#25_4 #66xy+9x^2"y^2"+121
#26_4 #2x^3"-28x^2"+98x
#27_4 #5z^2"-30wz+45w^2
#28_4 x^2n"+2x^n"y^n"+y^2n
#29_4 (3-x)^2"+8(3-x)+16
#30_4 #25-30(2x-3y)+9(2x-3y)^2
#31_4 a^3"-8
#32_4 #1+(8_/x^9")
#33_4 #8x^6n"+27y^3m
#34_4 x^3"-(y^3"_/64)
#35_4 #27(x-y)^3"-8(x+y)^3
#36_4 #5(a-2b)^3"-625(a-2b)^3
#37_4 x^2"-7x+12
#38_4 y^2"-2y-8
#39_4 a^2"b^2"-ab-20
#40_4 #2x^2"+8x+6
#41_4 #35x^2"-24x+4
#42_4 #3y^2"-y-10
#43_4 #6a^2"+7a-20
#44_4 #2x^2"-23xy-39y^2
#bd4 #d;a9#b "- #ab;ab "6 #i;b9#b c#ej
#be4 #ffxy "6 #ix9#by9#b "6 #aba
#bf4 #bx9#c "- #bhx9#b "6 #ihx
#bg4 #ez9#b "- #cjwz "6 #dew9#b
#bh4 x9<#bn> "6 #bx9ny9n "6 y9<#bn>
#bi4 "<#c "- x">9#b "6 #h"<#c "- x">
"6 #af
#cj4 #be "- #cj"<#bx "- #cy">
"6 #i"<#bx "- #cy">9#b
#ca4 a9#c "- #h
#cb4 #a "6 "<#h_/x9#i">
#cc4 #hx9<#fn> "6 #bgy9<#cm>
#cd4 x9#c "- "<y9#c_/#fd">
#ce4 #bg"<x "- y">9#c "- #h"<x "6 y">9#c
#cf4 #e"<a "- #b;b">9#c
"- #fbe"<a "- #b;b">9#c
#cg4 x9#b "- #gx "6 #ab
#ch4 y9#b "- #by "- #h
#ci4 a9#b;b9#b "- ab "- #bj
#dj4 #bx9#b "6 #hx "6 #f
#da4 #cex9#b "- #bdx "6 #d
#db4 #cy9#b "- y "- #aj
#dc4 #f;a9#b "6 #g;a "- #bj
#dd4 #bx9#b "- #bcxy "- #ciy9#b
#fe
#45_4 (x+y)^2"-7(x+y)+10 #ea
#46_4 (y+z)^2"+(y+z)-42
#47_4 #2(2x+y)^2"-(2x+y)-10
#48_4 #6(x+y)^2"+5(x+y)(y+z)-6(y+z)^2
#49_4 #12(a+b)^2"-14(a+b)(c+d)-10(c+d)^2
#50_4 #4(x-2)^2"+5(x-2)(y+4)-21(y+4)^2
,"! >e _m algebraic expres.ns :1
0prop] grp+1 c 2 put 96"o (! =ms 9 !
previs examples & !n factor$4
,,example #8_4 ,factor ax-ay-bx+by_4
.,solu;n4 ,if1 0! associative axiom1
we grp ! f/ two t]ms tgr1 &! la/ two
tgr1 & (use ! 4tributive
#de4 #ea
"<x "6 y">9#b "- #g"<x "6 y"> "6 #aj
#df4 "<y "6 z">9#b "6 "<y "6 z"> "- #db
#dg4 #b"<#bx "6 y">9#b "- "<#bx "6 y">
"- #aj
#dh4 #f"<x "6 y">9#b
"6 #e"<x "6 y">"<y "6 z">
"- #f"<y "6 z">9#b
#di4 #ab"<a "6 b">9#b
"- #ad"<a "6 b">"<c "6 d">
"- #aj"<c "6 d">9#b
#ej4 #d"<x "- #b">9#b
"6 #e"<x "- #b">"<y "6 #d">
"- #ba"<y "6 #d">9#b
""=;
,"! >e _m algebraic expres.ns :1 by
prop] grp+1 c 2 put 9to "o ( ! =ms 9 !
previs examples & !n factor$4
^7,example #h4^ ,factor
ax "- ay "- bx "6 by4
.1,solu;n4 ,if1 by ! associative
axiom1 we grp ! f/ two t]ms tgr1 & !
la/ two tgr1 & "<use ! 4tributive #fg
axiom) factor ! -mon t]m1 we a#ea
trans=m ! expres.n 96! =m ( ,example
#2_4
ax-ay-bx+by
_l a(x-y)-b(x-y)
_l (x-y)(a-b)_4
,,example #9_4 ,factor
#4x^3"-12x^2"-x+3_4
.,solu;n4 ,ag we grp ! f/ two t]ms &!
la/ two t]ms4
#4x^3"-12x^2"-x+3
_l #4x^2"(x-3)-(x-3)
_l (x-3)(4x^2"-1)
_l (x-3)(2x+1)(2x-1)_4
,9 bo? ^! examples we cd h grp$ ! f/ &
?ird1 &! second & fr? t]ms1 & obta9$ !
same result4
,,example #10_4 ,factor
#4x^2"-12xy+9y^2"+4x-6y-3_4
.,solu;n4 ,if we grp ! f/ ?ree t]ms1
! solu;n 2comes cle>4
axiom"> factor ! common t]m1 we a#ea
trans=m ! expres.n 9to ! =m ( ,example
#b4
;;;ax "- ay "- bx "6 by
_= a"<x "- y"> "- b"<x "- y">
_= "<x "- y">"<a "- b">4;
^7,example #i4^ ,factor
#dx9#c "- #abx9#b "- ;x "6 #c4
.1,solu;n4 ,ag we grp ! f/ two t]ms &
! la/ two t]ms4
;;;#dx9#c "- #abx9#b "- x "6 #c
_= #dx9#b"<x "- #c"> "- "<x "- #c">
_= "<x "- #c">"<#dx9#b "- #a">
_= "<x "- #c">"<#bx "6 #a">
"<#bx "- #a">4;
,9 bo? ^! examples we cd h grp$ ! f/ &
?ird1 & ! second & fr? t]ms1 & obta9$ !
same result4
^7,example #aj4^ ,factor #dx9#b
"- #abxy "6 #iy9#b "6 #dx "- #fy "- #c4
.1,solu;n4 ,if we grp ! f/ ?ree t]ms1
! solu;n 2comes cle>4
#fi
#4x^2"-12xy+9y^2"+4x-6y-3 b#ea
_l (2x-3y)^2"+2(2x-3y)-3
_l @((2x-3y)+3@)@((2x-3y)-1@)
_l (2x-3y+3)(2x-3y-1)_4
,,example #11_4 ,factor
x^4"+2x^2"y^2"+9y^4_4
.,solu;n4 ,if ! coe6ici5t (! second
t]m 7 #6, ! expres.n wd 2 a p]fect
squ>e4 ,"!=e1 if we add (and subtract)
#4x^2"y^2, r solu;n 2comes evid5t4
x^4"+2x^2"y^2"+9y^4
_l x^4"+6x^2"y^2"+9y^4"-4x^2"y^2
_l (x^2"+3y^2")^2"-(2xy)^2
_l (x^2"+3y^2"+2xy)
(x^2"+3y^2"-2xy)_4
;;;#dx9#b "- #abxy "6 #iy9#b b#ea
"6 #dx "- #fy "- #c
_= "<#bx "- #cy">9#b
"6 #b"<#bx "- #cy"> "- #c
_= .<"<#bx "- #cy"> "6 #c.>
.<"<#bx "- #cy"> "- #a.>
_= "<#bx "- #cy "6 #c">
"<#bx "- #cy "- #a">4;
^7,example #aa4^ ,factor
x;9#d "6 #bx9#by9#b "6 #iy9#d4
.1,solu;n4 ,if ! coe6ici5t ( ! second
t]m 7 #f1 ! expres.n wd 2 a p]fect
squ>e4 ,"!=e1 if we add "<& subtract">
#dx9#by9#b1 r solu;n 2comes evid5t4
;;;x9#d "6 #bx9#by9#b "6 #iy9#d
_= x9#d "6 #fx9#by9#b "6 #iy9#d
"- #dx9#by9#b
_= "<x9#b "6 #cy9#b">9#b
"- "<#bxy">9#b
_= "<x9#b "6 #cy9#b "6 #bxy">
"<x9#b "6 #cy9#b "- #bxy">4;
#ga
,,pro#ms #eb
,factor ! foll[+ expres.ns4
#1_4 ax-ay-by+bx
#2_4 ax-2ay-6by+3bx
#3_4 x^3"-2x^2"+4x-8
#4_4 y^3"-2y^2"+5y-10
#5_4 #2a-6-ab^2"+3b^2
#6_4 x^3"+3x^2"-9x-27
#7_4 x^2"-2x+1-y^2
#8_4 xy^3"+2y^2"-xy-2
#9_4 #4x^2"-y^2"+4y-4
#10_4 x^6"-7x^3"-8
#11_4 x^2"+2xy+y^2"-z^2"+2zw-w^2
#12_4 #4a^2"-x^2"+b^2"-y^2"-4ab-2xy
#13_4 x^2"+4xy+4y^2"-x-2y-6
#14_4 x^3"-5x^2"-x+5
#15_4 x^4"-7x^2"y^2"+9y^4
#16_4 y^4"+y^2"+25
#17_4 a^4"+2a^2"b^2"+9b^4
#18_4 x^4"+5x^2"+9
,,problems #eb
,factor ! foll[+ expres.ns4
""=;;;
#a4 ax "- ay "- by "6 bx
#b4 ax "- #b;ay "- #f;by "6 #c;bx
#c4 x9#c "- #bx9#b "6 #dx "- #h
#d4 y9#c "- #by9#b "6 #ey "- #aj
#e4 #b;a "- #f "- ab9#b "6 #c;b9#b
#f4 x9#c "6 #cx9#b "- #ix "- #bg
#g4 x9#b "- #bx "6 #a "- y9#b
#h4 xy9#c "6 #by9#b "- xy "- #b
#i4 #dx9#b "- y9#b "6 #dy "- #d
#aj4 x9#f "- #gx9#c "- #h
#aa4 x9#b "6 #bxy "6 y9#b "- z9#b
"6 #bzw "- w9#b
#ab4 #d;a9#b "- x9#b "6 b9#b "- y9#b
"- #d;ab "- #bxy
#ac4 x9#b "6 #dxy "6 #dy9#b "- x "- #by
"- #f
#ad4 x9#c "- #ex9#b "- x "6 #e
#ae4 x9#d "- #gx9#by9#b "6 #iy9#d
#af4 y9#d "6 y9#b "6 #be
#ag4 a9#d "6 #b;a9#b;b9#b "6 #i;b9#d
#ah4 x9#d "6 #ex9#b "6 #i #gc
#19_4 b^4"+6b^2"c^2"+25c^2 a#eb
#20_4 #25x^2"+30xy+9y^2"+15x+9y+2
#21_4 #3ax-6ay+4bx-8by+cx-2cy
#22_4 #20xy+7zw-5yz-28xw
#23_4 z^4"+4z^3"-2z-8
#24_4 x^4"+4y^4
#25_4 a^8"-b^8
#26_4 x^6"+1
#27_4 x^2"+2xy-z^2"-2yz
#28_4 (x^2"+2x-3)^2"-4
#29_4 (x-y-2z)^2"-(2x+y-z)^2
#30_4 #2(x+2)^2"(x-3)+3(x+2)(x-3)^2
#3-6 ,,simplific,n ,,( ,,frac;ns
,a basic pr9ciple = frac;ns1 algebraic
z well z >i?metic1 /ates t ! value (a
frac;n is n *ang$ if xs num]ator &
denom9ator >e bo? multipli$ or bo?
#ai4 a#eb
b9#d "6 #f;b9#b;c9#b "6 #be;c9#b
#bj4 #bex9#b "6 #cjxy "6 #iy9#b "6 #aex
"6 #iy "6 #b
#ba4 #c;ax "- #f;ay "6 #d;bx "- #h;by
"6 cx "- #b;cy
#bb4 #bjxy "6 #gzw "- #eyz "- #bhxw
#bc4 z9#d "6 #dz9#c "- #bz "- #h
#bd4 x9#d "6 #dy9#d
#be4 a9#h "- b9#h
#bf4 x9#f "6 #a
#bg4 x9#b "6 #bxy "- z9#b "- #byz
#bh4 "<x9#b "6 #bx "- #c">9#b "- #d
#bi4 "<x "- y "- #bz">9#b
"- "<#bx "6 y "- z">9#b
#cj4 #b"<x "6 #b">9#b"<x "- #c">
"6 #c"<x "6 #b">"<x "- #c">9#b
""=;
#c-#f ,,,simplifica;n ( frac;ns,
,a basic pr9ciple = frac;ns1 algebraic z
well z >i?metic1 /ates t ! value ( a
frac;n is n *ang$ if xs num]ator &
denom9ator >e bo? multipli$ or bo? #ge
divid$ 0! same quant;y (not b#eb
z]o)_4 ,? pr9ciple 0 /at$ 9 ,!orem
#2-8_4 ,h;e1 ! simplific,n or reduc;n (a
frac;n 6l[e/ t]ms is alw possi#4 ,factor
bo? ! num]ator & denom9ator 96_! prime
factors &1 0us+ ! basic pr9ciple1 divide
! num]ator & denom9ator 0! product ( all
_! -mon factors4
,,example #1_4 ,reduce
(8x^4"y^7")_/(12x^6"y^3") 6l[e/ t]ms4
.,solu;n
?8x^4"y^7"/12x^6"y^3"#
_l ?2^3"x^4"y^7"/2^2"*3x^6"y^3"#
_l ?2^2"x^4"y^3"*2y^4
"/2^2"x^4"y^3"*3x^2"#_4
,0divid+ bo? num]ator & denom9ator by
#2^2"x^4"y^3, we h
?8x^4"y^7"/12x^6"y^3"#
_l ?2y^4"/3x^2"#_4
-------------------------------------#ec
,,example #2_4 ,reduce
divid$ by ! same quant;y "<n b#eb
z]o">4 ,? pr9ciple 0 /at$ 9 ,!orem
#b-#h4 ,h;e1 ! simplifica;n or reduc;n (
a frac;n to l[e/ t]ms is alw possible4
,factor bo? ! num]ator & denom9ator 9to
_! prime factors &1 by us+ ! basic
pr9ciple1 divide ! num]ator & denom9ator
by ! product ( all _! common factors4
^7,example #a4^ ,reduce
"<#hx9#dy9#g">_/"<#abx9#fy9#c"> to l[e/
t]ms4
.1,solu;n
;;;(#hx9#dy9#g./#abx9#fy9#c)
_= (#b9#cx9#dy9#g
./#b9#b"4#cx9#fy9#c)
_= (#b9#bx9#dy9#c"4#by9#d
./#b9#bx9#dy9#c"4#cx9#b)4;
,by divid+ bo? num]ator & denom9ator by
#b9#bx9#dy9#c1 we h
;;;(#hx9#dy9#g./#abx9#fy9#c)
_= (#by9#d./#cx9#b)4;
-------------------------------------#ec
^7,example #b4^ ,reduce
#gg
(x^2"-7x+10)_/(2x^2"-x-6) 6l[e/ a#ec
t]ms4
.,solu;n4 ,if we factor bo? num]ator &
denom9ator1 we h
?x^2"-7x+10/2x^2"-x-6#
_l ?(x-5)(x-2)/(2x+3)(x-2)#,
& divid+ bo? num]ator & denom9ator by
x-2, t is1 apply+ ,!orem #2-8, we get
?x^2"-7x+10/2x^2"-x-6# _l ?x-5/2x+3#_4
,! elim9,n (a -mon factor 0divid+ !
num]ator & denom9ator (a frac;n 0?
factor is call$ .multiplicative
.c.ell,n4 ,s* a process %d 2 d"o ) c>e1
= ,!orem #2-8 is true only :5 x /.k #0_4
,9 ? case ! id5t;y is true = all values
( ;x except x .k #2 or x .k -?3/2#, : >e
n p]missi# values4
,,example #3_4 ,reduce
(12x^2"+30x-72)_/(52x-8x^2"-60) to
"<x;9#b "- #gx "6 #aj"> a#ec
_/"<#bx9#b "- ;x "- #f"> to l[e/ t]ms4
.1,solu;n4 ,if we factor bo? num]ator
& denom9ator1 we h
;;;(x9#b"-#gx"6#aj./#bx9#b"-x"-#f)
_= ("<x"-#e">"<x"-#b">
./"<#bx"6#c">"<x"-#b">)1;
& divid+ bo? num]ator & denom9ator by
;x "- #b1 t is1 apply+ ,!orem #b-#h1 we
get
;;;(x9#b"-#gx"6#aj./#bx9#b"-x"-#f)
_= (x"-#e./#bx"6#c)4;
,! elim9a;n ( a common factor by
divid+ ! num]ator & denom9ator ( a
frac;n by ? factor is call$
.7multiplicative c.ella;n4. ,s* a
process %d 2 d"o ) c>e1 = ,!orem #b-#h
is true only :5 ;x "7@: #j4 ,9 ? case !
id5t;y is true = all values ( ;x except
;x "7 #b or ;x "7 "-#c/b1 : >e n
p]missible values4
^7,example #c4^ ,reduce
"<#abx9#b "6 #cjx "- #gb">
_/"<#ebx "- #hx9#b "- #fj"> to #gi
l[e/ t]ms4 b#ec
.,solu;n
?12x^2"+30x-72/52x-8x^2"-60#
_l ?6(2x-3)(x+4)/4(3-2x)(x-5)#
_l ?3(x+4)/2(5-x)#_4
,? id5t;y foll[s f ! fact t
#2x-3 .k -(3-2x)_4 (,recall ,pro#m #4,
,>ticle #2-4_4)
,,pro#ms
,reduce ! foll[+ 6l[e/ t]ms4
#1_4 ?28/63#
#2_4 ?27x^3"/225x^5"#
#3_4 ?a^4"x^3"y/a^2"xy^3"#
#4_4 ?a^2"+ab/3a+2a^3"#
#5_4 ?a^2"x-a^2"y/ax^2"-ay^2"#
#6_4 ?24a^2"/6a^2"-9a#
#7_4 ?x^2"-1/x^2"-x#
#8_4 ?x^2"-4x+4/x^2"-4#
#9_4 ?x^2"-16/x^2"-8x+16#
l[e/ t]ms4 b#ec
.1,solu;n
;;;(#abx9#b"6#cjx"-#gb
./#ebx"-#hx9#b"-#fj)
_= (#f"<#bx"-#c">"<x"6#d">
./#d"<#c"-#bx">"<x"-#e">)
_= (#c"<x"6#d">./#b"<#e"-x">)4;
,? id5t;y foll[s f ! fact t
#bx "- #c "7 "-"<#c "- #bx">4 "<,recall
,problem #d1 ,>ticle #b-#d4">
,,problems
,reduce ! foll[+ to l[e/ t]ms4
""=;;;
#a4 #bh/fc
#b4 (#bgx9#c./#bbex9#e)
#c4 (a9#dx9#cy./a9#bxy9#c)
#d4 (a9#b"6ab./#c;a"6#b;a9#c)
#e4 (a9#bx"-a9#by./ax9#b"-ay9#b)
#f4 (#bd;a9#b./#f;a9#b"-#i;a)
#g4 (x9#b"-#a./x9#b"-x)
#h4 (x9#b"-#dx"6#d./x9#b"-#d)
#i4 (x9#b"-#af./x9#b"-#hx"6#af)
#ha
#10_4 ?a^2"-3a-4/a^2"-a-12# c#ec
#11_4 ?y^2"-y-6/y^2"+2y-15#
#12_4 ?2x^2"+5x-12/4x^2"-4x-3#
#13_4 ?6a^2"-7a-3/4a^2"-8a+3#
#14_4 ?ax+ay-bx-by/am-bm-an+bn#
-------------------------------------#ed
#15_4 ?14x-24-2x^2"/x^2"+x-20#
#16_4
?(4x^2"-9y^2")(18x-12)/(2x-3y)(12x-8)#
#17_4 ?x^2"-36/x^3"-216#
#18_4 ?2x^2"-14x+20/7x-2x^2"-6#
#19_4 ?2(x^2"-y^2")xy+x^4"-y^4
"/x^4"-y^4"#
#20_4 ?y^6"+64/y^4"-4y^2"+16#
#21_4 ?4a^2"-1/12a^2"+a-4a^3"-3#
#22_4
?a^2"-2ab+3b^2"/a^4"+2a^2"b^2"+9b^4"#
#23_4 ?(x^2"-16)(x^2"-4x+16)/x^3"+64#
#24_4 ?15ab-20a-21b+28/21-a-10a^2"#
#aj4 c#ec
(a9#b"-#c;a"-#d./a9#b"-a"-#ab)
#aa4 (y9#b"-y"-#f./y9#b"6#by"-#ae)
#ab4 (#bx9#b"6#ex"-#ab./#dx9#b"-#dx"-#c)
#ac4
(#f;a9#b"-#g;a"-#c./#d;a9#b"-#h;a"6#c)
#ad4 (ax"6ay"-bx"-by./am"-bm"-an"6bn)
-------------------------------------#ed
#ae4 (#adx"-#bd"-#bx9#b./x9#b"6x"-#bj)
#af4 ("<#dx9#b"-#iy9#b">"<#ahx"-#ab">
./"<#bx"-#cy">"<#abx"-#h">)
#ag4 (x9#b"-#cf./x9#c"-#baf)
#ah4
(#bx9#b"-#adx"6#bj./#gx"-#bx9#b"-#f)
#ai4 (#b"<x9#b"-y9#b">xy"6x9#d"-y9#d
./x9#d"-y9#d)
#bj4 (y9#f"6#fd./y9#d"-#dy9#b"6#af)
#ba4 (#d;a9#b"-#a
./#ab;a9#b"6a"-#d;a9#c"-#c)
#bb4 (a9#b"-#b;ab"6#c;b9#b
./a9#d"6#b;a9#b;b9#b"6#i;b9#d)
#bc4 ("<x9#b"-#af">"<x9#b"-#dx"6#af">
./x9#c"6#fd)
#bd4 (#ae;ab"-#bj;a"-#ba;b"6#bh
./#ba"-a"-#aj;a9#b) #hc
#3-7 ,,a4i;n ,,( ,,frac;ns a#ed
,! algebraic sum ( two or m frac;ns
hav+ ! same denom9ator is a frac;n )!
-mon denom9ator &a num]ator : is !
algebraic sum (! num]ators (! frac;ns
3sid]$4 ,? 0 prov$ 9 ,pro#m #13, ,>ticle
#2-4_4
,,illu/r,n4
?2x^2"/x-4#-?3x/x-4#+?5/x-4#
_l ?2x^2"-3x+5/x-4#_4
,6f9d ! algebraic sum ( two or m
frac;ns ) di6]5t denom9ators1 we m/
replace ! frac;ns ) equival5t frac;ns
hav+ ! same denom9ators4 ,x is pref]a#
6use ! .l1/ .-mon .denom9ator (,,lcd)_4
,! ,,lcd ( two or m frac;ns 3si/s (!
product ( all ! unique prime factors 9 !
denom9ators1 ea* ) an expon5t equal 6!
l>ge/ expon5t ) : ! factor appe>s1 & is
re,y a result
""=; a#ed
#c-#g ,,,addi;n ( frac;ns,
,! algebraic sum ( two or m frac;ns hav+
! same denom9ator is a frac;n ) ! common
denom9ator & a num]ator : is ! algebraic
sum ( ! num]ators ( ! frac;ns 3sid]$4 ,?
0 prov$ 9 ,problem #ac1 ,>ticle #b-#d4
^1,illu/ra;n4
;;;(#bx9#b./x"-#d) "- (#cx./x"-#d)
"6 (#e./x"-#d)
_= (#bx9#b"-#cx"6#e./x"-#d)4;
,to f9d ! algebraic sum ( two or m
frac;ns ) di6]5t denom9ators1 we m/
replace ! frac;ns ) equival5t frac;ns
hav+ ! same denom9ators4 ,x is pref]able
to use ! .7l1/ common denom9ator.
"<,,lcd">4 ,! ,,lcd ( two or m frac;ns
3si/s ( ! product ( all ! unique prime
factors 9 ! denom9ators1 ea* ) an
expon5t equal to ! l>ge/ expon5t ) : !
factor appe>s1 & is r1lly a result #he
(! foll[+ important !orem4 b#ed
,,!orem #3-5_4
?a/b#+?c/d# _l ?ad+bc/bd#
(;b, d /.k #0)_4
.,pro(4 ,we h
?a/b#+?c/d# _l ?ad/bd#+?bc/bd#,
0,!orem #2-8_4 ,if we n[ use ,pro#m #13,
,>ticle #2-4, we h
?ad/bd#+?bc/bd# _l ?ad+bc/bd#,
: is r requir$ result4
,,example #1_4 ,f9d ! ,,lcd (! frac;ns
?3x/x^2"-4x+4#,
?5x^2"/3(x^2"-4)#,
?2/2x^2"-x-6#_4
-------------------------------------#ee
.,solu;n4 ,factor+ ea* denom9ator1 we
h
( ! foll[+ important !orem4 b#ed
^7,!orem #c-#e4^
;;;(a./b) "6 (c./d) _= (ad"6bc./bd)
"<b1 d "7@: #j">4;
.1,pro(4 ,we h
;;;(a./b) "6 (c./d)
_= (ad./bd) "6 (bc./bd)1;
by ,!orem #b-#h4 ,if we n[ use ,problem
#ac1 ,>ticle #b-#d1 we h
;;;(ad./bd) "6 (bc./bd)
_= (ad"6bc./bd)1;
: is r requir$ result4
^7,example #a4^ ,f9d ! ,,lcd ( !
frac;ns
""=;;;
(#cx./x9#b"-#dx"6#d)1
(#ex9#b./#c"<x9#b"-#d">)1
(#b./#bx9#b"-x"-#f)4
""=;
-------------------------------------#ee
.1,solu;n4 ,factor+ ea* denom9ator1 we
h
#hg
x^2"-4x+4 _l (x-2)^2, a#ee
#3(x^2"-4) _l #3(x+2)(x-2),
#2x^2"-x-6 _l (2x+3)(x-2)_4
,! ,,lcd is #3(x+2)(x-2)^2"(2x+3)_4
,af ! ,,lcd has be5 det]m9$1 equival5t
frac;ns may 2 =m$4 ,divide ! ,,lcd (a
giv5 frac;n 0! denom9ator ( t frac;n1 &
!n multiply bo? num]ator & denom9ator (!
giv5 frac;n 0! result4 ,! equival5t
frac;ns may n[ 2 a4$1 z 9 ! illu/r,n
abv4
,,example #2_4 ,*ange ! foll[+ frac;ns
6equival5t "os1 ) _! ,,lcd z denom9ator1
& f9d _! sum4
?4/x+2#,
?x+3/x^2"-4#,
""=;;; a#ee
x9#b "- #dx "6 #d
_= "<x "- #b">9#b1
#c"<x9#b "- #d">
_= #c"<x "6 #b">"<x "- #b">1
#bx9#b "- x "- #f
_= "<#bx "6 #c">"<x "- #b">4
""=;
,! ,,lcd is #c"<x "6 #b">"<x "- #b">9#b
"<#bx "6 #c">4
,af ! ,,lcd has be5 det]m9$1 equival5t
frac;ns may 2 =m$4 ,divide ! ,,lcd ( a
giv5 frac;n by ! denom9ator ( t frac;n1
& !n multiply bo? num]ator & denom9ator
( ! giv5 frac;n by ! result4 ,!
equival5t frac;ns may n[ 2 add$1 z 9 !
illu/ra;n abv4
^7,example #b4^ ,*ange ! foll[+
frac;ns to equival5t "os1 ) _! ,,lcd z
denom9ator1 & f9d _! sum4
""=;;;
(#d./x"6#b)1
(x"6#c./x9#b"-#d)1 #hi
?2x+1/x-2#_4 b#ee
.,solu;n4 ,! ,,lcd is (x+2)(x-2)_4
,"!=e1
?4/x+2# _l ?4(x-2)/(x+2)(x-2)#,
?x+3/x^2"-4# _l ?x+3/(x+2)(x-2)#,
?2x+1/x-2#
_l ?(2x+1)(x+2)/(x+2)(x-2)#,
&
(#bx"6#a./x"-#b)4 b#ee
""=;
.1,solu;n4 ,! ,,lcd is
"<;x "6 #b">"<x "- #b">4 ,"!=e1
""=;;;
(#d./x"6#b)
_= (#d"<x"-#b">
./"<x"6#b">"<x"-#b">)1
(x"6#c./x9#b"-#d)
_= (x"6#c./"<x"6#b">"<x"-#b">)1
(#bx"6#a./x"-#b)
_= ("<#bx"6#a">"<x"6#b">
./"<x"6#b">"<x"-#b">)1
""=;
&
#ia
?4/x+2#+?x+3/x^2"-4#+?2x+1/x-2# c#ee
_l ?4(x-2)/(x+2)(x-2)#
+?x+3/(x+2)(x-2)#
+?(2x+1)(x+2)/(x+2)(x-2)#
_l ?(4x-8)+(x+3)+(2x^2"+5x+2)
/x^2"-4#
_l ?2x^2"+10x-3/x^2"-4#_4
,,pro#ms
,reduce ! foll[+ 6s+le frac;ns &
simplify4
#1_4 ?2/3#+?5/6#-?3/10#
#2_4 #5-?4/9#-?7/15#
#3_4 ?3x/4y#-?4y/3x#
#4_4 ?a^2"/b#-?b^2"/a#
#5_4 ?2x+3/6#-?4x-7/9#
#6_4 ?3x-1/5#+?4-5x/6#
-------------------------------------#ef
#7_4 x+y+?x^2"/x-y#
;;;(#d./x"6#b) c#ee
"6 (x"6#c./x9#b"-#d)
"6 (#bx"6#a./x"-#b)
_= (#d"<x"-#b">./"<x"6#b">"<x"-#b">)
"6 (x"6#c./"<x"6#b">"<x"-#b">)
"6 ("<#bx"6#a">"<x"6#b">
./"<x"6#b">"<x"-#b">)
_= ("<#dx"-#h">"6"<x"6#c">
"6"<#bx9#b"6#ex"6#b">
./x9#b"-#d)
_= (#bx9#b"6#ajx"-#c./x9#b"-#d)4;
,,problems
,reduce ! foll[+ to s+le frac;ns &
simplify4
""=;;;
#a4 #b/c "6 #e/f "- #c/aj
#b4 #e "- #d/i "- #g/ae
#c4 (#cx./#dy) "- (#dy./#cx)
#d4 (a9#b./b) "- (b9#b./a)
#e4 (#bx"6#c./#f) "- (#dx"-#g./#i)
#f4 (#cx"-#a./#e) "6 (#d"-#ex./#f)
-------------------------------------#ef
#g4 x "6 y "6 (x9#b./x"-y) #ic
#8_4 ?x+1/x+2#-?x+3/x# a#ef
#9_4 ?3x-2y/5x-3#+?2x-y/3-5x#
#10_4 ?2/12x^2"-3#+?3/2x-4x^2"#
#11_4 ?5/x#-?4/y#+?3/z#
#12_4 ?4/x^2"-4x-5#+?2/x^2"-1#
#13_4 ?2x-1/4-x#+?x+2/3x-12#
#14_4 ?x+5/x^2"+7x+10#-?x-1/x^2"+5x+6#
#15_4
?x-1/2x^2"-13x+15#+?x+3/2x^2"-15x+18#
#16_4 ?2x+3/3x^2"+x-2#-?3x-4/2x^2"-3x-5#
#17_4 ?3/a-3#+?a^2"+2/a^3"-27#
#18_4 ?2xy/x^3"+y^3"#-?x/x^2"-xy+y^2"#
#19_4 ?2/x^2"+3x+2#-?3/x^2"+5x+6#
-?4/x^2"+4x+3#
#h4 (x"6#a./x"6#b) "- (x"6#c./x) a#ef
#i4 (#cx"-#by./#ex"-#c)
"6 (#bx"-y./#c"-#ex)
#aj4
(#b./#abx9#b"-#c) "6 (#c./#bx"-#dx9#b)
#aa4 (#e./x) "- (#d./y) "6 (#c./z)
#ab4
(#d./x9#b"-#dx"-#e) "6 (#b./x9#b"-#a)
#ac4
(#bx"-#a./#d"-x) "6 (x"6#b./#cx"-#ab)
#ad4 (x"6#e./x9#b"6#gx"6#aj)
"- (x"-#a./x9#b"6#ex"6#f)
#ae4 (x"-#a./#bx9#b"-#acx"6#ae)
"6 (x"6#c./#bx9#b"-#aex"6#ah)
#af4 (#bx"6#c./#cx9#b"6x"-#b)
"- (#cx"-#d./#bx9#b"-#cx"-#e)
#ag4
(#c./a"-#c) "6 (a9#b"6#b./a9#c"-#bg)
#ah4 (#bxy./x9#c"6y9#c)
"- (x./x9#b"-xy"6y9#b)
#ai4 (#b./x9#b"6#cx"6#b)
"- (#c./x9#b"6#ex"6#f)
"- (#d./x9#b"6#dx"6#c)
#ie
#20_4 b#ef
x+6+?5x+1/12x^2"+5x-2#-?x/3x+2#
#21_4 #2y-3+?y-2/4y^2"-12y+9#
+?y+2/2y^2"-y-3#
#22_4 ?1/(x-y)(y-z)#+?1/(y-z)(z-x)#
+?1/(z-x)(x-y)#
#23_4 ?x/(x-y)(y-z)#+?y/(y-z)(z-x)#
+?z/(z-x)(x-y)#
#24_4 ?2x-1/2x^2"-x-6#+?x+3/6x^2"+x-12#
-?2x-3/3x^2"-10x+8#
,,5d ,,( ,,algebra ,,sample
,,9 ,,neme? ,,code
#bj4 x "6 #f b#ef
"6 (#ex"6#a./#abx9#b"6#ex"-#b)
"- (x./#cx"6#b)
#ba4 #by "- #c
"6 (y"-#b./#dy9#b"-#aby"6#i)
"6 (y"6#b./#by9#b"-y"-#c)
#bb4 (#a./"<x"-y">"<y"-z">)
"6 (#a./"<y"-z">"<z"-x">)
"6 (#a./"<z"-x">"<x"-y">)
#bc4 (x./"<x"-y">"<y"-z">)
"6 (y./"<y"-z">"<z"-x">)
"6 (z./"<z"-x">"<x"-y">)
#bd4 (#bx"-#a./#bx9#b"-x"-#f)
"6 (x"6#c./#fx9#b"6x"-#ab)
"- (#bx"-#c./#cx9#b"-#ajx"6#h)
""=;
,,,5d ( algebra sample 9 uebc,
#ig
,sample #c4
,calculus
,? sample is transcrib$ us+ ..,!
,neme? ,brl ,code = ,ma!matics & ,sci;e
,not,n #aigb .,revi.n 7on left-h& pages7
&! ,unifi$ ,5gli% ,brl ,code z ( ,june
#bjja 7on "r-h& pages74
#ii
,calculus ,sample 9 ,neme? ,code
7777777777777777777777777777777777777777
,neme? ,symbols
_ 7#def7 punctu,n 9dicator
_ 7#def7 boldface lr 9dicator
" 7#e7 2g9 modifi$ expres.n
% 2g9 modifi] 2l
< 2g9 modifi] abv
] 5d modifi$ expres.n
; 7#ef7 2g9 subscript
;; 7#ef1 #ef7 2g9 subscript )9 subscript
^ 7#de7 2g9 sup]script
^; 7#de1 #ef7 2g9 subscript )9
sup]script
[ -ma )9 subscript or sup]script
" 7#e7 return 6basel9e af subscript or
sup]script
> 2g9 squ>e root
] 5d squ>e root
? 2g9 frac;n
_? 2g9 frac;n por;n ( mix$ numb]
/ horizontal frac;n l9e
# 5d frac;n
_# 5d frac;n por;n ( mix$ numb]
, 7#f7 ma!matical -ma
,calculus ,sample 9 ,,uebc
7777777777777777777777777777777777777777
,,uebc ,symbols
.=; grade "o symbol
.=;; grade "o ^w
.=;;; 2g9 grade "o passage
.=; 72f a space7 5d grade "o passage
.=""= dot locator 72f l[] symbol on a
l9e 0xf7
.=,,, 2g9 capitaliz$ passage
.=, 72f a space7 5d capitaliz$ passage
.=.1 italic ^w
.=.7 2g9 italic passage
.=. 72f a space7 5d italic passage
.=^2 bold lr
.=^1 bold ^w
.=( 2g9 frac;n
.=./ frac;n l9e
.=) 5d frac;n
.=< 2g9 -p.d item
.=> 5d -p.d item
.=% 2g9 squ>e root
.=+ 5d squ>e root #aja
,neme? ,symbols 73t47
. 7#df7 decimal po9t
+ plus sign
- 7#cf7 m9us sign
* "ts sign 7dot7
.k equal sign
"k less ?an sign
"k.k less ?an or equal sign
( op5+ p>5!sis
) clos+ p>5!sis
@( op5+ bracket
@) clos+ bracket
@# a/]isk
! 9tegral sign
!@$c$59o] 9tegral sign ) sup]impos$
circle be>+ c.t]clockwise >r[
!@$c$[59] 9tegral sign ) sup]impos$
circle be>+ clockwise >r[
,= 9f9;y sign
' 7#c7 prime sign
^.* degree sign
$o "r >r[
v]tical b>
$_4 solid squ>e
.,d ,greek capital delta
,,uebc ,symbols 73t47
.=9 sup]script next item
.=5 subscript next item
.=.5 next item directly 2l previs item
.=.9 next item directly abv previs item
.=& sup]impose next symbol on previs
symbol
.=^: >r[ ov] previs item
.=4 decimal po9t
.=444 ellipsis
.="6 plus sign
.="- m9us sign
.="4 "ts sign 7dot7
.="7 equal sign
.=@< less ?an sign
.=_@< less ?an or equal sign
.="< op5+ p>5!sis
.="> clos+ p>5!sis
.=.< op5+ bracket
.=.> clos+ bracket
.=,- da%
.="9 a/]isk
.=! 9tegral sign
.=#= 9f9;y sign #ajc
,neme? ,symbols 73t47
.,s ,greek capital sigma
.a ,greek alpha
.f ,greek phi
.p ,greek pi
.y ,greek psi
.s ,greek sigma
gggggggggggggggggggggggggggggggggggggggg
,,uebc ,symbols 73t47
.=7 prime sign
.=^j degree sign
.=o "r >r[
.=_ v]tical l9e
.=_$#d fill$ squ>e
.=@$cc transcrib]-def9$ %ape3 circle )
c.t]clockwise >r[
.=@$cl transcrib]-def9$ %ape3 circle )
clockwise >r[
.=: 5d %ape
.=,.d capital ,greek delta
.=,.s capital ,greek sigma
.=.a ,greek alpha
.=.f ,greek phi
.=.p ,greek pi
.=.y ,greek psi
.=.s ,greek sigma
gggggggggggggggggggggggggggggggggggggggg
#aje
#5 ,vector ,9tegral ,calculus #bgi
,"p ;,i_4 ,two-,dim5.nal ,!ory
#5.1 ,,9troduc;n
,! topic ( ? *apt] is ..l9e & 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
"!%,c]ds_4 ,! subscript ;,c 9dicates t
"o is m1sur+ ! l5g? (a curve ;,c, z 9
,fig4 #5.1_4 ,if ;,c is giv5 9 p>ametric
=m x .k x(t), y .k y(t), ! l9e 9tegral
reduces 6! ord9>y def9ite 9tegral3
"!%,c]ds
.k !;t;;1^t^;2
">(?dx/dt#)^2"+(?dy/dt#)^2"]dt_4
,if ! curve ;,c repres5ts a wire ^:
d5s;y (mass p] unit l5g?) v>ies
#e ,vector ,9tegral ,calculus #bgi
,"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;
,if ! curve ;,c repres5ts a wire ^:
d5s;y "<mass p] unit l5g?"> v>ies #ajg
al;g ;,c, !n ! wire has a total a#bgi
mass
,m .k "!%,c]f(x, y)ds,
": f(x, y) is ! d5s;y at ! po9t (x, y)
(! wire4 ,! new 9tegral c 2 express$ 9
t]ms (a p>amet] z previsly or c 2 ?" (
simply z a limit (a sum
"!%,c]f(x, y)ds
.k lim ".,s%i .k #1<n]
f(x;i^@#, y;i^@#").,d;i"s_4
------------------------------------#bhj
,"h ! curve has be5 subdivid$ 9to ;n
pieces ( l5g?s
.,d1s, .,d2s, ''', .,d;n"s, &! po9t
(x;i^@#, y;i^@#") lies on ! ;ith piece4
,! limit is tak5 z ;n 2comes 9f9ite1
:ile ! maximum .,d;i"s approa*es #0_4
,a ?ird example (a l9e 9tegral is t (
."w4 ,if a "picle moves f "o 5d ( ;,c 6!
o!r "u ! 9flu;e (a =ce _;,f, ! "w d"o 0?
=ce is def9$ z
"!%,c],f;,t"ds,
": ,f;,t denotes ! -pon5t ( _;,f on !
tang5t _;,t 9 ! direc;n (
al;g ;,c1 !n ! wire has a total a#bgi
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 previsly
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
;;;,.d5#as1 ,.d5#bs1 4441 ,.d5ns1; & !
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
;;;!.5,c ,f5,t ds1;
": ,f;5,t denotes ! compon5t ( ^2;,f on
! tang5t ^2;,t 9 ! direc;n ( #aji
mo;n4 ,? 9tegral c 2 ?" ( z a a#bhj
limit (a sum z previsly4 ,h["e1 ano!r
9t]pret,n is possi#4 ,we f/ rem>k t ! "w
d"o 0a 3/ant =ce _;,f 9 mov+ a "picle f
;,a to ;,b on ! l9e seg;t ,a,b is
_;,f*",a,b<$o]_2 = ? scal> product is
equal to _;,f*cos .a*",a,b<$o], .a
2+ ! angle 2t _;,f & ",a,b<$o], & h;e 6!
product ( =ce -pon5t 9 direc;n ( mo;n 0!
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 #5.2_4
,if ^! 4place;ts >e denot$ by
.,d1_;r, .,d2_;r, ''', .,d;n"_;r, ! "w
d"o wd 2 approximat$ 0a sum ( =m
".,s%i .k #1<n]_;,f;i"*.,d;i"_;r,
------------------------------------#bha
": _;,f;i is ! =ce act+ =! ;ith
4place;t4 ,! limit+ =m ( ? is ag equal
6! l9e 9tegral !,f;,t"ds, b 2c (! way !
limit is obta9$1 we c al write x z
"!%,c]_;,f*d_;r_4
mo;n4 ,? 9tegral c 2 ?" ( z a a#bhj
limit ( a sum z previsly4 ,h["e1 ano!r
9t]preta;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
;;;,.d5#a^2r1 ,.d5#b^2r1 4441 ,.d5n^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 equal
to ! l9e 9tegral ;! ,f;5,t ds1 b 2c ( !
way ! limit is obta9$1 we c al write x z
;;;!.5,c ^2,f "4 d^2r4; #aaa
,"o c ?us write a#bha
work .k "!%,c],f;,t"ds
.k "!%,c]_;,f*d_;r_4
,if ! 4place;t vector .,d_;r & =ce
_;,f >e express$ 9 -pon5ts1
_;,f .k ,f;x"_;i+,f;y"_;j,
.,d_;r .k .,dx_;i+.,dy_;j,
! ele;t ( "w _;,f*.,d_;r 2comes
_;,f*.,d_;r .k ,f;x".,dx+,f;y".,dy_4
,! total am.t ( "w d"o is !n approximat$
0a sum ( =m
.,s(,f;x".,dx+,f;y".,dy)
.k .,s,f;x".,dx+.,s,f;y".,dy_4
,! limit+ =m ( ? is a sum ( two
9tegrals3
"!%,c],f;x"dx+"!%,c],f;y"dy_4
,! f/ 9tegral repres5ts ! "w d"o 0!
;x-compon5t (! =ce2 ! second 9tegral
repres5ts ! "w d"o 0! ;y-compon5t (!
=ce4
,x ?us appe>s t "o has ?ree types (
l9e 9tegrals 63sid]1 "nly1 ! types
,"o c ?us write a#bha
;;;work "7 !.5,c ,f5,t ds
"7 !.5,c ^2,f "4 d^2r4;
,if ! 4place;t vector ,.d^2r & =ce
^2;,f >e express$ 9 compon5ts1
;;;^2,f "7 ,f5x^2i "6 ,f5y^2j1
,.d^2r "7 ,.dx^2i "6 ,.dy^2j1;
! ele;t ( "w ^2;,f "4 ,.d^2r 2comes
;;;^2,f "4 ,.d^2r
"7 ,f5x,.dx "6 ,f5y,.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
,x ?us appe>s t "o has ?ree types (
l9e 9tegrals to 3sid]1 "nly1 ! types
#aac
"!%,c]f(x, y)ds, b#bha
"!%,c],p(x, y)dx,
"!%,c],q(x, y)dy,
: >e limits ( sums
.,sf(x, y).,ds, .,s,p(x, y).,dx,
.,s,q(x, y).,dy_4
,! =ego+ gives ! basis =! !ory ( l9e
9tegrals 9 ! plane4 ,a v sli<t ext5.n (
^! id1s l1ds 6l9e 9tegrals 9 space3
"!%,c]f(x, y, z)ds,
"!%,c]f(x, y, z)dx, '''_4
,surface 9tegrals appe> z a natural
g5]aliz,n1 )! surface >ea ele;t d.s
replac+ ! >c ele;t ds_3
"!!%,s]f(x, y, z)d.s
.k lim .,sf(x, y, z).,d.s_4
,"! >e correspond+ -pon5t 9tegrals
"!!%,s]f(x, y, z)dxdy,
"!!%,s]f(x, y, z)dydz, '''
------------------------------------#bhb
&a vector surface 9tegral
"!!%,s]_;,f*d_.s
.k "!!%,s](_;,f*_;n)d.s,
": d_.s .k _;nd.s is ! 8>ea
;;;!.5,c f"<x1 y"> ds1 b#bha
!.5,c ,p"<x1 y"> dx1
!.5,c ,q"<x1 y"> dy1;
: >e limits ( sums
;;;,.s f"<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]aliza;n1 ) ! surface >ea ele;t d;.s
replac+ ! >c ele;t ds3
;;;<!!>.5,s f"<x1 y1 z"> d.s
"7 lim ,.s f"<x1 y1 z"> ,.d.s4;
,"! >e correspond+ compon5t 9tegrals
;;;<!!>.5,s f"<x1 y1 z"> dx dy1
<!!>.5,s f"<x1 y1 z"> dy dz1 444;
------------------------------------#bhb
& a vector surface 9tegral
;;;<!!>.5,s ^2,f "4 d^2.s
"7 <!!>.5,s "<^2,f "4 ^2n"> d.s1;
": d^2;.s "7 ^2;n d;.s is ! 8>ea #aae
ele;t vector10 _;n 2+ a unit a#bhb
normal vector 6! surface4
,x w 2 se5 t ! basic !orems--^? (
,gre51 ,gauss1 & ,/okes--3c]n ! rel,ns
2t l9e1 surface1 & volume (triple)
9tegrals4 ,^! correspond 6funda;tal
physical rel,ns 2t s* quantities z flux1
circul,n1 div]g;e1 & curl4 ,! applic,ns
w 2 3sid]$ at ! 5d (! *apt]4
#5.2 ,,l9e ,,9tegrals
,,9 ,,! ,,plane
,we n[ /ate 9 precise =m ! def9i;ns
tl9$ 9 ! prec$+ sec;n4
,0a .smoo? .curve ;,c 9 ! xy-plane w 2
m1nt a curve repres5ta# 9 ! =m3
(5.1) x .k .f(t), y .k .y(t),
h "k.k t "k.k k,
": ;x & ;y >e 3t9us & h 3t9us
derivatives = h "k.k t "k.k k_4 ,! curve
;,c c 2 assign$ a direc;n1 : w usu,y 2 t
( 9cr1s+ ;t_4 ,if ;,a denotes ! po9t
@(.f(h), .y(h)@) & ;,b denotes ! po9t
@(.f(k), .y(k)@), !n ;,c c 2 ?" ( z !
ele;t vector10 ^2;n 2+ a unit a#bhb
normal vector to ! surface4
,x w 2 se5 t ! basic !orems,-^? (
,gre51 ,gauss1 & ,/okes,-3c]n ! rela;ns
2t l9e1 surface1 & volume "<triple">
9tegrals4 ,^! correspond to funda;tal
physical rela;ns 2t s* quantities z
flux1 circula;n1 div]g;e1 & curl4 ,!
applica;ns w 2 3sid]$ at ! 5d ( ! *apt]4
#e4b ,,,l9e 9tegrals 9 ! plane,
,we n[ /ate 9 precise =m ! def9i;ns
tl9$ 9 ! prec$+ sec;n4
,by a .7smoo? curve. ;,c 9 ! xy-plane
w 2 m1nt a curve repres5table 9 ! =m3
;;;"<#e4a"> x "7 .f"<t">1
y "7 .y"<t">1 h _@< t _@< k1;
": ;x & ;y >e 3t9us & h 3t9us
derivatives = ;h _@< ;t _@< ;k4 ,! curve
;,c c 2 assign$ a direc;n1 : w usually 2
t ( 9cr1s+ ;t4 ,if ,a denotes ! po9t
.<.f"<h">1 .y"<h">.> & ;,b denotes !
po9t .<.f"<k">1 .y"<k">.>1 !n ;,c c 2
?" ( z ! #aag
pa? (a po9t mov+ 3t9usly f ;,a b#bhb
to ;,b_4 ,? pa? may cross xf1 z =! curve
,c1 ( ,fig4 #5.3_4 ,if ! 9itial po9t ;,a
& t]m9al po9t ;,b co9cide1 ;,c is t]m$ a
.clos$ curve2 if1 9 a4i;n1 (x, y) moves
f ;,a to ,b .k ,a )t retrac+ any o!r
po9t1 ;,c is call$ a .simple .clos$
curve (curve ,c2 ( ,fig4 #5.3)_4
,let ;,c 2 a smoo? curve z previsly1
) positive direc;n t ( 9cr1s+ ;t_4 ,let
f(x, y) 2 a func;n def9$ at l1/ :5
(x, y) is on ;,c_4 ,!
------------------------------------#bhc
l9e 9tegral "!%,c]f(x, y)dx is def9$ z a
limit3
(5.2) "!%,c]f(x, y)dx
.k lim ".,s%i .k #1<n]
f(x;i^@#, y;i^@#").,d;i"x_4
,! limit ref]s 6a subdivi.n ( ;,c z
9dicat$ 9 ,fig4 #5.4_4 ,! su3essive
subdivi.n po9ts >e
;,a_3 (x0, y0), (x1, y1), ''',
;,b_3 (x;n, y;n")_4 ,^! correspond
6p>amet] values3
pa? ( a po9t mov+ 3t9usly f ,a b#bhb
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$. curve "<curve ,c;5#b (
,fig4 #e4c">4
,let ;,c 2 a smoo? curve z previsly1
) positive direc;n t ( 9cr1s+ ;t4 ,let
f"<x1 ;y"> 2 a func;n def9$ 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 "<x5#j1 y5#j">1 "<x5#a1 y5#a">1
4441 ,b3 "<x5n1 y5n">4; ,^! correspond
to p>amet] values3 #aai
h .k t0 "k t1 "k ''' "k t;n a#bhc
.k k_4 ,! po9t (x;i^@#, y;i^@#") is "s
po9t ( ;,c 2t (x;i-1, y;i-1") &
(x;i, y;i")_2 t is1 (x;i^@#, y;i^@#")
corresponds 6a p>amet] value t;i^@#, ":
t;i-1 "k.k t;i^@# "k.k t;i_4 .,d;i"x
denotes ! di6];e x;i"-x;i-1_4 ,! limit
is tak5 z ;n 2comes 9f9ite &! l>ge/
.,d;i"t approa*es #0, ":
.,d;i"t .k t;i"-t;i-1_4 ,simil>ly1
(5.3) "!%,c]f(x, y)dy
.k lim .,sf(x;i^@#, y;i^@#").,d;i"y,
": .,d;i"y .k y;i"-y;i-1_4
,! signific.e ( ^! def9i;ns is
gu>ante$ 0! foll[+ basic !orems3
;,i ,if f(x, y) is 3t9us on ;,c, !n
"!%,c]f(x, y)dx
&
;;;h "7 t5#j @< t5#a @< 444 a#bhc
@< t5n "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 ,.s f"<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 3t9us on ;,c1
!n
;;;!.5,c f"<x1 y"> dx;
& #aba
"!%,c]f(x, y)dy b#bhc
exi/4
,,ii ,if f(x, y) is 3t9us on ;,c, !n
(5.4) "!%,c]f(x, y)dx
.k !;h^k"f@(.f(t), .y(t)@).f'(t)dt,
(5.5) "!%,c]f(x, y)dy
.k !;h^k"f@(.f(t), .y(t)@).y'(t)dt_4
,=mulas (5.4) & (5.5) reduce !
9tegrals 6ord9>y def9ite 9tegrals & >e
?us ess5tial = -put,n ( "picul>
9tegrals4 ,?us let ;,c 2 !
------------------------------------#bhd
pa? x .k #1+t, y .k t^2,
#0 "k.k t "k.k #1, direct$ ) 9cr1s+ ;t_4
,!n
;;;!.5,c f"<x1 y"> dy; b#bhc
exi/4
,,ii ,if f"<x1 ;y"> is 3t9us on
;,c1 !n
""=;;;
"<#e4d"> !.5,c f"<x1 y"> dx
"7 !5h9k f.<.f"<t">1 .y"<t">.>
.f7"<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 = computa;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
#abc
"!%,c](x^2"-y^2")dx a#bhd
.k !;0^1"@((1+t)^2"-t^4"@)dt
.k ?32/15#,
"!%,c](x^2"-y^2")dy
.k !;0^1"@((1+t)^2"-t^4"@)2tdt
.k #2_?1/2_#_4
,x is logic,y easi] 6prove ,,ii f/1 =
;,i is an imm 3sequ;e ( ,,ii_4 ,6prove
,,ii, "o notes t ! sum
.,sf(x;i^@#, y;i^@#").,d;i"x c 2 writt5
z
".,s%i .k #1<n]
f@(.f(t1^@#"), .y(t1^@#")@)
?.,d;i"x/.,d;i"t#.,d;i"t_4
,n[ .,d;i"x .k x;i"-x;i-1
.k .f'(t;i^@#@#").,d;i"t 0! ,law (!
,m1n4 ,h;e ! sum c 2 writt5 z
".,s%i .k #1<n],f(t;i^@#")
.f'(t;i^@#@#").,d;i"t,
": ,f(t) .k f@(.f(t), .y(t)@) & t;i^@# &
t;i^@#@# >e bo? 2t
""=;;; a#bhd
!.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
""=;
,x is logically easi] to prove ,,ii
f/1 = ,i is an imm 3sequ;e ( ,,ii4 ,to
prove ,,ii1 "o notes t ! sum
;;;,.s f"<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.5<i"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 #abe
t;i-1 & t;i_4 ,x is easily %[n b#bhd
@(see ,,cla1 ,sec;n #12-25@) t ? sum
approa*es z limit ! 9tegral
!;h^k",f(t).f'(t)dt
.k !;h^k"f@(.f(t), .y(t)@).f'(t)dt
z requir$4 ,=mula (5.5) is prov$ 9 !
same way4
,we rem>k t ! value (a l9e 9tegral on
;,c does n dep5d on ! "picul>
p>ametriz,n ( ;,c, b only on ! ord] 9 :
! po9ts ( ;,c >e trac$4 (,see ,pro#m
#5_4)
,9 _m applic,ns ! pa? ;,c is n xf
smoo? b is -pos$ (a f9ite numb] ( >cs1
ea* ( : is smoo?4 ,?us ;,c mi<t 2 a
brok5 l9e4 ,9 ? case1 ;,c is t]m$
.piecewise smoo?4 ,! l9e 9tegral al;g
;,c is simply1 0def9i;n1 ! sum (!
9tegrals al;g ! pieces4 ,"o v]ifies at
once t (5.2), (5.3), &! !orems ;,i &
,,ii 3t9ue 6hold4 ,9 (5.4) & (5.5) !
func;ns .f'(t) & .y'(t) w h jump
4cont9uities1 : w n 9t]f]e )! exi/;e (!
9tegral (cf4 ,sec;n
;;t5<i"-#a> & t;5i4 ,x is easily b#bhd
%[n .<see ,,cla1 ,sec;n #ab-#be.> t ?
sum approa*es z limit ! 9tegral
;;;!5h9k ,f"<t">.f7"<t"> dt
"7 !5h9k f.<.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>
p>ametriza;n ( ;,c1 b only on ! ord] 9 :
! po9ts ( ;,c >e trac$4 "<,see ,problem
#e4">
,9 _m applica;ns ! pa? ;,c is n xf
smoo? b is compos$ ( 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 .f;7"<t"> & .y;7"<t">
w h jump 4cont9uities1 : w n 9t]f]e ) !
exi/;e ( ! 9tegral "<cf4 ,sec;n #abg
#4.1)_4 ..,"?t ? book all pa?s c#bhd
( 9tegr,n = l9e 9tegrals w 2 piecewise
smoo? un.s o!rwise .specifi$4
,if ! curve ;,c is repres5t$ 9 ! =m
y .k g(x), a "k.k x "k.k b,
!n "o c reg>d ;x xf z p>amet]1 replac+
;t_2 t is1 ;,c is giv5 0! equ,ns
x .k x, y .k g(x), a "k.k x "k.k b
9 t]ms (! p>amet] ;x_4 ,if ! direc;n (
;,c is t ( 9cr1s+ ;x, (5.4) &
------------------------------------#bhe
(5.5) 2come
(5.6) "!%,c]f(x, y)dx
.k !;a^b"f@(x, g(x)@)dx,
(5.7) "!%,c]f(x, y)dy
.k !;a^b"f@(x, g(x)@)g'(x)dx_4
,! ord9>y def9ite 9tegral !;a^b"ydx, ":
y .k g(x), is a special case ( (5.6)_4
,simil>ly1 if ;,c is repres5t$ 9 ! =m
x .k ,f(y), c "k.k y "k.k d,
&! direc;n ( ;,c is t ( 9cr1s+ ;y, !n
#d4a">4 .7,"?t ? book all pa?s c#bhd
( 9tegra;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 ! equa;ns
;;;x "7 x1 y "7 g"<x">1 a _@< x _@< b;
9 t]ms ( ! p>amet] ;x4 ,if ! direc;n (
;,c is t ( 9cr1s+ ;x1 "<#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 !5a9b f.<x1 g"<x">.>g7"<x"> dx4
""=;
,! ord9>y def9ite 9tegral ;;!5a9b ;y 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
#abi
(5.8) "!%,c]f(x, y)dx a#bhe
.k !;c^d"f@(,f(y), y@),f'(y)dy,
(5.9) "!%,c]f(x, y)dy
.k !;c^d"f@(,f(y), y@)dy_4
,9 mo/ applic,ns ! l9e 9tegrals appe>
z a -b9,n1
"!%,c],p(x, y)dx+"!%,c],q(x, y)dy,
: is a2reviat$ z foll[s3
"!%,c]@(,p(x, y)dx+,q(x, y)dy@)
or
"!%,c],p(x, y)dx+,q(x, y)dy,
! 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 (5.4)
2comes
(5.4') "!%,c]f(x, y)dx
.k !;k^h"f@(.f(t), .y(t)@).f'(t)dt_4
""=;;; a#bhe
"<#e4h"> !.5,c f"<x1 y"> dx
"7 !5c9d f.<,f"<y">1 y.>,f7"<y"> dy1
"<#e4i"> !.5,c f"<x1 y"> dy
"7 !5c9d f.<,f"<y">1 y.> dy4
""=;
,9 mo/ applica;ns ! l9e 9tegrals appe>
z a comb9a;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.>;
or
;;;!.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 !5k9h f.<.f"<t">1 .y"<t">.>
.f7"<t"> dt4; #aca
,! l9e 9tegral is "!=e multipli$ b#bhe
by -#1_4 ,(t5 x is 3v5i5t 6specify ! pa?
0xs equ,ns 9 "s =m & 69dicate ! direc;n
0us+ ! 9itial & t]m9al po9ts z l[] &
upp] limits3
"!%,c];,a^,b",pdx+,qdy
or
"!%,c];(x;;1;[y;;1;)^(x^;2^[y^;2^)
",pdx+,qdy_4
,x w 2 se5 lat] t "u c]ta9 3di;ns1 "o
ne$s only prescribe 9itial & t]m9al
po9ts3
!;,a^,b",pdx+,qdy_4
,,example #1 ,6evaluate
"!%,c];(1[0)^(-1[0)"(x^3"-y^3")dy,
------------------------------------#bhf
": ;,c is ! semicircle y .k >1-x^2"] %[n
9 ,fig4 #5.5, "o c repres5t ;,c
p>ametric,y3
x .k cos t, y .k sin t,
#0 "k.k t "k.k .p,
,! l9e 9tegral is "!=e multipli$ b#bhe
by "-#a4 ,(t5 x is 3v5i5t to specify !
pa? by xs equa;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;
,,example #a ,to evaluate
;;;!.5,c5<"<#a1#j">>9<"<"-#a1#j">>
"<x9#c "- y9#c"> dy1;
------------------------------------#bhf
": ;,c is ! semicircle
;;;y "7 %#a "- x9#b+; %[n 9 ,fig4 #e4e1
"o c repres5t ;,c p>ametrically3
;;;x "7 cos t1 y "7 sin t1
#j _@< t _@< .p1; #acc
&! 9tegral 2comes a#bhf
!;0^.p"(cos^3 t-sin^3 t)cos tdt
.k ?3.p/8#_4
,"o c use ;x z p>amet]1 &! 9tegral
2comes
!;1^-1"@(x^3"(1-x^2")^?3/2#"@)
?-x/>1-x^2"]#dx_2
? is cle>ly 9 a m awkw>d =m = 9tegr,n4
,! sub/itu;n x .k cos t br+s "o back 6!
p>ametric =m4 ,"o c use ;y z p>amet] b
has !n 6split ! 9tegral 96two "ps1 f
(1, 0) to (0, 1) & f (0, 1) to (-1, 0)_2
!;0^1"@((1-y^2")^?3/2#"-y^3"@)dy
+!;1^0"@(-(1-y^2")^?3/2#"-y^3"@)dy
.k #2!;0^1"(1-y^2")^?3/2#"dy_4
,note that x .k >1-y^2"] on ! f/ "p (!
pa? & x .k ->1-y^2"] on ! second "p4 $_4
& ! 9tegral 2comes a#bhf
;;;!5#j9.p
"<cos9#c t "- sin9#c t">cos t dt
"7 (#c.p./#h)4;
,"o c use ;x z p>amet]1 & ! 9tegral
2comes
;;;!5#a9<"-#a>
.<x9#c"<#a "- x9#b">9#c/b.>
("-x./%#a "- x9#b+) dx2;
? is cle>ly 9 a m awkw>d =m = 9tegra;n4
,! sub/itu;n ;x "7 cos ;t br+s "o back
to ! p>ametric =m4 ,"o c use ;y z
p>amet] b has !n to split ! 9tegral 9to
two "ps1 f "<#a1 #j"> to "<#j1 #a"> & f
"<#j1 #a"> to "<"-#a1 #j">2
;;;!5#j9#a
.<"<#a "- y9#b">9#c/b "- y9#c.> dy
"6 !5#a9#j
.<"-"<#a "- y9#b">9#c/b "- y9#c.> dy
"7 #b!5#j9#a "<#a "- y9#b">9#c/b
dy4;
,note t ;;;x "7 %#a "- y9#b+; on ! f/ "p
( ! pa? & ;;;x "7 "-%#a "- y9#b+; on !
second "p4 _$#d
#ace
,,example #2 ,let ;,c 2 ! b#bhf
p>abolic >c y .k x^2 f (0, 0) to (-1,
1)_4 ,!n
"!%,c]xy^2"dx+x^2"ydy
.k !;0^-1"(xy^2"+x^2"y?dy/dx#)dx
.k !;0^-1"(x^5"+2x^5")dx
.k ?1/2#_4 $_4
,if ;,c is a .clos$ curve1 !n "! is no
ne$ 6specify 9itial & t]m9al po9t1 ?< !
direc;n m/ 2 9dicat$4 ,if ;,c is a
simple clos$ curve (trac$ j once), !n "o
ne$ only specify : (! two possi#
direc;ns is *os54 ,! not,ns
(a) !@$c$59o],pdx+,qdy,
(b) !@$c$[59],pdx+,qdy
ref] 6! two cases ( ,figs4 #5.6(a) &
#5.6(b)_4 ,! c.t]clockwise >r[ ref]s
6:at is r<ly a c.t]clockwise direc;n on
;,c_2 ? w 2 t]m$ ! .positive direc;n (as
= angul> m1sure)_2 ! clockwise direc;n w
2 call$
,,example #b ,let ;,c 2 ! b#bhf
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 ,! nota;ns
""=;;;
"<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$ #acg
! .negative direc;n4 ,x %d 2 not$ #bhg
t ! direc;n c 2 specifi$ 0ref];e 6! unit
tang5t vector _;,t 9 ! direc;n ( 9tegr,n
&! unit normal vector _;n t po9ts 6!
tside (! region b.d$ by ;,c_2 =!
positive direc;n1 _;n is #90^.* 2h _;,t,
z 9 ,fig4 #5.6(a)_2 =! negative direc;n1
_;n is #90^.* ah1d ( _;,t z 9
,fig4 #5.6(b)_4
,,example #3 ,6evaluate
"!@$c$59o]%,c]y^2"dx+x^2"dy,
": ;,c is ! triangle ) v]tices (1, 0),
(1, 1), (0, 0), %[n 9 ,fig4 #5.7, "o has
6-pute ?ree 9tegrals4 ,! f/ is ! 9tegral
f (0, 0) to (1, 0)_2 al;g ? pa?1 y .k #0
&1 if ;x is ! p>amet]1 dy .k #0_4 ,h;e !
f/ 9tegral is #0_4 ,! second 9tegral is
t f (1, 0) to (1, 1)_2 if ;y is us$ z
p>amet]1 ? reduces to
!;0^1"dy .k #1,
s9ce dx .k #0_4 ,=! ?ird 9tegral1
! .1negative direc;n4 ,x %d 2 #bhg
not$ t ! direc;n c 2 specifi$ by ref];e
to ! unit tang5t vector ^2;,t 9 !
direc;n ( 9tegra;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
;;;!&@$cc:.5,c y9#b dx "6 x9#b dy1;
": ;,c is ! triangle ) v]tices
"<#a1 #j">1 "<#a1 #a">1 "<#j1 #j">1 %[n
9 ,fig4 #e4g1 "o has to compute ?ree
9tegrals4 ,! f/ is ! 9tegral f
"<#j1 #j"> to "<#a1 #j">2 al;g ? pa?1
;y "7 #j &1 if ;x is ! p>amet]1
dy "7 #j4 ,h;e ! f/ 9tegral is #j4 ,!
second 9tegral is t f "<#a1 #j"> to
"<#a1 #a">2 if ;y is us$ z p>amet]1 ?
reduces to
;;;!5#j9#a dy "7 #a1;
s9ce dx "7 #j4 ,= ! ?ird 9tegral1 #aci
f (1, 1) to (0, 0), ;x c 2 us$ z a#bhg
p>amet]1 s t ! 9tegral is
!;1^0"2x^2"dx .k -?2/3#,
------------------------------------#bhh
s9ce dy .k dx_4 ,?us f9,y
"!@$c$59o]%,c]y^2"dx+x^2"dy
.k #0+1-?2/3# .k ?1/3#_4 $_4
#5.3 ,,9tegrals ,,) ,,respect
,,6,,>c ,,l5g?--,,basic
,,prop]ties ,,( ,,l9e ,,9tegrals
,=a smoo? or piecewise smoo? pa? ;,c,
z 9 ! prec$+ sec;n1 >c l5g? ;s is well
def9$4 ,?us ;s c 2 def9$ z ! 4t.e
trav]s$ f ! 9itial po9t (t .k h) up 6a
g5]al ;t_3
(5.10) s
.k !;h^t">(?dx/dt#)^2"+(?dy/dt#)^2"]
dt_4
,if ! curve ;,c is direct$ ) 9cr1s+ ;t,
!n ;s al 9cr1ses 9 ! direc;n ( mo;n1 go+
f #0 up 6! l5g? ;,l ( ;,c_4 ,let ;,c 2
subdivid$ z 9 ,fig4 #5.4 & let
f "<#a1 #a"> to "<#j1 #j">1 ;x c a#bhg
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 f9ally
;;;!&@$cc:.5,c y9#b dx "6 x9#b dy
"7 #j "6 #a "- #b/c "7 #a/c4 _$#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 "<;t "7 ;h"> up
to a g5]al ;t3
;;;"<#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 #ada
.,d;i"s denote ! 9cre;t 9 ;s f a#bhh
t;i-1 to t;i, t is1 ! 4t.e mov$ 9 ?
9t]val4 ,"o !n makes ! def9i;n
(5.11) "!%,c]f(x, y)ds
.k "lim%n $o ,=%%max .,d;i"s $o #0]
".,s%i .k #1<n]f(x;i^@#, y;i^@#")
.,d;i"s_4
,if ;f is 3t9us on ;,c, ? 9tegral w
exi/ & c 2 evaluat$ z foll[s3
(5.12) "!%,c]f(x, y)ds
.k !;h^k"f@(.f(t), .y(t)@)
>.f'(t)^2"+.y'(t)^2"]dt_4
,? is prov$ 9 ! same way z (5.4) &
(5.5), )! aid (! =mula
?ds/dt#
.k >(?dx/dt#)^2"+(?dy/dt#)^2"]
.k >.f'(t)^2"+.y'(t)^2"]_4
,"o c 9 pr9ciple use ;s xf z ! p>amet]
on ! curve ;,c_2 if ? is d"o1 ;x & ;y
2come func;ns (
;s_3 x .k x(s), y .k y(s)_4 ,! po9t
@(x(s), y(s)@) is !n ! posi;n (! mov+
po9t af a 4t.e ;s has be5 trav]s$4 ,9 ?
case1 (5.11) reduces 6a
,.d;5is denote ! 9cre;t 9 ;s f a#bhh
;;t5<i"-#a> to t;5i1 t is1 ! 4t.e mov$ 9
? 9t]val4 ,"o !n makes ! def9i;n
;;;"<#e4aa"> !.5,c f"<x1 y"> ds
"7 lim.5<no#=>.5<max ,.d5iso#j>
,.s.5<i"7#a>.9n f"<x5i9"91 y5i9"9">
,.d5is4;
,if ;f is 3t9us 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
;;;(ds./dt)
"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$4
,9 ? case1 "<#e4aa"> reduces to a #adc
def9ite 9tegral ) respect to b#bhh
;s_3
(5.13) "!%,c]f(x, y)ds
.k !;0^,l"f@(x(s), y(s)@)ds_4
,if ;x is us$ z p>amet]1 "o has
(5.14) "!%,c]f(x, y)ds
.k !;a^b"f@(x, y(x)@)
>1+(?dy/dx#)^2"]dx_2
"! is an analogs =mula = ;y_4
,,5d ,,( ,,calculus ,,sample
,,9 ,,neme? ,,code
def9ite 9tegral ) respect to ;s3 b#bhh
;;;"<#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]1 "o has
;;;"<#e4ad"> !.5,c f"<x1 y"> ds
"7 !5a9b f.<x1 y"<x">.>
%#a "6 "<(dy./dx)">9#b+ dx2;
"! is an analogs =mula = ;y4
,,,5d ( calculus sample 9 uebc,
,,5d ,,( ,,volume #a
#ade
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