math-203 exercises ( solved )king saud university college of sciences department of mathematics...

King Saud University

College of sciences

Department of Mathematics

Math-203

Exercises ( solved )

The exercises from :

Calculus by Swokowski, Olinick, and Pence (6th Edition)

Prepared by:

Lecturer: Fawaz bin Saud Alotaibi

find the hmit or the sequence, il It exists.

24 {In{,:'~ I)} 26 {co:n} 27 fe"/n4 } 30 {{- l)"nJr"} 38 {n2/2"} 42 {n[ln{n + I)- Inn] }

S.o W:\ GIY.l~ . ··~ .. ...... .. .... ... .......... ...... .... ...... ... ....... ... ....... .... .... .... .............. ................... .... .. .... .... .. .. .. .. .. .

····· ·· ··· ······· · · ···· ·· · ··· · ··· · · · ····· ·· ···· · ·~ ·· ····· · ·· · · ··· ·· ··· · · · · ·· ·· ·· · ·· · ·· ·· · ·· · · ·· ·· ··· ·· ·· ·· · · · · ··· · · ························ · ··· · · · ·· · ·· · ·· · · ··

2: -'!7 ·· · · ··'1i1· ·~ · ···· Y'1 ... .. , L\·~·· · l1"" · · · -- .. ~--·<· \.\.'ti ·Jr\~'(''t\.4~eeA. · -~-Y~ . f~~~ . ·~···it~:~~~~ I ~£~ ············~···· .••.•.••.••.. •. •...•....••.•••••••. .•.•.••.••.••• ••. . f\ . r "' ) _ l . ?t. - ( OG r _ , 1 , J ... ~.""'····t-· ~ · :1' ... ..... _ .. .. ··\ """"'.... ..... . ... .... .. . ·· ·~· ··"f'-fl'l·IA"\.7 ... \:AY\I'!·~t.Y.M\w-.\.

. X.-:7. .. .oo .. .. ...... ......... .. .. .. .X-::?. .eN. .. . ! . . .. .. -~.C.x+J) ....... .. ... ... .. ..... ... .. ... ... ...... .. .... ..... ............... .... ... .

~~ ~~B~fi£~ '~ ~~ -- !:;~ ~ic

~~~?::~~~~~ = ~·~= = s~~ (\ lrl\( ) ... .. ...... ... ... ..... .... ....... ... .. ........ .. ....... . Y.l.:-+..1 . .................... .... .. ........ .... .... .. ..... ... .... .... ...... .......... ... .............. .

~9 ~y, ~ ~ ! !'-_..""...~~ '-"'~ffl~~: ....... ........... : ......... ........................... -=.t .. ~ .. C0. .. . ~ .. ::. ... b.. .......................................... .......... ......... .

-::::::::::::::::::::::::::: : ::::::::::::::::::;r: : ::~ :: : : ·: : · 6.6. ::n :· : ::~ · ··~: : · :: ::·:·· :::: :· : · ::::: · : ·: .:: :::·:::::: : ::::::: ::: :::::: V) - V\ ""

!~~~~~~~~}· : :~ :0::::::::~~ :;.c:~~I: : :: :ne~:~::: ;:: · ::P.;~ ... :: ·~::~:~:::; : :a. : : . :: :: :: :: ·: :: :

ra~Ft6>S~J~J; ({,; .z:~ I~ a: . .... ... ··· ··· ··· ·· ···· ·· ······ ··· ···· ······· ··· ·· ·· ······ ····· ······ ······ ·········· ··· ··· ·· ·· ··· ····· ··· ···· ··· ·· ·· ·· ···· ········ ···· ····· ····· ········· ····· ····

( )

---- - - - - --

find the hmll of the sequence, 11 it exists.

{ 11

2 } 2 {cos n} 24

In {n + I) 6

-n 27 {e"/n4

} 30 {{ _ l)''nJr"} 38 {n2/2"} 42 {n[ln {n + I) - In n]}

......... l ....................... x. .. ... ..... .. .. .................................... ............... .. ............. .................. .. .................. ... .. ... . _ , e _ oo ··..:,..~o:..:. ··· ·· · ··~ ···· ........ ......... .... _ .. . ~ ... .. . .... ... . . . .. .. .. ... ..... ..... .. .. . ... .. . .. . . ... . .. .. .. .. . .. . . .. .. . .. . ..... . . .. . . . . . ... .. .. .. .. .

... .... )( .-c;>O\I ..... !t:x .~ ..... ......... P.'?. ..... ..... .. .... .. .. ..... .... .. ........ .... ... ..................... .. .. .... ....... ................ ... . . ·· ··· ··· n ··· ~ ····· ··· · ········ e_·x··· ··· ·· ···· · · ···· · ·· ·· ·· · ·· · ····· ··· · · ···· ···· · ············ · ······················· · ················ · ···· · ···· · ·· · ············ •

~ ... ,._,,_.._ ...... ~ .. .... ... .. .. .( ... ~·j· ·· · · .. 5'f~e ...... .v~.le) ............... ................ ........ .. . ·x ·~··~ .... .l.2... .;c... ... .... ... .. ...... ....................................... ....... .... .... ........ ..... ...... .. ... ... .... ... ....... .. .. .. .. .... .

- Do . .. .,_..... .... ~········ · · ···· ·· ···· · ··· · ·· · · · ·· ·· · · ·· ·· · · · ······· · ·· ·· ·· · · ·· ·· · · · · ····· ·· ·· · ···················· ···· ······ ·· ·············· ·· ····· · ···· ··· · · ··· · · ·

t:>O . ···· ··········· ·· ·· ··· ·· ······ ·· ····· ·X ··· ···· · ··· · ··· ·· ············ ····· ·· ··· ··· ·· ·· ·· ······ · ····· ····· ·········· ····· ·· · ·· ·· ·· ·· ··· ········ ······ ········ ·· ··· ··

··~ .. tf~···w e .,. .... :::. .. _: ............. ( ... ~.j· · ·· ·· ·· ·S ·~~ ·~ ·· · ·· v.w.l.'kj. ... .. ..... . . ... "" )(:::sj'06''' ~._x .. .. ....... ....... ... .. .... ... .... ....... ............ .... .. .. .. .................... ....... .................. ........ .... .

;:: :tl;~:: : : :::·::c.~:: : . : ::: . : ·: : :::·::::::::::::: : :::: ::. · ,.c: :: ::[i .j ,. ·. · :: ;s~· .~:·~ :: :: : :: ·;:;.le) ::::::::: : .... ............ .......... ... .').,J. .L······· ··· ··· ····· ·· ········· ····· ···· ·· ··· ··· ·· ······· ·· ·· ·· ·· ·· ·· ··· ·· ·· ·· ······ ···· ··· ·· ·· ···· ······· ········ ·· ·· ·· ·· ·· ··· ·· x__., oo -, ... ..... .... .... .. ... ....... ...... ........... .. ... ... ............................................................... ...... .... ..................................... .. .

&><> - ,ex:, .. .. = .. .. ....... '2-q .. ............... . ........... ................. ....... .. .. .................................... ............ ... ... ..... .... ...... .

~ ~-~ %& ~~<

;;~w~;;~ jj;~~~~S ~~ ;~ f~~?~~ > ~iE ~~~ ~<~ £~~) ~~~~~i~·~~~ if,.j ITH~>M5 VJt!.· ;e;;:,.. ~ . ......................................... ..... ... ................................... ..... x ... P.\). ...... 3 ......... ........ ........ .. .. ....... ........ .. .. . ... ........ z:··· ............ } ·x ·l:· ...... ... ... . = ...... ~ ... .. .. .. ....... .. ... .. ........... ... .. ............... ... ................ .......... .. .

::~x..~::: :: :: ~:#J~:i : : : : :::::= : :: ::: :~: : : : :: :::::· . : : :: : : . : : : . · . :: . :: :: ::: : : . ::::::::::: : ::::::::::::::::::::.:: :::::: .. :::

•r~:~• •••~~··•··~~:••••j• •·_;:~~~J~•• • •~···~••• •~•~o x ->o.o ).X(/..,!)~ pa

(2, )

~--------------------------------------------~-

I CHAPTER 8: Infinite Series!

S.l: SEQUENCES

lind the limit of the sequence, if it exists.

;::a~: :r ········;. :a: .::::: :;;;p. ::~:~:: -:c=v.:~ : :n : '.:···········;·:·~::: :· :· :::::: ::: : :: :: : :: ::: ::::::: : : :: : :: : : 11

····· ··········· ···· ··· ··· ·· ············ ····· ······················· ······· ··· ·· ··· 3 ···· ··· ········· ······ ··············· ······· ···· ······ ·· ···· ··· ·· ·· ...... .. .. .

..... .. ........ ....... .. .. ;. .. · · l=f)11"F,?> ······· · ····· ············ ········ ~······· · · ·· · A · · ·· · · · ·· ; ···········; ······ · ················ · ··· · ·.. .. .... . • -~ ..... q.tl····- ·· ···\.:: ... . ·· ······ ·· ·· ·C~n -"--1 ··'-K··- ft.,~ ., . /.. ... lS. .... .. o ......... .......... .. .. .. . ·········· · · · ······· · ····· ···· ·· · · · ···· ··~~- - -······ ·· · ·· ·· ··········· ···· ··· · ·· · ········ ···· ··········· ······ ·· ···· ··································;'······· ·· ·

.. 2·C6·)· ·····tl ·····-=·- ... J1 .~()··· }F ·k~ .C\V\ ... -:;:: .. ~ .. .. ?. .. ~A"\J .~~-~'('M.'~ .. .. :~d'~ ? 1 I" "" ..... ~ ()Q •

:::~:;;: :~.:::: :*::::::::; :: :::~::~: :·;L ::: : : : ::::::::·. : :.: : :. :: :: :: :: ::: :: : · :: · : :: :: . · :· .. :::·:.·:.::::::··::·::::::::·:::::: : :::~.~::::f. :~;::::~:.: :l,~::::::: :~~::: ::~~.:: .. :f~~:;;. ::~~:k_\~(~:~~~J ~-~· ·O;r · ······ ··· · ···· ········ ·x··~··!JO······ z. ... .......... .......... ... ... ............ ........ .......................... .... .. .... ........ .

~~ L'h~"fiS£'!~~ _: £~;;~ :~: ::~ ;;::::::: · ~f 'j'( · ····· · ·· ·: : : :: :: ::: :: : :~· : ··· :: :: :: ·: ·: :: : :·· :::: ·::: : : ::: ::::: : ·: : :: :: : :::: .: :. · : :: ::::: : :::::::::: : : : :: :: :: : :: :: · ·· ><·-Y ··~· ·· · ·:£: -... e\,\.2 ...... .... ....... ... ~ ................................................ ........................... .......... .. . ..

·~~·~····0:~~····•Tt1·-~~" ··~~· ·· · ··::~~;·z_ ·~· ·£· ··~·~· ~ ~~s ~; ~~~ g~ ~~ f~~I~I~~I ;~, ~~ ~~-~;± I~ ~ • '

it&) "''ii"= n{ ~(n+I}-Jl" 'IJ w •

}~~~; iE~~t~<~~~=~(~I

~ ~ - ~=~J:t~~=.-~~i i=

· ·· ·~(t:~~·~~:~~~~····~· ·~~~· · ·( · t ·~ ·~)•=:••l·" ·~ ·~:·~~· -x

I CHAPTER 8: Infinite Series I 8.1: SEQUENCES

lind the limit of the sequence, tf it exists.

·~s~ ~?( ·~ ~?:e& ~:;••• ~~0~· ·~~: ;~~·~•• •• • ><.. · ·· ·· ······· ·· · ··· · ·· · · ······ i · · · ···· ·· · · ·· ·· · · · · ; · ; ······· · r · r · · ··· · ····· ·· · · · · · · ·· · ·· ~· ·· · · ·n · : · ············~·· ct ··· · ··;t·· · ······· · · ·· · · · · · · ·· · · •

..... ~ .. , ..... .. .L .... f.f..~ .~ ~ ·~· · ··J ····· YL.J.( · ~ .. ··· .tt ·~···· ···· ··· · ··· · ·· ··· · t. .. x) ... .. ............ .

....... .. ...... .. ..... ............ ... ....... ... .... .. .. .............. ........................... X ·_, .. &>.) ... ... ...... ~ ... .. ................. ........ . ~ .

:::::···:::··: .. .. ~ .. : : ::::~: : : ::::: : :: ~::::::.:::: : : : : .::::: : . : ::::: :: : · : : . : .::. :·: .. ::.::::::::: .. :::::::::::::::::::.::::::.::::·:::.:::::::: .. [ .: ...... .... .... .. .. .. ... .. . J ... + .. ~ ... .. ......... .. ~ ·; · ·· : .......... ( ... ~··'···r .. ·\·· ....... .. ... .... .,. .............. ~ .............. . ~=p;, ~f~ X~~ ~~~ ~ ·; ·~~ =1

... .. .. ...... ......... ..................... .... .. .. ........... .. .. .. .. ... /j ........... ... .......... .. ..... ..... ...... ...... ..... ...... ... ....... ..... ...... ... .... ..

· · · ···~~W'<,: · · ·Ci·i\ · · .. ::=····O:A·"" ....... ~E·~t- ·n-t-1) ·· ·- · ·~ .. .. "'J····:= ... L ........ ..... ....... . ····tr~··~ · ·· · ··· · · · · ·· · · · · · · ·~=9 '0{) ''''····· · · ·· ·· · · · · ·· ·· ·· ·· ·· ·· · · · ·· ·· ·· · · · ··· · ································· · ··········· · ··· · · ··· · · · ··

: : : :{4~: : .. :: ::~: .. : :::~t~~:c. : :~:~j:;:=::~::.~.J: :~~;: · :/~ :~; : :-:;;;~~~+ : : . ~ :~: : i . ~~1~1~~~~~~~~~~~~~~~~~~

... ............... ... .... ......... ....... ..... .... ................ ..... ... .... .. ................... ..... .... ..... ...... ..... ....... .. .. .. ........... ...... ...... .......

... ................ ............ ... .. .. .. .... ...... ... ............... .................... ... .... ....... ........ ..... .. ........ ..... .. .......... ...... ....... .... .. .... . ,

...... ......... .... .... .. .. ... .. ............. .. .. ............... .. ..... .. .. .. ... ... .. .............. .... .......... ... ... ... ......... ............ ......... .. .... .. .......

Determine whether the following series converges or diverges

"" 1 "" (I I ) "" [ I 4] .., 1s I 19 I -+ 26 I - - 28 I<2-· - r 3") •= 1 1 + (0.3)" ~ •= 1 8" n(n + I) Do() •= 1 n(n + I) n •= 1

\ 8) ~I tck~ ·3)~ =~ ~~~

:<~;~§ ~~; .~?~ 1~; => ~~ ~ .... ;:=) .. .. .. ..... .. ~. .. . ............. &al .. \.v.e~t5· ···· · ··(· ·ne··· ·s·""~· · · ·· · · ··· · ·· ·· ·· · ··· ············· ··· ·· · · · · ¥-) ·.!:.· t ·····~o. •. y .. ~ .............................................................. ........ .... ......... ................... . ··············· ····· ············ ···· ·· ···· ·· ················ ··· ········ ··· ··· ·· ····················· ·· ················· ·· ··· ·············· ···················) ···

C::t~~::~: ~Y.:?::~::~:: ::?.:: : ::~ ::~ :~: :~: : : : : :J :~: ~ :~:::r :~::~:~ :~:::: ::::: : .. ::: 1~x~ ·= .. r~ ~ ~J~~ciY" · ·

::ai~:: : ::~ :: ::: :=::[~r::£::·~:: :~ : :c···· · ··· : · :: :: :;;::~::+:: :J:: :~:~ ::+:~ :::c ?? ., ,., ""! -:. lli~LCe] "''"'J "" ··········· ······························· ·· ······························································· ··································· ·············· ··· ······· .. LaoJt-:./.q: .... AMrher:.: ... wt:. .. : ... ............. ................................................. .......... . ~ l..: ~ I ::::::.:_]~ :t: l:.. ;::~ Tm .

,_1 Lsn 11-f'~-t!}) (%j Yll»+l) . .. ~ .... .................... ...... .. ............. ........ .. ..... .. ..... r..:::::.\ .................... ..... .. ~.~1 ...................... ..... ........ .... .. .... .

cx.s

~tt:f= ;!:, ~ t == 1 r~=t.,;~~ney ... :; .. .. .................................... := .... ~ .. .. ........... ..... ~g .... .. ............... r. ............. l~1<-\-- .. ... .. :2-: ::=:r:::: : : :: : : : : : :~::~:· : :~: :=: : : ~:c .... · ~: .. : :::::: :: :: : :::: : : ::: : ::::{.rel.~i :c~p.;~: :~e}l(oy fi ::: t")(>l+ 1) i1::;: {() ""-t i} . . . . .... . . . . . . .... ~. 1 .......................... ......... ~. 1. .................... .. .................... .. ..... .. ........ ......... .. .... ........... .. ...... ......... ...... .... .. . .......... ................................ 1 ................................. ................................................. ...... ... ................ .......... .... . ..... }1, .. =····· ··} ·.-..·M ·····*S,.::.. ... br··+·~+ .... , ....... + ··bV\ ......................... .

1 ~E±E<I f!~::; iEE£J

........... ....... .. .. .. .... .. ....... .. ... ................ .. .... ............. 1··- ··-' -.............. ....... ...... .. ........ .................... ..... . - 'J'\+ I eou

::t~:.~: :: :~~: : :: : :; :::~~:::(j: ::: ::l=: : : ·::: :~ : : :£:::~:2:: :::: :::r :: : ::::: .. : : ::~r::(co~"~J ... Y.\~ .. QO .... ...... .... .... . ~ . .ry.()Jo .......... ..... t.)±.l) .... ... ....................... t'\=-J ··~ -~-~±D .................... . ::;;~rr~ + s =-1 ~~v; ~h s~~~ L:+t~~

1\.:::.l LB ~CV\+\2.) ~ ( 5 )

Inti CONVERGENT OR DIVERGENT SERIES

Determine whether the fo llowing series converges or diverges

.., I .., ( I I ) "' [ I 4] "' 1s I 19 I -+ 26 I -- 28 I<2-·- r 3·>

•=I I+ (0.3)" • =I 8" n(n +I ) • = t n(n +I ) n •= I

~t.)i:f I - ·· -lL},i:t~mm mmmmmm ·· ··· ········il\::= · r · · · · ·~(~ .. t.l). ............ ~. ·· ·· · -····· · · Y\ · ·~r · ·· ~ ·· · · · ·· ·· · · · · · · ······· ·· ··· · ·· · · · ·· · · ···· ·· ···· · ·················· -

:::t~:~: :::~: : : : : : ::~::t :~:~:::f· : ·····r · ····· · · · ······ ···· ·: := : ::;ij·:_ : : : :;: : : :~: : :::: : : : ;; : ::;;::; : :: : : : :2 :2:: : Y\ "" V)l V\+ \) r') ~ .. . . . . . . . . -::'-. ,QI;) ... .. ...... .... . .jl.) _,..()g ............... ~.. . . . . . . . . . . . .. .. . . . . . . . .. . . . . ............................................................. .

L-ouk.······.for···· · · t.t¥1-8-f ·heY:· ·· · ·· te-s- ·t· ~~·········· · ·· · · · · ·· · ·· ·· · · · · ······ · · ·· · ·· ··· ·· · · · · ··········· · ······· · ······ ·

~ T :=:£1~~ T m :..::4 :.t.I.. rr:::r£"1 (n+iJ ... .. . .. "!] ..... ·;;;·;;:T·· ·"' '"' ·~··l}·· ........ ... ·>~=1·11. ... .... . . .... .

... . ~ ............................ ... .......... ................ ... .... .... .... .. .............. ... .. ................... .. .... ....... .............. .... ... .... ...... .

6 ..... ..L ..... el\ve.rv.e..s ... 1 ... s.)~-c .~ .... U;:: .... {s ...... ~f.~6· f}\~'"=- -··ser.f.e.J n .:: . ~ · ····~· - · ··· · ·· · · ····· · ··· ·· · · ·· · ····· · · ······ · · ···· · ········· · ·· · · · · · ··· · · · ··· · ··· ·· ········· ·· ······ · · · · · · · · · · ·· · · · · · · ······ · ············ · ·· · · · · ····· 11re4Yfw.,. · · · ··· · · ··· ·~- - - - - r · ·· · l ··· · · · ·· · ····· · ·~· - ·±1······· ·, ··; · · -· ·· · · · · · - .............. .. ...... ,~-~ ... ~\A~·)· . H .... ........... . . .. fo> ····b ····L....-.,-t"f'HoJ)······ .. · ·~· ··· ·~J.V~ .. ~ ... ...... L! ... ....... .. ..... .. ..... .

·· ······· ··· ··············· ······Y\··- ·1····· ·· ························ ·· ···· ····· ·· ······································ ········································ - \

~~) ~~~~ ~v ~~r~~ ·~ ::[{~:: :~::::::= :~~: · : · ·I: ·~:::· : :I·: :~: :I::~: :c; ::=::~: :=·c;:::::: :?-:: :7 "'""'"'" H~Lt~j Ccaj J · · ·············································· ···I··..L.!··<r··········j··±::J··z.J········ ·· ·············· ........ .. .. .......... ........... .. .... . ......... .... ................................... f .. .2.. ...... .... .. .. J .... .. ~ ............... ............................ ............................... . p(l(})(-.. ~- .. t;tlfO'TheY .. ·~t$·f -~ .. ....................... ...................... .. .. ................................. . ·· ············· ···································· ·· ·· ··· ·········· ·· ················· ······· ········ ······························ ··· ·· ········· ·········· ·· ······ ·· ~ ~ ~ ~ ... ~ ... -{~=~ ......... -~:; ........... '"" .................... ~ ..... "" .... : .. T ....... "'l ............... ... ...... .. .......... .

..... & ... 0 .... ·~- ... ~ ......... :::: .. 6 . ./.± .. L-.~.f~) ...... .. .. .. .. ..... .... ......... .. ... ... ..

... ~.~ - ! ... .. .. ..................... ... ..... .. ...... ,~. 1 ... \. ... ;l ....... l'\.?::1 .. \:Z./ .................... ..................... .

~ ~~ ~=t ~~~cr~0~;;8 ~~~~=\~~~~~~~ i ~ m

L:(Zc-2, &•rrJcX!Jeb ;;ts>uw. \s ~ Yl:::::-1 ~

( 0.. )

CHAPTER 8: Infinite Series 8.3: POSTIVE-TERM SERIES


16 I I 19 ;. I 25 I 26 I ~ 14 I - 1 -

. ~2 np-=1" .:-, n3" n•2 J 4n 3- Sn .~4 2n - 7 •= I Jn + 9

00 I + 2" ~ 1 00 n + ln n 00 ln n 29 " - 35 L- ~2 40 " -- 43 " -L... 1 + 3" • = 1 Sn + 1 L... n2 + I L... n3

n :=-: 1 N::. l n • l

·············· ······ ····· ····· ···· ······ ······························ ......... .... ........ ... ... .. .. ........... ........ ...... .. ..... .... .......... .. ..... ...... .... .

-1 --9-- -~_:J -·---r~r.ea--~- --- -1:B- -f. --:-- - ··· · · ·· · ······ · ··· ··· · ... -.. ................ .... .. .... .. ........................ .. ·· ·········· ···· ·············· ···· ····· ·· ······· ··· ········ ···· ················· ··· ·· ··· ·· ···· ········ ··· ·············· ········· ··· ·········· ····· ··············· ···

e.(;..?{) ....- ) ;,) ~7 ~ ............. .... ........ .... ....... ....... .. .... .. ....... ....... . . ::~: .. · .. .. ::: . ·::~ : :::;_:~:~,.-~, . ::: : : : · :: .·: :· : :·· ___ : ·· :~:: .. :·.·.:·.: ..... ........... ............ .. ~ - --······· · · .. ······ · .. .

t ·l) ·· · ·fcx) ···">·cr ···· ··· ·'cj · ··~-~- '.L.- -- · (I ·D····· f-·- · ·· l ·> ···~"'h-~- \t- o- "115- - · ... om.[2.?C J .............. ..... .. ... ...... .. ....... ........... .... .......... .. .. ..... .... ........ ...... ........... ....... ...... .... ...... ..... ...... .. ... ... .. ....... ....... ......

(-\--i· \) ····f· · ·· i ·t,· · · · ·-ol-ecYt!:C(-- :S -/r1. 't) -· ·· · ··cM···· ·( ·21 -ev) · · · -;- ···<;"- -{f\ ·· ·C: --~ --- ·· ···· ·· ··· ··

. .z. ....... f1<3 ... .................... ....... ......... l ......... · ...... ;.t .......... ................ ......... ................... ........ ... r .......... ... t ··=·····J··· ·· ···· -- -·J:>< ----- -=-.. - ~ ··~s- - - ... ... d.)( ...... .. ... ... ~- - -\(~ - .. --- s-e:z-lk>)-··········:J-,··· ·····)c·Jxl;.) ............... t..~ ....... '2: ...... x--rx2;;.:.,. ···-----t -.,J.').«l ........ .......... .. ....... 'l..

..: t~ @~Z'Li)= s~~~~<l.{) ;. s~~'c_;,;,) ..:.. ;~z~~) ·--~~~- - -·· · ······ · ·· ······ ········· ··· ·· · · · ········· · ····· · ·············· · ·········· · ········· ·· · · ······· ·· ···· ·· ·· ······ · ··· ···· ·· ····· ·· · ·· ·

:..1~.{: ~ ;,; ~ ~;; __ (J:.:~;).J; ~;.:~e~~.s ....... .. .............. ........ > ........... ............... .. ..... ~ ...... ....................... ......... ....... .. ...... .. .... .

~?~~~k::-1 ~~~~~~~· 13-) .... f .... ~ .. .. ·~-~-Ot ........ < .. ·~--~ .. -=- ~--{- --~ .. )~ .. ~ .. b ... ............... V\:il ''''n-·$Y)··· ···· .. ·· Y\ ·~t ... ~--- .. ~ --Y\··:;_ ·y .. · 3~ - --·· .. ·· · · V\·"'!tll_.3/··-~-~~i- .... .. '1 ... .. .. ...................................... .... .. .. .. ..... .. .... ... .... ......................... ...... .. ...... .. .. .. c ·aY\V"j ···<_3ectrh'e-+YI .. c_

~)··~j ····~~·;··,··~····_;··· ~- ··· · ···~·L~····· · ·· · ······ ·· ···· ····· l~l· ·~···l~· · , ··· ·~~·~ I ................ ~ ...... ...... ........ ..... .. Y) 'O::'i"''h '?1· ................ .... ... ..... .. ..... ... .......... ................ ... ....... ..... ..... ... .... .. .. .. .

~?)!_~ ,~itl~ ( ~

CHAPTER 8: Infinite Series

8.3: POSTIVE-TERM SERIES

· Determine whether the following series converges or diverges

"' I "' 1 ~ I 16 I 19 I 25 L

• = 2 np-=1" •=1 n3" • =2 ) 4n 3- 5n

"' ~ "' I 26 I - 2- 14 I - -

• = 4 2n - 7 •= 1 Jn + 9

29 I I + 2" 35 f ~ 40 I n + In n 43 I ~ • • 1 1 +3" • = 1

35n

2+ 1 •=1 n2 + l •• 1 n3

··GhoG~· · · · 2:: · ·b~ · ·~ ·£· · ·= \ ·· · :JZ·l: · · ·.=L-.· · ··· · · ·· · P.-->c.·h~ ·~·> ·· · · · ···· · · · · · ·· ·

~~~ n~~~~l ~ ?>~ ~~~~~ ··V··)e;··· ··L..C.T .... ... t.,"M ... :.~\II\····· · ::::.··t~ · ·· · ······ 1· · : ·· :::: ·· · ::: · :: :j.· : .::1 lq :·.:.::···.:::··::::· .... .... ..... ........... ................... ..., .~ ·~· ··· ~V\ .. ...... . V\~~00 JLtV\' -?" V) f'J .t') ;;, ·· ···· ··· ······ ··· ······ ··· ··· ······· ·· ·· ·· ···· ········· ·· ·· ··· ··· ·· ···· ···· ····· ···· ····· ······ ··· ····· ············ ······· ·· ······· ·· ····· ··········· ··· ······ ····· ··

=l·= ~ =i· ~~ 9t:,;~c 1.~)~,_ ~~ ... ~~.f-Vl::--~: ~F~'t-~>:~~ ~... ~~ ~ _/.. I -) .. 1-= · ·· · · ···~· · · ·>··D ....... ............ ........... ....... .......... .. ........... ... .............. .. .... ...... .... .. . .... \: .. t .-o/··· ......... ............ ......... .. .. ........... ............ .. ..... ........... ...... .. .......................... ............ .... .. .. .. .. . · · · · · ·· · · · · · ··· ·· · ········ ·· · · · · · · ···~ · · · · · · · ···~··· ·· ·· · · · · ··· ··~··· · ·· · · ····· } ··· ······ · · · · ··· ·· · ·· · · · · ···· ·· · · ·· · ··· · · ··· · · ·· · ·· · · · · · · · · ·

. .S.a.r .. t.?j··· ··L.C.l···f .. ... ~ .. a:;, ... ~ .. ~..... . . .. ... eoV\V.. .... .. .............. .

......... .. .... ... .... .. ............. ..... ..... .... n;o"l:··· .. .. .... ... n:::·;z:; · "'- ·~.t· t')>-5· ·Y\·· · · · · · ..... .............. ...... ........ .. .. .

Ooo ~

~~?~~ ~~~~t ~~~~ · · ··· · ··· · · · · · · · · · · · ··· · · · · · · ·~ · · ··· ···· · · · · ·· ·~ · · ········· ·· ~ · ·· · · ···· · ······ · · ·· · · · · ·· ·· · · · · ·· ··· · · ·· · · ···· · ·· ·· · · · ·· · · ········ · ····· · · · ···· · ·· · · ·· · ·

. c,. hc.,os,~ .... E .. ·hti\ .. ~I.. '?> {. .. .. . · ~ .. ~~ .. . ~ .... J.\.Vr~·C>\Y\'\'t~~· f c..

....... .... ......... ...... . V\. ==·If· .... ....... Vl~f . 2-n ........... ... .. .. ... ...... .... .. ... ... .. .. ..... .... .. ..... .... .... .. ........ .. .. ..... .. ...... .. .

P~ F~~·~~~ :: ~ E~ ~:~~~ ~~~ ~~~~~~ :;o-~ ~~

• . . ... • . ... .. . ... . ... . .. .• . ... ... ......... .. .•. . ~ .. . . .. .. .. .. .. .. ~ . .. . .. •... . ...• .. .... .. . . ••••... . ... ... .. .. ••.. . . .. . • . . • . .•• •. . ..•. •.. •.. . •

...... ).o/ ··~)· ·LC . .{ .. :- ...... 2-.q~.==-· · ·~ ........ : ... J .:Y' .... J. .. \v ........ ..... ......... . ····:· .... ... ....... .... .... ............. ... ... .. .. vt~tf-.. .. ..... .. ..... it~ .. .... z.~~.~~ .... ............................ .. .. .. .. ..... ..

~ . ~--------------------~~=-./Lf) .. .. £ .. ~ .... ;:; ..... 2 ····qV\ . ............................. ......... ........................ ... ........... ... .... ... . ............. n~ ., .... 5.~ .. lj. .. ....... ...... V\.~t ·· ·· .. ·· ...... .. .. ... .............. .... .. ... ...... .. ..... ..... ... ..... ... .. .... .... ... .. .. .... .

... .. ..... ... .... ... .. ............ ~ ... ..... .. ... ...... .. .. ...... ~ .. ... .. ........ ....... ..... ....... ...... .......... ..... ..... ... ... ... ...... .... ...... .... .. .

···Gh6(f)G· .. ·2 · .. ;··\p- .. :;:; .. L .__\_ .. ;:: .. 2: ... ~ ...... , .... .(?~.~ .. ~ ......... . ... .... .. .... ... .. ........ ... . V"\·::::: ·1 .. . - 'l>. ........ .. .. n ·<;:!·I .. .J.:t\, .. .. .. ..... .... .... .. )1 .. 1.-: ........ ....... p . .,.::. .... ~<: ... J ... .. . ....... .. ........... .......... ......... .. .. ....... ...... .............. ..... ...... .. ......... ..................... ......... ..... J'v ........ .. .. ........ .. .

( 8 )

~----------- - --··- -·

I CHAPTER 8: Infinite Seriesl 8.3: POSTIVE-TERM SERIES


00 3n 00 I 26 L: - 2- t4 L: --

.~4 2n - 7 . ~ 1 Jn + 9

00 I + 2" ~ 1 00 n + In n 00 In n 29l: 35£...3~ 40l: - -43L:-

•=11+3" •=1-J-5n2

+ I . ~ 1 n2 +1 • * I n3

... U.se, .... LC.T-: .. ..f/t)'r · · ·~· · · · ·-= ·~w"' ·· ······L · · ······/·· ··± ···· ··· ····· ·· ·O··· ···· ·· ·· ·o··o····· ···

.......... ... ..... .. .................. . ·t~ 4

.00

.... \o. . .1. .. ....... . n _,. 00 ... .r;(.. +'1·. . .. ,(.;\ ... .... . 0 •••• •• •• 0 •• 0 • • • •• • 0 • • 0 •• 0 •• • 0 •

. :: ·~:.·.r ,::::::::·· \Tvi ··· · ···· · ::: .. : .::::. :.(~_:.: : : : :.···· · · · ·· l ····· · ····· · :.:: : ~ . : : . ·· · - ,······· · ··:= : : 1 : . : : : :; :~ . : : ·····~ -V)~ ·OV··j· V\··· ·-t ·!:J ········· · ··· l"'~Oo ···· l ·+ ·~· · · ·· ··· ··· ····· ' ··+.CJ .... .. .. ········· ·· ········ ··· ................ .............. ............ .. ... ...... ............. ........ ............ ...................... ......... ........ ................ ......... ... .. .. ...... .. .

- 5o) ··YJ·~ ·-··- L..CT ... : .... ~ .. -'\~ · -::: ··L: \ · ·· ····cl·\V······o··o··· ··o··o ·· o·· ··· ·· ··o··o ··o ··

...................... ............................. .. ........................ ...... ...... W-+:9·············· ··············· ...... ......... .... .......... .

· :(~·)· ... . ~ .. ·· (,\~··· =: · Z ''"I"T '2:Vl:.:. ... ;z · ·· ·=····~+i.'4 ' " '"''""''"' ' "" 'i"\ """ ' '"' ' ' .. ...... ......... ~ ..................... .. ....... r-.+ .. 3

-vr ... - ... :L... V\ ...... ~ .. 2.r .. ~ ....... ... .. . ............. . ...... . ............ 0 ............ . . ...... . .. . . .. .. . . ....... . . . . ..... .. . .. ... . ................. 3 ...... ............. ... ... .. ...... :1 .. .... .. ..... .... .

~2~Cf~ :. : ~ l~l~Jfl:f<J

... S·uT-~'J ..... rs.cT .. ~ .. .. 2 .. ·-1 + ~ ..... cd'f\,·" ........ .. .... .. .. .... ................. .. .. ...... 0 .. .. ..

............... .......................... ............ .. ............. J.+3·· ................. ........................ .. .. .. .... .. .. .. .... ...... o .. .... .... . . .. .

. 55)·· 2:ir · ·~~ ······· · · · <= · 2:q~············ ········· · ···· · ······· · w ••• • •• •• • ••••• • •• •• •• •• w • •• • • • •• •• • •••••• • ••

......... ................. .. ,.n···+r····· ......................................... .................. ........ ............ o .. . . .. ........ o . . . . . . . ... .. ...... .. .

~r~s0 ~ ~~ ~ r ~}~;~ ~~+;~~ ~ ~i~ .... l)Se ... .. LCT:- - - .f.-tV)\··· ·~·-~· .t\-l¥1 ·· ·· .. ·· ··\ ... }1/ ................. .... .... .. ................. .. . ....................................... V}-9·~····· .. ~ ·· · · 1··~fl) ... lf5·t11:rt .. lf·- /~\... .... ... ....... ... ...... .... o .. .... .. .

I CHAPTER 8: Infinite Series! 8.3: POSTIVE-TERM SERIES


"' I "' I "' 3 "' I 16 L:

2 19 ~ _1 2s L:

3 26 L: -

2-n- 14 L: -·-

•=2 n~ .~1 n3" n c2 j 4n - Sn . ~ 4 2n -7 • = I Jn + 9

~~> ~9;~~~:j:"l~

····~~~······~··~··~·~······~·· ·· ·~· ·~··~··••;:~:·i~7~:~ ~~ ~~! ·· E~ ~: ~~~:i~j~}k

=~~~ ~~+-~~(;) ~~; ;~t : 1i: ~~ F? ~~~~ ~ · ~

· ::~~ .; .. b;:::T,c: ty~ :.:.J:;t;~: . :~:z:···· -n! ·p~ ·Vt· ·.:: f}(v~ : : .:::.:::::::.::::: .. :·.:.::: ·: :::.::: .... ... .... .. .. ...... ......... ...... ... ... ... .. .... .... .............. .... ... ... .... i.J ... . +../. ............................. " ............... .. ........... ...... .. .

· +->)····=·· t;t~· ·~ ·E·· · ..I.~ .r······s ··E· ·-T·· ·==··L···~··-=· ·2·· ·~~:::::::: : ..................................... ... ................. n .... .................. .. .. ~ ........... ... .... ........ n ... .............. ..... .... .. .. ........ . ......... ................................................................. .................................... p. . .,.,..se:.y.feS

7 ... 'f.= .2...? .. \

..................... ... ........ ......... .. .... .. .. ... ....... ... ................................ .. ....................... Cav\'J········· ··· ····· ·· ···· .. ·····

······························ · ··· ··· ······ ···· ·· ·· ······ · ······ ·· ·· ·· · ·· ·· ·· ········ · ···~······· · ·· ·· ·· ·· ······ · ······ · · · ······· · ·········· ·· ···· ·· ·· · · · · ··· · ····

···So;··Y:>j ·····f5·C·T,···· ··2:tth···::: ··L ···..- t1 j ..... cov\;~· ~···· · · · ... ...... ... ....... ... ... ...... . ... ..... ... ..... ....... ....... ................ ..... ......... .. .. .. .. ... .... .. ... ... .......... Y.l ... ....... ..................... ... ..... ........... ..... ...... ...... .

............. ................................ .. .... ····•· ·•··· ·· ·· ·: ········· ·· ··········· ·· ... ..... .... ............ ... .......... ... .... ·· ·· ··· ······· ····· ·· ·········


00 I "' n" 6I~23I -·= 1 e" n = I 10"

··· ···· ·· ··· ····· ····· ·· ······ ··········· ···· !··· ········· ·········· ·········· ··· ·· ·· ··· ······ ·· ·· ·· ···· ····· ··· ···················· ··· ··· ··· ··· ····· ······ ··· ·· ···· ·· 4 Y)-t Q.O

:: :.~ :~ .~~~::::: :: :~:e:: :· · .. :.~::::: · : :: : .: : ::·:::::····· . :·: :. : .: ·: .: .· :: . : · :: .. ::::·::: ::.:::::.:. ::.·::.::: ::::·: :: :. ::::.:: :.::::: :::: .9;;~.1 ~;J~ze;.t.; :La.,~2~ J(J

:~:~: :: : . :'§ : : :~:~ ::~: :::~:::·::~~: :. :. : · :::: ·: :: .. : ... : .. :·::::::.::.:.:::::.::::: ·:.:::::.:::·:::::::: .. :.·::::::.::: :: ::.:: ·· ··········· ·· ··· ·· ·· ···· ····· Mo ·· ················ ···· ····· ·· ······· ·· ·· ·· ····· ·········· ' .... .. •.. .... .. ... . .... . , ... . ··' ·· · ·:·· ··· ··· · ······· ·······

~-, .-... . t(.t1 "" :::' ·· ··· · ·~1'\· ····· ·' ~ _· ··. :~ ··_·. · · · · · · .. .. : . · .. ..... ~ · .. .. ... .. .. .... . ·· ··· ·· ···· ···· ·· ··· ··· ·· ········· ··········· ········ ····· ·· ·· ····. .. ... ........... ...... .. ... .......... .. . . ... ........ ........ . .

.... u.r~·····~lJO.c. ...... ~;ed .. ~ .................................. ............ ......... ................... .... .. ..... .. .... ....... ..... .

~: ~ e_.; / Vlvt__)J;, : ViJ~ ~~ ~~ ~~~~5/~~ ?~~~~jo7):~

:;:L~·::::: ····Y'···· . :::~:.::.; . : :::~::::·: : : .: : .: : ::: :: :. :: : : :: :: : .. .. :::·:.·:.:: :: ::::: ·::: :.:·:::: :·::::::::::::.:: :: ::::·: :: ::::::: · ·· ··~--~·· 06 ········ 1 .. 0. .... ............ .............. ........ .... ........... ......... ........... ..... .... .. ..... ... ..... ........ ... ...... .......... .. . · · ······· ·· · ···· ·· ···· · ·· · ···················· · ·· ·· ··· · · · · ·· · ········ · · · ········ · ·· · · ·· ··· · ········ ····· · ·· ·~·· ·· · · ···· · ··~· ···· ·· ··············· ·· ········ .. · · ·SoJ · ·· ~(J · · ···~· ·· ·~~+· ·:·· ·· ·Z:·q~---~· ·2=· ·· · ·2" Y\ .... .. .. cJiv .. .. .. .... . .............................................................. ......... ........ ................... .......... ... .... ........ jCJ ............ ... .... .. ... ..... .. .. .. .

CHAPTER 8: Infinite Series ~~:f: ALTERNATING SERIES AND ABSOLUTE CONVERGENCE


0. ...,. _,

1. 2:. (_- \) -'1.~' -~= ' V\+1--

oo n s I(-t)"-

• = l Inn

.. ... -· -- . -· . ·, determine whether (a) a series which contains both positive and negative terms is absolutely convergent, conditionally convergent, or divergent ; or (b) a positive term series is convergent or divergent

38 f cos (~rt/6) n=! n

2 ~" ~ oon~+f j ~ ?'~ ·t:f~;;-~-o ------V-Y\· "7;-1---I --itn, ·-· 4y, · --=--~·------- ~ ... ;:::!_ .~·-- ······ ·· ···· · · · · ·· ··· ·· · ··· .. ,, ... ,. ,,,, .. ,,., ...... ,,., ........................... ,._,.f>O ............... l'\-~ot>-· · Y.\ ... +~--······ · ·············· · · ········· · ···· · · · · · ·

t:{n -- i>--···de.crea-y.,hJ··7 -- ·S· }.~ . .ce... .. .... ~.st.t -ti)1· C - ---·f<?<) ···E: --· { ... ; x~1 ····· ............. .. .. ,, ., ... , '' '' .. ... '' .... . , ... , ., ,,, .. ,, ...... ''" ., ... .............. ............... .. ....... ,,,, .. , .. , ., .,, "' ' .. .. '''' .. ,. , . .X .. r.f-::., ''

· f~~T~ ·~~~~~~· · ··~~·· ~ ~J's ~~~~i~~~ ···· · ··············· ... .... ,, .... .... , .... .. .. ... ,. ,. , , . , , ., , . , ,, ..... , ... ,~ ........ , .. .. ~.-\· ·· · ······ · · ·· ···· · ··· ···· ·· ·· · · · · ···· ······ · · · ·· · -- ·· ····-- · -· ·--· · ··· · ·--·· · ·· · ·· · · -

.S -o-/ .... bJ····itS·T···:··L -G··..\) .... .. _ 1 .. .. .. /5 ------ C.Vl~t!'f{J-~'Y:l-f:. ............. .

..... .... .. .. .. ... ..... ........... ... ... ..... .. .. n·?-{ --···--···-- · ·--· · · Y.J .?:.-r.f.~~ .......... ... ........ ........ ... ... ..... ...... ... ................... ..

_____ ......... ......,._ .......... ......,......,.. .. . . . .. ... .. .. .. ..... .. ..... . . . ...... . . .. .. .. . , . .. , . . , ... , , .•• . . . • •• • , ...... . , .•••. . •• , . .••. • . ,W

- S:~ · ·· ·Q ·· · ··~ · · ··· .n ..... .. .•. , ... , .. Jtl -- ? ·1 ·-- ·· -···- ···· ··· ······--·-········· ···· ······· ·· ·· ·· --· ·· ·· ·· ·-··· ···--· ··· -·-· ···· ·· ·· ····· .. , ~ ~ j ~ . ••••• ~~~~;~~~••s~ • • ~:~• • • • ~ ~ ~?~\~~\)~~~ ~~ •·• D

~?~ =~~~I~~: ~~ ~~~~~~~;;' :;. ~~ ~~~ ; ~~ ~rJ? ~·~ ~~~ ~~; :i .~~ i AC

~

(j2-)

CHAPTER 8: Infinite Series ········· - · lui: POWER SERIES

•--:;-ind theinterval of con vergence of the power series

. 00 " Yl "" . . ~ n

2 - ~ 111 '. ~-~ '--~) • _}_r::>C_.It..)

21 .'=:0

-23• (x + 4)" . 1 L- \.:: V\ I '-' --r 't.. n.:::o -

- .. ,· ) h . Y1 .- / )·~ . -~J .. ,;,11' ... --. "t)f+- ·* · ... ............... ... .. .... .. ...... .. .. .. ........... .......... .......................... .... .. .. ... . . ~ :lo.., -

.. .. ..... ... ............... .2 ............. ................. .... .. ... .... .... ........ .... ..... ....... ....... ..... ........... .. .... ...... .. .. .......... ... ... .... . · ... '

........ ..... ......... ........... (ti.+l)~ .. .............. .. ..... ~f-l ··· · ····· · ···YI'l+-L:V\ ·~ ·\ ·· · · ·· · · ·· ·· · · ·······n · · ············ · ·· · ·

.......... '1+r=- .. .. ,_> c "-+' > ... _ .. .(.x~lf). :: .. :::=;:.. 1 n 'l .. (><-·++) ... (><-.+.4-) '""'''''''''' ' '''''''''''''''''''~'''' ' '"''''''' ''' ' '' ""' ''' '' '' ' ' ' ''' '''" ' ''''''''"''2: ··· .. ···~· ·· · ···· ·· ·· ·· · ····· ······························· ·

L·~f\4;.;\f ;,:~~ ~<~ii*r : c~;;,.)h(~;~j : t" ····· ········m ·· ~"-"' .. .... '4., J ...... ,.,. .. j ic" .... 9, .. .............•....•.... : ... : ................ ·..... .. l'I·'::(X.+If)." I . . .... .. ....... ...... ..... ......... .................... .... ...... : ............. ......... : ·.: ..... ..... .. .................... .... ........ - ~ ........ .. -...: ......... -~ ...... .

=9··t·-x·t+·I··C::: ··?> .. · ·~······- ·g .. L.x+lt· ·L:~ .. ~ .... - .l ··2 ·L. .. )<-'-!t···

:Ab:·: ::~:::4::; : :2=: ~~E:::~ :;g:~::~£::~:~\::: : ~t:i\i ::::; : :s.~:~ :~~ :::~i~~~:~ ~ ~ o ... ........................... .. ....... .. ............ 2: .......... ..... ...... ............... ......... ... .. ............. .. ........... ............................... .

. ~~ ~~-~ ; ~ i~ ~iil~?:~ ~~~~~ ?;~~ ~~~;~~~~= "' _ :_s.~ ·:: : :i~ :~~¥.~::~::~~~~:i§e~:~:::: ;~:::::c:= :ii;: :~j: : :: : : :: : : : :: : ~:::: : :·:: <~:: _

~~)i~k ~~,~=f~~~)~ ' ,, .. - ............... ................... .. : ·· · · · ~····~+J ·:···~· -t-I ··············· · ·At ·~ ··· : · · · ·:\)'\~·)· .. .. 3~ ·: .. ~()( .. ~f-Jl)(-Jr) - -· .

............. U ... .. .. .. _ ..... .. [. .. ,I). ......... 3. .... .. ... Cx.~). .... - ... t .... .... ....... -' ...... . _

....... : ..... ... ~i. l .... ~ ... ................ l_h · ·-t··~)'! ·· · · · ····· · · · ···· · ·····~··· · · · ~ · ·· ·· ·· · ·th+9.·v.\ · ~ ························ - ~ - . ~

~~~\~~:'·+~5;_~ ~~ ~~ ~~ = i: 1 ~=~I ~E :5~i~+.~ :~~: :~F::::~:k :(: ~ :::::~~J:~;: : ::::~s ·:::::\K : :~::i:~;:~:: :·::: :

:~~:~~~:~::::~~:::::~~~:~~~~:~: :: ::~~: : : :~= :~;:~5>:: :::: :: ::: : :: ::::::::::: ::-

r-4·~···1·· ... !.4~\olcr..Y.~.c.Ac..~ .... . \s ....... Do ..... ... ........ .. ...... ..... : .. ... .. .. ............. ... ... ... .. ...... . .

( 13 )

CHAPTER 8: Infinite Series

~-~ 7: POWER SERIES REPRESENTATIONS OF FUNCTIONS

· , find a power series representation for f(x) and specify the interval of convergence.

-~ ---- .. -5 f(x) = x 2f(l - x 1 )

t"7 f(x) = xe3x .,.

b. f(,Y ::: ilo\ (3 + 2x). f. ~ t?\) ::.X~! t1 n -t- XJ \<6. ~(.~) := S"i"\\(-5.x) .

Use infinite series to approximate each of the integrals in Exercise: ~ . to4 - decimal places. . - : ~~ ·- -

Jl /3 I

19 - 6 dx 0 I+ X

ao

5J.fQ{) ·:· --X~· -· \ . .......... ~ ~ ............. .. .. ..... .... ............ : .L:;, .u.~.:: .. _L_ .... j···· -\-l..\ \ <. \ .. /. ........ ........... .. ............ ! .~x~ .................... .................................... n :.o .. ............ i .~.~ --····· · ··········

- ................... ~ .. ~'1:-: ·~ · r-.·{Jl-)~---; ....... ,~-J -~·· J ·· · : .... .. ......... .... : ....... ;:: ..... ...... ~ ...... ... ......... ~ .. : ... :· ··· ·· ·· · · · · · ··· · ·· ···· · ··· · · · · · · · · · ··n·: · ~> · · ··· ······· · · · ···· · ··· · · ·· ···· · ··· ·· ········· ·· ······ ·· ······ · · ···· · ·· · · · · · ·· · · · ··· · ··· ·· ·· · · · ·········· · · · ··········

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()~ )

···~

_ CHARTER 8: Infinite Series ······· ···· .... -···

~-~7: POWER SERIES REPRESENTATIONS OF FUNCTIONS

· , find a power series representation for f(x ) and specify the interval of convergence.

5 f(x) = x 2j (l - x2)

17 f(x ) = xe 3"

' __ 6. fc~ ::dll\ (3 + 2x). f. ~t7\) =_ "'L! t1 (' t- x) \C6. ~- (.~) = S-i"\(-.S.x)-

Use infinite series to approximate each of the integrals in Exercises - ' ; to four decimal places.

Jl/3 1

19 - -tix 1 + x6

0 Doc. )' L "'+I Qa ~ "'+ \ -7:::)- .f.cxJ.~ .. ,t .. ~.L-' Cx) ... ; ... ,.X::\4.\ ...... : .... .. ...ln . (l+":} -~IJ:~~!.. ... ~ ... ; \U\ <1 ·

\'"\+I · t\:o h+l . ·· · · ·· ·· · · · ······· · ·· · · · ·· · ···· ··· · ··· "'' ·~·o ····· ·············· · ········································································· · ····· · · · ········· ·· · · ··

• ... -::r-. -

. ·· ···· · ········ · ··· · · :··· · · · ·· · ····· ·· ····:·· · ·""" ·· ··················· ~ ··········· ··········· ······ ······· · ······ ·· ·· · · ·· · · · · : t{ · ·· · ·· -~ · ·· · · ·· ·V)· ·· · · · .

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· ~f[if~X).,_=~<5XIl ; 0~~JE.tii ·· · · ·· · · ····· ·· ···· · ················· ·· ·· · ···h· ·;·~ ·· · ·· yl ·~ · ·· · ·· ·· · ·····f···yl ··~ ·· · ··J ···--··· SJ · ··t15 ·~--·~\\< ' .. ...... ......... .. .... ..... ....... ... ...... .. .............................. ...... .... ..... 9. ........... .... .. .. ..... ..... ..... ........ .. .. .. ... .... ... .. .. .. ... .... .

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. 0 ................ .. .... .... ... ... .... .. ........... .............. ... ..... ............... ..................... .... ...... ... .. .... ..... ............. .......... ...... ... .. ...

.CHAPTER 8: Infinite Series . --- ) :7;POWER SERIES REPRESENTATIONS OF FUNCTIONS

: . · ., find a power series representation for f(x) and specify the interval of convergence.

5 f(x) = x 2/(l - x 1 )

17 f(x) = xe3"

6. f(,Y :::: il.\ (3 + 2x). f . .f t?\) ::::. )1,.'2..! )1 (' t- x) \<6. \c~l ~ s,"\-$-5'.x) ·

Use infinite series to approximate each of the integrals in Exercises : ~ -: . to four decimal places.

Jl /3 l

. .19' o I + x6: -\'\· : . bV\i-\

, ·=·· ·E·(-$ · · ··c-~2- - · ·· · · ·=~ : . : .t· ·=· ·· ·±-c · · - ~t -,t· ··±· ··c·· ·~-)~: - -~ -·~ -:~ ·-· · · : · ·_ · Y\·~· cr · · ·· · · · · · ···tV\.+T · · · · ·~ ·-·· ··· · ··· ····· -.:> · · · · ·· · · ···· -1- ···· ·· ····~! - · ·· ·· :: · ·ri · ··· ·-.s ···· ·······:····:······

~~0 i(~; ~ ~ : ~~~~~~i.~ :~ ; < -. .

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... '~ ~ - ·- .

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-· ..


8.8: MACLAURIN AND TAYLOR SERIES.

h'"-cA- MQ.CkLAY,·~ 5e.Y\'e s ,feY t\...c:.. £.-~c.\\~:

" .- . : r ''3 finw.lthe Taylor Series for f(x) at the indicated number c. (Do not verify that lim"_ "' R"(x) = 0.)

IS f(x) = l fx; c = 2 . . ~ :· .

, · · · · ' _. '• find the fi[St four terms of the Taylor series

us_e an infinite series to approximate the

given number to four decimal places.

f'n ·

36 o x cos xJ dx 31 tan- ' 0.1

Approximate the integral:

to four decimal places. ' ' t' t'' """' ' .. '" ...... .... ... c . -• .. . .. _,, .... ............ ...... ·- ·--· -··· ··-· plac:es. Assume that if the integrand i~ f(x), then. /(0} .= lim,_, f(x). -

42 J' sin x dx 0 X

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: : . ~>,~::5~ ~£~ ~~ ; ! Tt'!l~ Z:J rf, "' ~ve11

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, ...

I CHAPTER 8: Infinite Serie~

8.8: MACLAURIN AND TAYLOR SERIES

0'i\rA Mo.ck&Ar,·V\ 5e-n'e .s

~6Y *"-t.. ~""""'c.\:\cM:

11 j(x) = cos2 x

. __ : __ ~ ·- . __ .. "l find the Taylor Series for j(x) at the indicated number c. (Do not verify that lim._.., R.(x) = 0.)

15 j(x) = I j x ; c = 2

· . ·. •, find the first four terms of the Taylor series

for f(x) at the indicated value of c. I 21 j(x) =sec x ; c = n/3

use an infinite series to approximate the


fl /2

36 o x cos x3 dx 31 tan- I 0.1


to four decimal places. • •Yt-'"'""'' .. '"' ..... ····-o·-·- ··· -·-···-- ·· .... ·- ... ... - ··-···-· places. Assume thai if !he integrand is f(x), then / (0) =

lim,-o / (x).

f1 sin x

42 -dx 0 X

(e.) ·1·5)-.. f{J( .) --~- ·~ · ·- · - · ·· · ···.f· (· 2-) ··~·-=k:=··:f ·(~·· · ··············-··· · ··············-··- · ··· · ·-- · -· ···· -··-· · · ·

6i> ~ ~~ f~~~~ f ,...,.,

' ' '' ' ' ' '' '' ' ' ''' '' ' ' ''''' ' ''' ' ''' ' '' ' '' '' ' ' '' ' ' '' '' '' ''' ''''~' ' ''' ' ' ' ''''' '' ''' ' ''''''''''' '' '""'''' ' '"''' ' ' ' ''''' ' ''''"' '''' '' ' " ''''''''''"''''' '' ""''"'" ' '' ' '

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.............................................. .... ....... ... ........ .. ....... .. ...... k ................... .. .... ....... .. .. .......... ...................... .... .. .. .

~~~L~ f~~ r~;J i: ~ • •

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~;?~ ~:(J ~-~ ?iJ~)f~c)~ ··~~· ··· · f~i·~ · ··~· =~~:l£~~ ~, , ~~~~~~2 · · · ~,~~(:;1?_)~ .... .... .... ... ... ...... ...... .. ... .... ....... ... .. .. .. ~.?.~ ...... ?.:-. .......... .... . ~ ... .... .. .. ..... .. ...... .... ... ... ~. ;;;;. 9 ............... ?: ....... .. .


8.8: MACLAURIN AND ~TAYLOR SERIES

f;·~J. - Mo.cku.r,·V\ 5e-n:t.> _

,f6Y t\...~ ~"""'c.'t\cM:

11 . f(x) = cos2 x

. . .. : ... ~-- .; r ' 'l find the Taylor Series forf(x)at the indicated number c. (Do not verify that-lim.-"' R.(x) = 0.)

15 f(x) = l fx; c = 2

· ' :- '• find the first four terms of the Taylor series



J•n ..

36 o x cos xJ dx 31 tan- • 0.1

.Approxi;nate the integral:

· to four decimal places. ''t"t'''-'"·'' ' ' ""' .. •• •~ •••••c,• -ov ' '' -•••• • •www •~ •w •• ·•- · • •••• ••-•

-- places. Assume- that i[ the integrand is f(x), _then f(O) = lim,_, f(x). -

-' J1 sin x 42 -dx

0 X

- for f(x) at the indicated value of c. · 1-_-. 21 f(x) =sec x; c = n/3 __ • _ \o)

-~---f-t)()~ -secx: -·· · ···-~{~)- :::: .. 2..;.~ .. f,_) ················ · · · ·_ ··········· ·· - · · ····· · ·· · "· · · ···· :~ ···· ····· ·· ··r ····· ········· ············ ············ ······ ················r ······· ··--- ----- ------------······ ···· ·-- -· .. .... .... .... .......... .. ... ... .. ... ..... .

------- - -,ft-')(}·"? ·--5ecx:~ -~~-)(--- ·- -·R ·C~)-~-~-~---····· ··· ............................ .. ............ . 3

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C) l>o ))'\ 'l.V'\ (19l CoS lk~ ~ (-\ JL , \J. f\~

il~o l2.nJ l,

I .• , - •

.. :

CHAPTER 8: Infinite Serie


h'J\~ f'J\o.c~&AY,·V\ 5eY\'e s ~ 6( * \...~ ~.,.._c.\: \ crt\ :

11 f( x) = cos2 x



fi / 2

36 o x cos x3 dx 31 tan- I 0.1


· . • . : _. M ·- • : _- 'l find the Taylor Series forf(x) at the indicated to four decimal places. number c. (Do not verify that lim"_"" R.(x) = 0.) ;~~~;_ ... ;\;,;~~-~h~;·;[·;h~· i-;;;~~;;~;j ·i~ i(xi: ·~;.~~ / (oi .. :·

lim, .. 0 / (x).

15 f(x) = l fx; c = 2

f1 sin x

42 -dx o X

' . ·. •, find the first four terms of the Taylor series

fot f(x) at tho indiootod "!"'of ' I 21 f(x) =sec x , c = n/3 c.o \I\ '2.V\

4)

31.) ..... .. .. ......... .. ...... ...... .. .. .. ................. .. .. .. ... .. .............. ........ .. ~;.·''(~};: .... L_ ... ~\) lA j u ~ [.-11 l]

~~~(~ . ~~ ~~~~\):~~r~~ ~~~ ~~~ ~ ............... ... ... ................................ .. ......... ..... ...... J> ... .. ... .. ............ .. ................ .... Q:s.. .............................. --5 .. .. ....... .. .. .. ... ........... ~ ... o ..... t .. - L9 .. ~ . ' - ......... ; ... J 1·ir t-e. .. .... ~~-L-~· (h{)t> -CJ -00 2. (I 0 .... ... .... ... .... .......... .. .. .... ........... ............. .. .. .. .... 3 ..... ................ .. ... .. .... .. .......... .. ... 2 .... ....... ...... .................... . . ... ...... .. .. ....... ........... ::: .. o ... o.f:fr:f., ..... ..... .. ................. ......... ..... .. ......... .. ......... .......... ...... ... .. .. .............. .

........................ .,_. .. .. ................. .... .. ........... .. ..I .. .. . I>!I ..... ...... . . Y.\ .. .. '2.1'\'fJ' ..... .... ... .... ............ ...... ... .... ... oo..... "~ l.V\+ I ··lt2)····f'·· ... <Z.J .~ .. x .... J.;x .. ··-J· · ·· · ·E~L~\J. ... x .... .... ,.Ji ................. S'i'i~u· · ==-·'£(~\) u. ; u~ ..... ..... .......... 0 ...... ... . X .... ......................

0.... .... .. .. ....... .<.~.lL ............ ..... ............ .. .......... t'l~ ·o- · (_'l.v\+\)!

.. ................. ... .......... .. .. ....... ... ....... .. ... .. .... .. ... .......... ......... :X ...... ...... .... ......... .. .. .. ... .... .... .................. .... ... .... . - ]~· · ! - .- ·~····=\M·· .. ·x-¥\ ··· ............. ......... .... ~ ... = ...... ~f ... .. .... ..\ .... ..... 2¥1 ...... .. ... .. ...... ... .. ..... .... . - ··· ... ~.~ .. ) .......................... olr···= .. z ···l ····\) .... ....... .. J ..... x ...... d·x ·· ····· .............. .

:~::: : . · : ... : .:::::: : : : ~ :~~ :~~ :!_:: :: : :::: : : : · : · :: : : : : :~·~o. ::C~~~.t :::~:: : : .. ·: ::.·::::::::: :: :::::::·:: ·:: ::::::.::: ::::: .: _2:I...:'5~ : X"+' J ..:. $ L..._Q., ~ .... .. ... .. .... ('k~ .. +\). ~ .. .. ........ . 2Y.\.+.). .. ... o ..... .. =.Y\·.=·6·· (~+\)- ... ('2.V):tl)t ......... .. .. .. .... .

. oz; ... •••

1= 1k · · ··~ ····~·~~· ·= ;;~g ·~···· ·· i ~:~ · ·· ·· ·· ·· · · · · ·· ············· · · ·· · ·· ··· · · ·· ··· (2o l

I CHAPTER 8: Infinite Seriesl


h'"-rA Mo.c~«AY,·\1\ 5eY\'e > rfe-r *""-~ ~~c.\\&M:

11 f(x) = cos2 x

. ___ __ . ___ . : _-- 'l find the Taylor Series forf(x) at the indicated

number c. (Do not verify that lim"_ "' R.(x) = 0.)

15 f (x) = 1/x; c = 2

·. •, find the first four terms of the Taylor series

21 f(x) = sec x; c = n/3



Jl /2

36 o x cos x3 dx 31 tan - I 0.1


to four decimal places. 0 &1"'1" ' ......... ... .... ···- ·· ··-t:o·-·- ... -·-· -·--- .. ·- .... ·--· -·--···-· places. Assume that if the integrand is f(x) , then f(O) =

lim,_, f(x).

J1 sin x

42 -dx 0 X

for f(x) at the indicated value of c. I .. S:tlflce.::---- ··· L --· ·-< -- -o ---.--o -o -o-a .. z .. g ... 7 .. . # .e .. s..~ .~----~--- - 1-~~ ...... .... ... ............ .... .. .3 ... ?,.2.8-o. ... ... ..... ...... ........ .......... ......... .. .... ... ... ... ................... .. .. .... ..... .... .............. . · ·cdJeynJl+r)1; [;) · · · · · · ·.ser·; e J· · · · ~ 't>· · ·· · · - 1/;t.tY.~-pf-~Le: · · · ·ctcct.>t.Yq-c::J · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... :,q: ...... #Q.: ···{i·-nf- ····fk.yeJ!--... fey.m.s ... 4_.y.e. ... ~umw.t.ed. ....... .... ...... .. . ... .... ..... .... ... .................. ....... .. .. ... ...... ..... .. ... ............. ......... .. ...... .......... .............. ........ ...................... .. .. .. ... .... ..

>f.it~; :::: ·· :J.: : .:. ~ ·s~- ~ -k< :;;(;; :::~: : : ;.: ~ :::~+:t.i: :~ : .::: : :::·::: :::: : :: :::::::·:: :: ::::: :: :::::::::::::::: : 6 2£ """ J

........ ...... .... ...... ............... .. .. .... .................. ..... ....... .. .. .... .. ............. ...... ...... ....... ... ... ..... .. .... ................. .............

...... ... .. ....... ... ...... ...................... ... .............. ... ..... ... .... ........... .. ............ ..... ......... ............ ..... ................ .. .... ... . .

(2\ )

Evaluate the iterated integrals

7 J2 Jx eytx dy dx I xl

f• /6 J•/2 8

0 0

(x cosy - y cos x) dy dx

-. express the double integral over R as

an iterated integral and find its value.

21 JSR xi dA , where R is the triangular region with vert ices (0, 0), (3, 1), and (- 2, ! ).

23 JSR x 3 cos xy dA, where R is the region bounded by the graphs of y = x2

, y = 0, and x = 2.

)2..~ ~ 'L :><. ~

~ IE ~?~? ~~ ~ !:~ EI ~><?? ~F =t~~h ~ ~ J~~~~=~~?~

'L

... ... ........... ...... .. ·· ····· ··············· ··········· ······· ···· ·· ...

~?s~ s~,.~~; .-. ; ~~i;i~ ........... 9 .. ....... .... . () ............. .... .... .. .. ... ................ ...... .. ......... ................ ........ ... ............................. .. ..... ........ .

~ S1 ~~;~; ....: ~G.SXj~~;. · ·· ··········u ·· ······ · · · ·················· · ····· · · · ·· ··· ··············· · ········· · ··· · ·······~ ········· · ·· · ················ · · ·· ·· · · · ···· ·· ···· ·· ··· · ··············

~ 1~i~=~6.i~=~~]i~ ····· ··· ··· ··········· ······ ··· ·· ·········· ········ ·· ·· ···· ··· ···· ················· ···· ·· ·· ··· ··· ·························· ······· ·············· ···· ··· ···············

- · '1.. ................................. .. ....... '1e ....... .............. J'. -~ ..................... ... -z.:. . ·.-r."· ........... ... . ........ ...... ..... .... . ... ... . 1._ - I I( = ~--= ~ .,; ,·~"" ~G = m

Evaluate the iterated integrals

7 I2 Jx eytx dy dx I :cl

I CHAPTER 13: Multiple integrals\ 13.1!: DOUBLE INTEGRALS

f• /6 J•/2 8

0 0

(x cosy - y cos x ) dy dx

.. . express the double integral over R as

an iterated integral and find its value.

21 JJR xi dA, where R is the triangular region with vert ices (0, 0), (3, 1), and (- 2, ! ).

~-~~~)< 23 ~~;:;~~?· ;?'~ .. :,::h~. ;'••on .boond~ . by . t:j C;·f(,··· · · ·· · · ·· ·~ .. ~~~.~~.J j

..... .......... ... ..... ... .... .. . ····· ········ (--'l:·J)· .. LJ,. ~J. .... ..... ..... ... ............ .... .. r..~ .rJe {1'3.3) ············ ···· ·················· .~ .......... '~ ~~~ ~-x:· · ·· · ·· ·· ········ · ·N· ·~·cJ'j oL~.

······· ·· ······· ······ ·································· ··· ~}~; · · ··· ·· ·· · · ··· · · ·~ ;_i;: ••··········· :ii''i<i-...:,0. ~"- ol ':1

~~~~j~~·~·· · ·~t · ·~i~~~~·~~··~··~·· · ·~· ·!:t~· ·~~· · ••;;• J@f,;.2j'}4j;;J~•;td· • • ; •••Is•]·· ' ~ f .. .......... () .... ................... ............ ..... .. ...... .... .... () ........... .. ........ 3. ...... ........... ~ ... 1... ············· ······ ··········

0

~~: :: : : : :::: :: :::: : : :: :: : :: : : : : : ::.:: :: :: · : :· : . : :~ : :: .::: . :: ?. :: ·> : : :: : :: : :]~ ::::: :: . ::: : :. : :::: :: .. : : . : . :: ·~· · ·. :.:::.:. ::.::::: : : :: ::: :: :: :: U:;x_ ~ ... ..tS:

)f~;~~;;~;;··~·· · · ··· · · · ··· · · ·· · ·· ··· · · · · ·· ~R-~~· · · · ·~ ·· · ·~· ·· · ····· · · · · · · ·· · ·· · · · ·· · · · · · · · · · · · · · · · · · · · ··"B·· ······· ··· ····· ·· ··· ·· ·· ····· ······ ··· ··· ····· ····· ···· ··· ·· ··· ······· ···· .... ..... .. .. .... ... ... L ... .... .... ... .... ...... ... .... ..... ... .... ... ... .... .

·~ 2 ~

ff;~~~~~~_g~ ~~~ • ' l. L ~-=:x..4oo

• •=•• • •I~r~•~•~±~~~·~·•~• •~•l• •• • =; • ~,j•~~~~~~~• X

,;. }~I-:[S;~•<X5•~·~···• · • • ••~• • • •.;• ••• If~;J;;,(;J)d~ " J 16 ...... ..... .. .... ......... ............... ...... ... ·1..: .......... ..... .. .. .. ...... ....... .... ..... .. ... ... .... ..... ..... ... ..... .. .... ...... .... .. .... ... ..... .. ..... .

:z .... -..l .... ~{?c?J·) ·· · ·-= · · · · ·· -l ·· ·· _.J ... ~(B)·· · · · · · .. ... .......... ....... ......... ............ .. .. . ............. 3 .......................... ...... 0 ··· ·· ···· ·····3> ... ....... ]> ................. .... ....................... ... ... ........ .... ... .. ... ....... .... .

(21, )

sketch the region bounded by the graphs or the given equations and find its area by means of double integrals. · · .. : '·" a ... , . :·

. sketch the solid in the first octant that is bounded by the graphs of the given equations and find its volume.

·. find the volume of the solid bounded by the graphs of the given equations.

21 z = x 2 + 4, y = 4 - x 2, x + y = 2, z = 0

':1

~X -·····= ······ ···· "4··· ··· ··· ·2.············ ···· ···· ······ ··· ······ ···=······ · ··v, ·;rs··~r · ··· · ··· · ···· ·· ··········;r · · ···· -~- ·

· · ···-~·-··x· ·+· ·x.· · - ·~·1 · · · ····-~···· · ct: ·-· ··-·. .... 2... ···· :;- ·· ··h ··= ·· - -~~· · · · ·\ ::::::::::::.:.::. :· ·::::··::.::::.::::::::::·: .:.: .. ::·: .. ·: .. :-·k:.:.· .. : .A~::::~ : :· : : :· :: .. :·::::.:::::::.::.:::::: :::: :: :::::::. ·:: .. ::: :····_:::·:: A _ ·-r ·-r 1 \ A ·r r >< 1 ~ A H ·· .. . ·- J ·j ····· ··· ~ - - 0\.. ·····::=·) ···· ····s ·· ·· .. ···············d\j ·· O '- X · .. ·· ····· ··· .. .... .. ......... " ···· ···

····· ··· ····· ·· ··· ·· O ·· ·· ····· ··· ························o.;· ······ · ·· ·· ··z.:.··· ········· ······ ····· ···· · ·· ···· ··· · ··· ·· ·· ···· ·· ·· ·· ··· ·· ··· ·: ············ ···· ·· ,_. ')(. .

= s~ ~JF;:,_i~ = c~ 1 = ~~ ~~ J j .. ·[ ::-1 uv--..... .... ... G\_ ...... ... ... .... .. ... :x::l. ... .............. ... .. .. .. ... ... ~... ····1-f-;Xl..: ....... ..... .. J ....... ......................... .... . ······±· ... _ ............ .. .................. ... ... ... ..... .. ... .................................... ;.:.J ............. ........ -3 .... .. ... \J- .... .. ............... .. 2;;· ·· .... . t .. .. .. ·- ····~-x-1--.. · ·cJv< ····::: · ···'~·E ··&v\ ·· -v9 ·,_- --·~· - · · · ···················· ········· E,+x'- j 3j

0 ~ ..................................... .. .. .... .... .... ... .... ... .... ................ .. ............... .... .... .. ................ ~ ........ .. .. .. ............... .. .. .. .

"' frs:::.f \ i.:.T rs= 'J·· ·· ·- -?i: .. -- ............. .... .. .... ..... .. -~ .... ~ .... kV', .. ( .. .. .. \J .. 52- ... / ............... 3C . .v.2. 2- ................ ...... ...... .. ............ .. ... ..................... ................ .. .. ..... .... .................................................... ..... ... ..... . ...... ..... .. .... .... ... ....... ... ..... ... .. .

·rh/$···· /)··· ·· ··fhe· .. ·~ ·¥1-t·· ··4-.. .. ... ft,e_ .. .... . bqs~ ... o( .. .. f.Ju_ ..... f'i:Jl4r:~ ... ~ t'h.t. .. · ?e._y ·=f~·t\·E:; ·· .. ·=rh:e· .. ·:fiiiY~ce. .. . x ... wk,~ - ck. ..... ..fdy.~ .. .. f.tte:: .... . ·f:vf' ·· · .. t;f ... · .. th~ .... >a·lt•cJ...·j · .. i>··· .. ~·~··x:> -..... T-huJ-·;'··· ······· ........... .. .. ..... ..... .

(2.'fl

1 I CHAPTER 13: Multiple inte_gralsJ 3.2: AREA & VOlUME

sketch the region bounded by the graphs or the given equations and find its area by means of double integrals. · · _ : ·-·" :'! ···' . : ·

. sketch the solid in the first octant that is bounded by the graphs of the given equations and find its volume.

. find the volume of the solid bounded by the graphs of the given equations.

21 z = x2 + 4, y = 4 - x 2, x + y = 2, z = 0 ~ Vx

tr "" >< 3. ·. · . I 2.8 - -v- -~~-~ - - -· · · -~---o\ ··A- ····-~---S ····· ·· · & · · ·· · ·· ······· · · · )<---· -d· -~.1-o\x - -~--- · • •... ¢.---

\"<~ ~~ • . ~

~? Yii;i ~~< ;J= ~~;~ .. ............ q~ ;'-'-; ~~; ... ~. ?····· .. .... t ••• ?" •••• •••• ~~···· ··· · · ···· · · ··· ···· ·· · ···················· }k.,_ ?'- - "

;;~:~~;cf~~~ ~~ .. ........ ... ... ...... ........ ....... .. .. .. .. .. .. .. .. ... .......... .... ... .. ..... ....... ~ .. ... ... . ~ .. ..... ...... ... ... ... ... ... .. ............. .............. .. .

v;JSi . ~A ~ t£~{;1~~ ~;~ l ~~j ~I'-:~i~ ~

....... ............... ...... .. .... .. ... .... = ...... ..... .. -= ... ..... +:-2.3> ........... ............... .. ............................... .. ............. .

. :::::: ::: .. :::::::·:::.::::: : :::::: ::::: : :·:: · : · :::: : : : :: :: :·: : :: : :::= :~:f:: : : : ·: : :· : ::: :::::: ::::· :::::::::· : : : : : :: ::.:.:::::::::: :: : : :· .: ::: . :: :: .

(Z.$)

~-------------------------~

I CHAPTERl13: Multiple integraiJ .. .... . . liD: DOUBLE INTEGRALS IN POLAR COORDINATES

Evaluate the integrals in Exercises · by changing to polar coordinates.

f2 fx 1 21 dy dx

1 o jx2 + y2

f2 f~ 23 0 0

cos (x 2 + /) dx dy

27 Find the volume of the solid bounded by the cone Z~= x'+ 'J 'l.. and the cylinder •>f .;r.!J ~._ 2. :x:..

2·-') .... +·~··?Cf··2 ·yD·~· ·'J ·L ·k .. ............... .'-. .. l··· · · ··~ ....... ... .. ............................... .. . x-:: · · 2.: -=9· " Y'C:.v~ · · er~··z· ·· · ·· · ····· .. ·· ·······!J~K · ·4:-~ ~· "?C· -::: ·'2.;······ ... ........... .

~:~:1::~;~~:~:~:: :~ ::::::::: ::: : : :·:::: :::: : :: ...... ... ....... /. ..... .... ~ .. .... ~~: :::: : : :::::::::: :::::::::: : : :::: . X"'l ~ - o

~··r· -:= ···S· er:· e- ·· ·· · ·· · .......... ... .. ...... ..... .. .. .......... ... .. ... ... .. ..... .. ..... .. ..... .. ... ............. .... .......... ..... .. ... ..... .

~;;~t~c~,~~ ~-~""-i r;;s~s><-:. ~ ;LJ~~=s~fs:ec~;,i :~J;J~ ~I C} !J<':t!~ ;~~~... :

e··· f.. ·· ···~··S6121··-··Se .ceJ·· ·J.~ · ··~ ·····t· · ······S'ecoJ.e ......... ............... .......... .. .

~~~~i~~~~~~B~~ ~{~~i~ 2:3)- ··0 ·f:.··~J ·~·~··~ · · e-r~·X; ··~·J-4- ·-y'-···· ·· .... ................. /J, ,_: ·························

c~e.;f • ·· · ·· · ···· ~·~ ·· ··· ···· · ·· ····· ··· · ~:~;..;;t:~;.~~h-.re •• ••••• ••••••••••• $:;o • • •• • •• ••· •~•• • -~ ····· ·· · · ·· ···· ~:i'-!~~? ····· · ·· ····· · ······· ·· · · ··· ··· · ·· ·· · ····· · ·· ·· · · ·· ····· · ··· ·· ·· · ·· · · ·· ·············· · · ·· · ··· ·· ····· · ···· · ·· · ·· ·· · ··· ···· ·r~- ·'t ·-=->r.:= .. 2-..... . · · · ················ · · · · · ··········· ·· · · ·· ·· ···· · ·· ·········· · ········· · ·· ·· · · ········ · ··· · ··· · ········ ·O· ·~·r· ·~~·················· ····· ·····

······· ·· ······ ··· · · ··~· ······'2- ... ... ........ ... .... .. ,.,,.,,., .. ,, ..... .. ... .. .... ..... ..... ... .. ..... , .. ... .. .. , ... .... .. .. ..... ........ ... .... .. .. .. .... .

· ··I· ·= ·S···A · · ·S ·· · ··r-CDSc-v~J-d ·r·c!-e- ················· · ······· · ······ ·· · · ·· · ··· ············ · ·· · ····· · ··· · ········ ······· ·········· ·o ······· ·· ···a ······· ··· ·· ················ ······ ·· ······ ·······:·· ·· ········· ··········· ····· ······· ··· ···· ············ ······ ·· ···· ····· ········

:::: :: : ::~: : : ~ : ~ :~ :::::: : : ::if:::~~~~~?: ::~::::: : : :: :::: ::::: · : :: : ::::::: ::: ::: :::::: : :::::: · .: :: : ::· : : :: : : : : :::: : :: : : :: : :: ::::: :: :: :::

(.2.6)

Evaluate the integrals in Exercises · by changing to polar coordinates.

f2 fx } Zl dy dx

1 o j x2 + y2

f 2 f''4=}-i 23

0 0

cos (x 2 + y2) dx dy

27 Find the volume of the solid bounded by the cone z'2.= X2+) 1..

and the cylinder ;>( +"J 'l"= 2. :;x:..

... 2.. .?-) · · ····t:.'-::::.. x~t-::t~·~··"2z~Y~.:::;? .. ~ .. = .r. ...................... ....................... ..... ... ..... . -1-+-,~ . ·2:-:X.:· ·· -::::;;:>·· .... y~.:::=~y-C:.c::~S.·G· ·~>· . ·r·:.:=· .2.. .c.a S.6 .... .... .... ........... .

::~~:: : :::p(~r.:: . :h~Jj.::::~j : : :: fd..:~:::: ;~/;;;L : ::gi~s:::: ;;};~ :~ :~:\b:~:::x:~ ~ ···~··· · ·· h"''s ·, ·d~· ······-rh~······q~'('( ·{t· · ·· r: · :::::.. ·· 2: · C:<JS··s'J · · · ·· ······ · · · ·· ·· · ··· ·· · ··· ·· ········· · ··· ·

: : :::R :~:::~: : zr.; :;5:~~ :: ::~::~::~: :Q::~::~::::::: ::; ::~: : ::y.~:::~:~:6;:::;:: ~v\.ot

#t~ ~P 4 +~~ ~t~.I i~ tL ~~~ i4J1 &J ; r:

ij~~ ~ l:f-:;.tl!: ki~~ : ·····V-i · ··~····· J- ·· · ··· · .. ·· ·· · · f- ·· · ·· · · · · · ···· ( ·r-J · · ~···r·cl· ·'fJB···=- · ·· · ··· · ··~ ... ;;: ... 3~· · ··········· ·· · ······ · ···· · · ·· ···· · · ·· · ·-·~·· ·· · · · ···· ·· ······· · ···· ·· · ·· · · ·· ······· ·· · ··· ····· · ··· ·· · · ······· · ·· · ·········· · · .. ················ ···· ···· ·· ···· ·· .. ········ ···

2.. 0

. ::::;;.4~ ::::: :t~::;;~;;:::: ]4:~;;.:::::::~::::::f.~ ::::~~~::::::;;~ :t~:~::: : :(hj:::

::s.9 :;n~~t.r.;) :::;::4~4::::::~~::.;&; ::~;;;t:: :::;;~:i-~~~:::: ;.;~;::::fh~:~[;;yf!;

Y? ~Y~ ~ ~ · ~~ ~i· ~ii~i '2---& ...... .... .............................................................................................. ... 1i.r. . ........................... .. .. .. .... .

.. .... .. .. .................. .. .. ........................................................... ... .. ....... ... ...... .. . ............ ... ................... . .

...... .... .. .. .. ..... ........................... .. ...................................................... .. ' :: .. ~ ........ .. .. .. ........ . _ ... ;.; .. ..

CHAPTER 13: Multiple integrals

13.4: SURFACE AREA

( 28 )

SOLUTIONS:

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13.5:TRIPLE INTEGRALS

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( 29 )



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( 30 )



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( 32 ) ...................................................................................

. . . ' • • • • • • ' ~ ' I . : • . .. - · • - · ; . : ' I ... ..,. . .. '>

, I •. • .... .. ..... ........... ........ ......... . . . . . -

· • . •• , .' , , • . · • ·• ' ...:· · • • • ~ · • · ·; •• ·•• L. ,, · • ~ c• • • • " • ' . . . • . •• • •• • •• r

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13.6:MOMENTS AND CENTER OF MASS

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13.7:CYLINDERICAL COORDINATES

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13.7:CYLINDERICAL COORDINATES

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13.8:SPHERICAL COORDINATES

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CHAPTER 14: Topics in Vector Calculus

14.1:VECOR FIELDS

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14.2:LINE INTEGRALS

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14.3: INDEPENDENCE OF PATH

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14.4: GREEN’S THEOREM

INTEGRAL CAN BE EVALUATED EASILY BY POLAR COORDINATES

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14.4: GREEN’S THEOREM

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14.6:THE DIVERGENCE THEOREM

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14.6:THE DIVERGENCE THEOREM

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14.7:STOKES’S THEOREM

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14.7:STOKES’S THEOREM

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math-203 exercises ( solved )king saud university college of sciences department of mathematics...

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