474 Chapter 6 / Applications of the Definite lntegral in Geometry, Science, and Engineering

ffi puvpffiK&#*-Htr wruruflT*sru$ &Nffi H-fl&ruffifiNffi fi&ffiilffis

In fhis se{tis*tr,,<: r,t,ill ,ttutly t:err*.in to*tbitzr;tfiotos r:f <rr *rrJ e-r,, calleil ,'lt1:perboli<:

fu.w*io*,s." T?tase fietu:rions, *:lticit, uris<: is,t v*ricrvs engine:ering *ppfi*:ati<stts, rzrtt,e zrusx),prolsertie,s itt c*stz*ttsn u,itl.t tlt*. irigoxornerric .iuttrtitxts. F/tis sjrxiLrri4t is ,s**teu,ie*fst*pt,i,tiag. s!*ce rhere is litrle on rke .surf{r{:e to .eusg€st rh*.t r.here sh$uld be {sn.rt

ref*tiottsitip be*t'<:en e.xponettfi*l *xtf rrig*tztx*e*k: fi.r.ttt.tiotzs. Fr'*.s is bet*;ss<: rhere!<$iortsitip occurs withist lhe t**tett <sf c*t*pfe.x m*ebers, * trspit u:lci<:!e u-e v,ill ieave .i<sym<t x: ady utzt ed t.]#rsrs.

% *xr,ut*&Y|#. s ss'${ypHffi&#$"$( FiJ**CH#$S$-Ib

intr*rJur:e the hype:{:olic fufir:tiE:ns, E}beele fre}e1 tlxerr-i:e 55 in Seeiioa l}.2 that t}refunr-ticn e" ea* be expressed ir: ihe f.olic:-,r,ing wa,v as the sum *f a* *r-** f.uncti+* a*d ar:ocld fi.a*ction:

iii;e*

Thes* l'unctir:ns are suf{icientiv'irqx}d&nt thai ther* iire &nmcs and:rotalicn associals*l rvitirth*ra: thr: odd l*lrctir:n is callecl the lnperbclit saizs cl -:; a*rl thc even function il call*<J thehtperbolic c*sine cl-r. Thev ;rre d*:iotcri b;r

si*.lrx : {'' - €--' ancl

2

{'' +rj -c*shx:

?

r.vhere sinh is prcncuncetl "cin*,!r" iurc! cosh ih,vmes rvith "g*sh." F?*m th*sc tw* buiiili*glrlocks x'e cilr! cretltr: lour m*re fu:tctions tc pr*duc* tlre lollor.ving set of *i: feyperb*liepcrc*ri*ras.

6.$"I $$:s-xpilr.x$rd

Elyperls*lie si*e

EfyperFz*lic easirue

,l

T:he:,terrns 'tanh," "s6firr: and lc;ih"are pronounced "tanch," 'seech,' and"coseech,' respectively.

sinh,r :

t-*sh x :

tanh.v :

c*trh.v :

ser-:lt x :

csch .r :

e- -{ '

2

t' + (-':

sinh;iEtyp erfu *!ic t€dlsgs lrt

.f,f3per&* Jde e 0 t & EE g e ie d

E€yperbo{ie se{&?rt

E€j:perb*li* c$secant

c$sh itcosh,r

sinh;;I

{x 1 g-t

-2

"*th "I

-ir,tl-"2

e): - e-'\

T{€ t'tr N *i"s &Y sEf; sT'ffi ffi Y

Com'puter afgebqa systems have,,built-in capabilities for evaluating hypeibolicfunctions directly. but some calculatorsdo not. However, ifyou need to eval-uate a hyperbolic function on a calcu-lator, you can do so by eipressing itin terms of exponential tunctions, as in

Example l.

si*h0:

t-ash $ :

sirrh 2 :

-,u -.-0 1-1nL 2

!+!

-- a!

:}l

a' -f

l

1

,* 3.6269 <

r. $Newp$* i

W

6.9 Hyperbolic Functions and Hanging Cables t7s

ffiffiAp$"$$ s$3 T$SS $.$YpS${$SLE{ FA}rueT}*ruSThe gr*ph: r:ithe hyper$olic func: ions, rnhich are ehc,:a'* in F:igirre 6.9.1, ean be ge*eratedrvith a graphirig uiility, but it is rv*rthlvhile t* Ei-asen-* that the g*neral shape of tl"ie g*rph +f-r,': ccs} ,{canbeohtai*edbyskefchirrgti:eg:-aphsof y - 1e''an{i: : }r-' leparatelvand*ddingthecon'espondiri5y-r-oordi*;rie:l:e*pa$(a)of thefigurel. Si*riiarlv,the gener"al

shape ol the graph cf y : sinh x s"-an be atr-',ilined bv sk*tching the gr*phs s1 _y : + e' a*d

-, : -,1"-' separately and acl<Iing c*rresp+nding -u-csor{ii**tes !see part (tr-l1 of the ligure!.

;:;;;::..-"--1@\

i ,, = "**,' i:' i

{.d)

& Figure 6.9.1

ll:::"):'.*)

; = ;;;,;,;l.,,,,,,,,,)

{r')

i r; ;::l:.ii.e)

: 'lI 1' = CS(.11.i ii:, :(.l i

()leo l\liisoniStcn.ii,reti] lniiig.s

'|'ltt tlesign Of ll'tt {iatewc:t; .+,rch itiSl. Lpnis is has,:ct t;st ttst itt;eriei k;;lterr,,lr':,'<tA:' t:Ut". i i;L:I-tSL 7-1 ).

Observe that sinh x has a domain of (-oo, 1oo) and a range of (-oo, f oo), whereas cosh xhasadomainof (-oo, foo) andarangeof [1, +oo). Observealsothaty : le* andy : le-'are cumilinear asymptotes for y : cosh x in the sense that the graph of y : cosh x gets

closer and closer to the graph of y : )d as , -+ 1oo and gets closer and closer to the graph

of y - |e-* as x--> -.$. (See Section 4.3.) Similady, y : +r! is a curvilinear asymptote

for y: sinhx as x-++oo and y: -)e-' is a curvilinear asymptote asx-+-oo. Otherproperties ofthe hypertolic functions are explored in the exercises.

H&ru6aEluln { &ffi& F\ &ru!} &BAm&M APNB ${ frn&&UIUh

Ily'per'L'olic: furrciians arise in vibrirto$ m*{ion* insid* cl;r*{ic solids anii m*rc gelerall-v inmariy pr*b'lems rvhere mechanical energy is gr*dLralll; abs*r$cri l":y * suno**ding meriiu:r:,They ols* *ccur when * hom*geee*us, llexitrle cablc is suspr:*<ied betw'een tw'.c points. as

u,ith a telcphoae line hanging betw'ecn tr-r,r: p<;le*. Such * czrble {*rrlls a e $r?o, calleri ;r

c{rfe*{rzy {ficrx ihe l,att* ccte*c, r::etrning'thain"). lf, as in iigure 6.Q.J. a *oordinate$ystrn: il ir:tradcce{t so that the }*rx' poirri of the c*ble li*s on the v-aris. then it can he

shcia,r: *si*g princ,iples r:f phvsir-s that the cable has an etluation of the fbr*:

W

J':{icorih{-l+,,\u/

476 Chapter 6 / Applications of the Definite lntegral in Geometry, Science, and Engineering

where the parameters a arrd c are determined by the distance between the poles and thecomposition of the cable.

zz $-tYpsRffi&ilx( {ffillINY$Y$#S

1'h* h,r'perholic ft*ctions sirtjsfu r,arjous jdentities that are sin:ilar to ic{e ntities l'or trig$na-*retric funrtions. Ti:* lx*st &rndai:rerrtai *l thes* is

c*shzx -sirrlr:.r: !

which can be proved by writing

cosh2x - sinh2y : (coshx * sinhx)(coshx - sinhx)

(1)

:({#:.,+)(!#::e'.e-x:l =)

Other hypertrolic identities can be derived in a similar manner or, alternatively, by per-forming algebraic operations on known identities. For example, if we divide (1) by cosh2 x,we obtain

| - tanhz x: sech2x

and if we divide (1) by sinh2 x, we obtain

coth2.r - 1 : csch2x

The following theorem summarizes some of the more useful hypertolic identities. Theproofs of those not already obtained are lell as exercises.

LaryArr+p)./'1,'tjrJ c.r!l )i!!i1!l Rdir!,.rd In..

.4 jl,cxi!-.ie cabla t uspenied behteen txt>

1:tt.:,i et .{47 sss e1 x111s y11yt-,.

siiili;)

6"9.f 'c'st.hr$$e.F:$,s

cosh r ;F sinh * =j ex

coshx-sinhx:e-'

cos-h2x *siqfi2,,76 -.1..:. :::::::::::::]

I - dnh:,t..=..sech2 x

coth?x*1icseh2xcosh ( -.r) : cosh .r

sinh(-x): -sinhx

sinh(* t }l) : sinhx cosh y + eoshx sinh y

cosh(x * /) -- csih x cosh y * sinh x sinh y

sinhG * I) ,: sinh r cosh y r cosh x sinh y

eoat{x - y) : coshr coshy - sinhx sinh y

sinh 2x : 2 sinh x cosh x

cosh 2x : cosh2 x * sinh2 x

cosh2, : 2sinhzx 1 1 = Zeosh2 x - |

w w{F+Y T{qffiY &RH *&f;-H-Sffi *$vpsffiffiS$_6c F$Ner$*N$Rer-*l] tit;rt the pararxetric e<gu*tiolrs

,r : co$f , ], : sin, il) <.. I --: 3rr-'t

r"epresent the u*it circle .r2 I ,r,2 - I {Figur"e 6.9"-1*). *3 111ay he seen bv u,riting

);1 + .v2 : coslr + sin2 / : I

If t] < , < 2:r, then tle parameter / can he interpeterl as the angle in raq"iiar:g fi'cm thepe.:sitive .y-axis to the p*i*t {r-.*s r. sin /} e.:r', atrternatir.'e}1,, an tr-ir-:e t}re sha<letl area of theeector i* Figure 6.9.3n (veri8). Analcgously.^ the p:rar:reidc equations

& Flgure *.*.3

& Figere 6,9,-t,l;' : c$sh l. y : sinh r (-w .: r .: -i*)

6.9 Hyperbolic Functions and Hanging Cables 47i7

represent a portion of the curve *2 - y2 : 1, as may be seen by writing

*' - y' : cosh2r - sinh2r : 1

andobservingthaix: cosh, > 0. Thiscurve,whichisshowninFigure6.9.3b,istherighthalf of a larger curve called the unit hyperbola; this is the reason why the functions in thissection are called hyperbolic functions. It can be shown that if / > 0, then the parameter /can be interpreted as twice the shaded area in Figure 6.9.3b. (We omit the details.)

ffi sffi$e$vATtiifr &$ss $Nyfi&$e&L $sKi\{{J$"&slleriv*t:ir'* ibrntulas I'or si*h ,r anel cosh.r ca* be *btained b.v expresring thes* funcdons int*rrns of rr- **d e-':

d -. I [."'-e-'I t-'+r-'_lsinhxl : _ | _ I = ______ : ((]:h.ydr" Jul 2 j :

d J [i'+e-*l .'*r-', [co;h.ni= - l- i: ^ : 'i*irrr/.r ,l"il 2 I :

I)+rivatil'es of i.he rcmai*ing h3;pe$otic lunctions can h* obtained b"v expre ssing thern interms of sinh *n*i cosh and appl-ving *ppropriaie i*lenrities. For exiample,

J tt I.i*h,..! ..',,tr.r{!*i,,hr!-sinhr$l.o*L*ialtr-rrh "r: - I 'lll]' l: 4x 4arf.r'---.'."' d.r Icosl,r] coshi.r

r.-e::h2x-sir:h2 x 1..-:.-:Sei:h'.Yoosh'x cosht -r

-l'!:e foll*in'ing theorer:: pr*lieles ir conrplete list rlf th* generalieed elerivative fomurlas an<!

cr:neap*ncling :integratie.:n for:nulas f'or ihe i:;,perbsrlic functi*ns,

{i.S"S yg{Er}fic$.r

d... du f, liinha]:coshu, fcoshadu:5Y;14a6axdxJ::dduf

;[coshul:sinhuA; lsmhudu:coshu:_CI

d - -) du ff [t*tl u]: sech2 u;

;l sech2 u du : tanh z * C

firc*nu) : -csch2 uff l r"rt',' udu :-cothz *c

frfr*nul : -sech uann uff / r*.r', utanh u ttu : -ssgt1, a 6

dduf*[csch

r.rl: -csch ucothu dx ;[ csch ucothu du: -csch u *C

s" NNer*pE* ?

,1".jl1,joshi.i'll : sirh{r3) . *lrtl = l.i r si*ht.r.tjdr' ds'

d 1 J sechf;:

-I.{*{tanhxil:_-. . ltilnl.r.vl- __ - {

d.r ' t:tnh -r d.r ' ' lanh r

478 Chapter 6 / Applications of the Definite lntegral in Geometry Science, and Engineering

w #xermp$* 5

lfu'e r.vere ji:stified in dr+pping the *b:oiute value signs sinc* rrosh.r :. * f*r ail ^r. d

r $:xesmple 4 A ] 00 ft wire is attar-!:ed at its e r:cls to the tops of trvc 5* ft peles that are

positioned 1)0 ft apart. Ho:a' high ;rl;ove the grouncl is th* midel]* of the r.r,ire'l

,$$dle$i$$g. Fr*nt abar,*, i!:* rvire f.brms a cat€*a+.- cur:'e ix,ith *<luatio*

r, : # c()sh {l} *,rl&er"* the arigin i:, e.:n the ground nTidrvay betx'*e* the poi*s. Usir:g l'ormula (4) of Secticn

6.4 forth* le*gth of the i-ratsr!&ry, lv* har-*

{ , hxd sinh'.y cosh .r J.u : I siuir"-r *' C r: sD

J 6"'""..'- ;1t;**:i*'i" I sinh;:f tanh i,/r: ; . d^.J _l e\.,:h ,t

: ]* lcr.,sh xi + C , ,l ] TJlr, i

:ln(cesh.r)+C r"-.''..".'i

,1

By symetry Iahoutthe v-axis

r,.,ra I1\ ,1,, '-nv iif-J,r'. r*l !-"""\A.l *'- 'thalcoshr>0 l

100: "f dt1,Jr,\)

'*l*Jn.15

"J- o .1,,

^ l'*':, J,,

:, lt'*Jo

i---------------!,)i I aIV \

.il+{-i}d.r! \d.1:lt'.

,r ^,x''ll-i-sinh-l-ldxti \alI {4',

i r. = 5o rlt .ntui=*i -Ii.ns r

\]b.l; I J i

A Figurc 6.93

:J5: la sinli {1) l- : 2,,0;n* { i-1}..rrl}o \{?.r

Using a calculating utilit,v's numeric solver t* solve

10{) :2uri* ft] \"",,, \ * llora givcs s ry 56.01. Then

50 : l'(45) : 56.0! 'u,r, {-f- } u ., * 75.S8 +

"\.56.0t,,so c,y -25.*8. Thrrr, the nriddle of the in'ir* is -v(lli Ar 56.01 - 25.1)8 : -1*.93 iI above

the grou*d iligur'* 6.9,.{i, <

{N\dfr RSSS *F }iYpHe$S#$-$ fl $*U $WT'X S gid$

li.el'erring to Frigure 6.9. t, it is evielent that the graphs of si*It,r, ta*h x, coth .r, a*d e.-sch spass thc horizontal liee tcst, hut thc graphs ol cosh.r and sech -.r* {ir'} &t!t, ln the lottcr ease,

r*stricting .r to be nonnegative makes the tuncticns invcrtible {Figur* 6.9"5}. The grap}rs cF

r&c sir inverse hyperb*iio functions in Figure 5.9.5 werc obtiii*c<.i E:v rellecti*g th* grrphs

olthe ?ryperholic functions {rvith the appr*priatc restriclions; al:out ihc lin* r; : .1.

i With th- r..tri.ti.ilhrt r: 0l] the c*rres..; = c()sh.r nnd I

6.9 Hyperbolic Functions and Hanging Cables 479

Tirble 6.9.1 summarizes the basic properties of the inverse hyperbolic functions. Youshould conflmr that the domains and ranges listed in this table agree with the graphs inFigure 6.9.6.

-.". _;.":r =:llll---l

l

lr=:::*]

Table 6.9.1PR{IPERTIF.SOFINVEREF-}r{FFRR{}T.L]FTIN=-r]TION*S

i = sech,\: pass the horizonirl i

line test, I

& Figure 6.9.5

& Figure 6.9.6

i-*-.:;-l\' = sttf:ft 'r Il-:--^--. .s

Slni! '1 (-*, **) sinh-iisinh xi = .r if -* < ,r < +%

sinhisirh ixi =,r il -* <.Y { *m(**, +*)

ci)sl't ' J f 1. +*l [(i, +*)cosh-i(cosh e'i = :; il ,i; > ilcoslticosh-i.r) = .t il -:; > 1

lal1n -Y (-1, 1) (-*, +*) tanlr-i{ianh r} = x il -e (.r { *etanir(tanh-ix) =x if -l <x< i

fillll - r i--, -1) U il, +*) 1**, 0) U (ti. +*i c+th-1 (coth.t) =

cr-lhiccih-i.Y) =

r<0or;r;>iJ.r<-lorx>i

tifr if

sech*I x ({i, tl [0, +*i sech-1(sechx) =x if x>{rsec!r{sech*1x}=x il 0<x< -t

csch-i(csch "x) = .r il' "r < C i:r :r > 0

csch(csch*i.t) =x il x<{ii-:i -r > {i

t = ct>sh:t:

i._*-117"1,li:l*11-l i l;:*", ,.

,'lcscll ' .x; 1**, {i) LJ (i}. +*) i-*, i}i U ({i, f*)

480 Chapter 6 / Appllcations of the Definite lntegral in Geometry Science, and Engineering

W *#*&W&YY{.ft49( $S${MS SS *f\$Wffi${$fr ${YpS&mSL[{ SE$NflT$Sf\$$

Becaus* the htpe:*olir: f'unctions are expre ssihle i* te*::s of e'", it s!:oukl noi !:e suq:risi::gtl:et tfte i*verse it_vperholir- furrctions are expressible i* terms of naiural lo5arithnrs: ilxe **xttheorern shot,- rhlt this is so"

'**-.^*"*.\6.S,d TE{E{}KH},$ Tkc .fbltawing rtl*tiarcsltigzs !*t<l .fbr ullslated int,e rse hype rbolic J'uttctiorts:

x ist tke rksmai*s aJ'the !ii

,slfi-r,*

, h'xx

iri se€h'x

=.ud'.tfl.=P..'..: 1r, /1!\2 \l-x/

: : : I , ': : : :: :

: :: : : :

-,,/ri \\xl

iCoih,==1.x,: u{t,+, 1

corh-rx: l* (" *'\2 \x-l/

)++ x'?\csch-tx:rn(l+/l', I

\x lxl I\ ::'::' '

!1.e wiil show how to clerive the fir"st l'ol:lurla ili this iheorem arrd }eave tite r€st as exereis*s.

lhe basic idea is io -,r,rite €h* eqlratisri x : sinh ,r; i* ter:x: *i exponential iunctirins ar:d

solvethisequaiianlbryasafunctionof;. ll'!:isrvillpre:riur.*iheequaiie.:ny::inh-l ,twith si*h-l x expr'*sserl is: ter"rx: ol *atural logarithnts. Irlxpresli*g -r : sinh -y in {erms r:fexponenti;ris ,vieids eJ' * e-!

.r : sinit.r,

rvhicit can be rewritten as

e" -].x -r-):0fululiiply'ing this equatio* thrr:ugh bv g} r.ve trbtain

E3r_Z.vo,)_l:ti

anil applying the ciuari*rtic formul;r yiel*ls

2x + iQ a$

Si*ce <rr' > *,Thus,

tht: s*lu{ir:n i*v*lviirg t}re mi*us sigir is *.,itrirnecus and rnust br: discanJeel.

{:Y:aao{:11'l-aking *atcrai ie:garilhr::s _vi*Ids

I:l*tx*r,'x2*l) or sinh*l x : ln(.r +.r'?'+ t t

r ilxmrmp{e S

sinh*l

tanh-1

[ - ]*il11\g *t-\z t

+ 4P + t ) : Inil + yr2) ^, 0.8814

1 /!++\ t

; t" { ;* I : ; tn-r Ai 0."549.1 'd- \'--l L

$ # at e dedvfih;-€.,r#:the firrnc.

tion sinh-I i cdn also be obtained by

lqtlng y = riipb;I x and tlien dlfferen,

fi'ating x = "66'y ;milicitly.

,

6.9 Hyperbolic Functions and Hanging Cables 481

flHRnr&fi vg$ &Bd m * MrE& {e&!-s ! ru$sll{* N# * NVEffi $H F*YpHE{m #$. $ fr F{J ru fl TI0S$$l'ormulas for the derivalives of the inverse h.r,perholic tunctions r.:*n he *btained tior*'I"heoren: 6.9.4. Fbr example,

dd--1/t\a;tinh-l.r;:|11*{-i-r,1rl-! ri::{l *, - }d.r' ' d.r, ."+v.r:*l \ ,/r:+lf,&, + I +, j

{'v * 1I''I;TT; (v/xz + r )

[]ris conrputati*n leads t+ fiA'o integra] formu]as, a

equiv:rle*t ti:rnrula that involr,** logarithtns:

f-t

t *: s!:t!t-rx -F C: ln(, + r,''r7t + I J * CL.-.l ,irz1.1

'l'he fbllowing tr,r* theorcmr }ist the ge**raiized derivative t'*r::;*las and r.-**espa*ding

integra€i+n formulas f*r the i*verse h.vpertr-'*Iic tuncti*r:s. Sorne *f the prr:of-s appear ils

exercises.

/.a

r,';- - |

f'ortr:uia thal invoives sinh-l .r and a*

f6.$.s '*;"r-"*

d..,._, , 1 du--(Srnn - *): -p--1-

d.ldu-. (coah-' u): --p *, u > 7

dx ' {uz_14dldu-:lunh-t ut :

- ], lul o 1ax t-u"Qx

d .-. I du'|icoth-' 111 : 7--;-;-. lul > Idx L-u'axdTdu

-;-(sech-' u) : -----a',, U< r{ < Idx utfl-azaxt:

dldu-(Csch-r

U\: --;:-. , u +Udx lult/t I uz dx

6,9"e ?srfl{}tr{E&{ lf a > {}, t!*n

{#=cosh-r (,r;1*' *

r d* [ ],"*-' (i) * ,,J a-;:

l;;;,n=, (i) *r.

t -+: -1"..h-'l1l.,J uJa2 lu2 d....

ntu+uG3tii. u-o,,,

lul<a . ! r

o, !hl1:\, i:i, lul { a,,,- _ '2a ld-ullul>a

,--r---aror,-=L1 (o+t/-o'.-" )*a, o<far <aa\tull

+::e;,'il:+a

t ,, *_** ,"_-_, .,,,_ ***:tI

4a2 Chapter 6 / Applications of the Definite lntegral in Geometry, Science, and Engineering

*. *xssxrp$* $t {1"\ .1

llvalr:4l* i:-.tr> -.t /. , 2r \/+x,_ _y

dquecuu ($4ffi*${ Hxf;fficrsfs s"s

S*Jss$xrip. Lct * = 2.r. Thur. ,l * = 2J.v irnij

{dxlfZdtl{,}LtI _- _ I _

- _ I

--

i -*:- - - ,J r'l.i:-9 :,1 ,,4r2-q 2J 'i,i-;l: l..,rh-' {il} + c : l.nrh-, {I} *o: \3/ 2 \3,1

Alte*rativel.v. 1ve <;a* ase th* l+garithmi<-. eqxil'aleni af r:osh-l \zxl .2i,

."'n-' {'+ } : }r:r2v - o,"'I"r 91 - 1,, -1\'/(verif.v), anci express the ansiner as

L. coshx : sinh x :tanhx:

2. Complete the table.

3. The parametric equations

x:cosh/, y:sinhr (-m</ <*o)represent the right half of the curve called a

-'

Elim-inating the parameter, the equation of this curve is

-.flHtr$t{*S$} %ffi e"* E rJr;rykirrgttil"tv

d

;[tanhx]:

s. lcoshxttx:- /,mr,, dx:-

l u,nn*dx:-o. ft"out-'xl :- ft*irr,-'xl :-

!Ltuot'r-' *l:ax

d4.

;fcosh.r1d;[sinhxJ

1.

Approxirnatc the expression ta four dccir*al pl;rces. ;,I

(a) sinh3 (b) cosh(-2) (c) tanh(ln 4)(d) sinh-1(-2) (e) cosh-13 (f) tanh-l

o1

2. (a) csch(-l) (b) sech(ln2) (c) cothl(d) sech-r I (e) coth-13 (f ) csch-1(-J3)

3. Find the exact numerical value of each expression

ate identities to find the exact values of the remaining flvehyperbolic functions at x6.

(a) sinhxo :2 (b) coshxs : | (c) tanhx6 : I6. Obtain the derivative formulas for csch x, sech x , and coth r

from the derivative f'ormulas for sinh -t, cosh x , and tanh x.

7. Ffud the derivatives of cosh-l x and tanh-1 x by differen-tiating the formulas in Theorem 6.9.4.

8. Find the derivatives of sinh-1 x, cosh-1 x, and tanh-l x bydifferentiating the equations x : sinh), x : cosh), and

x:tanhyimplicitly.

9*3# l'ri*d d!ldx. i!,t'l

9. Y: sinh(4x - 8)

4. In eachpmt, rewrite the expression as aratio of polynomials

(a) sinh(ln3)(c) tanh(2ln 5)

(a) cosh(ln x)(c) tanh(2lnx)

(b) cosh(-ln2)(d) sinh(-3In2)

(b) sinh(lnx)(d) cosh(-lnx)

5. In each part, a value for one of the hyperbolic functions isgiven at an unspecified positive number x6. Use appropri- 10. y : cosh(xa)

ll. Y: coth(lnx) 12.

13. y: csch(l/x) 14.

15. y - V?, +..-t15r; 16.

17. y-x3tanh21Ji1 18.

19. y: sinh-1 (|x) 20.

21. Y : ln(cosh-l x) ?2.

1,t r,-tanh 'x

25. y : cosh-1(coshx)

27.Y:e'sech-118

***44 Eval*;rte the integrals.

29. I sinh6xcoshxdxJr-

3I. I Vtanhx sech'xdx

33. I anhxdxI

4ln 3

35. / tanh x sech'x dxJln2fdxt_J urt**,fdxI -

-

(x.0)l Jt-suf drt_I -t5*+-'f t/2 dx,_.ln l_ x2

y:ln(tanh2x)

)/: sech(e2x)

),: sinh3(2r)

y: sinh(cos3r)

y: sinh*l(Ux)y : cosh-l(sinh-l x)

y : (coth-l x)2

y : sinh-l(tanhx)

v:(1+xcsch-1x)10

1x :. 5/3)

(a) lim sinhx(c) lim tanhx

(e) lim sinh-1 r

(b) lim sinhx

(d) lim tanhn

(f ) lim tanh-l xr-1

6.9 Hyperbolic Functions and Hanging Cables 485

is 2 square units. Express your answer to at least flve deci-

mal places.

53. Find the arc length of the catenary ): coshx between

x :0 and x :1r,2.54. Find the arc length of the catenary, - a cosh(x / a ) between

x:0andx:xr(xi>0).55. In parts (a)-(f) flnd the limits, and conflrm that they are

consistent with the graphs in Figures 6.9.L and 6.9.6.24.

26.

28.

ffi

30. I cosh(2x -3)dxJ

sz. [ ,rrf3*\d*J

sa. fcoth2 xcsch2xtlxJ

71t3 "x

_ "-t36. I

-dx

Jn er + e-rf dr

3s. / l+ (x, il1r VX-_Zf sin9 d0

40. l:J ,fl + cos2 o

56.

57.

Explain how to obtain the asymptotes for y:1a6r,from the curvilinem asymptotes fo,r y - coshx and !

) : sinhx. i

Prove that sinh x is an odd function of x and that .o*h, i

is an even function of x, and check that this is consistent i

with the graphs in Figure 6.9.1. I1

&#*.4'41 TYup;*F*$ge lJeterl'niae rvhether tlr* stalenrert is i.r:rr* or

f;rlse. lixplai* ycw anslvor. :ilftl

45. The equation coshx : sinhx has no solutions.

46. Exactly two of the hyperbolic functions are bounded.

47. Thrre is exactly one hyperbolic function /(x) such that

for all real numbers a, the equation f(x) : a has a unique

solution x.

48. The identities in Theorem 6.9.2 may be obtained from the

corresponding trigonomekic identities by replacing each

trigonomekic function with i* hyperbolic analogue.

49. Findtheareaenclosedby y : sinh2x, y: Q,a1fla : ln3.

50. Find the volume of the solid that is generated when the

region enclosedby y : sechx, y : 0, x : 0,andx : ln2is revolved about the x-axis.

5I". Find the volume of the solid that is generated when the

region enclosed by y : cosh2x, y: sinh2x, x :0, and

x : 5 is revolved about the x-aris.

B 52. Approxirnate the positive value of the constant a such that

the area enclosed by y : cosh.rx, ) : 0, x : 0, and x : 1

58*$$ Pr*v* the id**{ities. :;;l

58. (a) coshx * sinhx : e'(b) coshx - sinhx : e-'(c) sinh(x + y) : sinhx cosh y * coshx sinh y(d) sinh2x : 2 sinhx coshx(e) cosh(x + y) : coshx coshy * sinhx sinhy(f) cosh2x : cosh2x * sinhzx(g) cosh2x:2sinh2x*1(h) cosh2x :2cosh2x - 1

59. (a) 1 - tanh2x : sech2 x. tanhx * tanh ythl tanh(-r -l r,) :

| + tanh x tanh Y2tanhx(c) tanh2.r

I * tanh2 x60. Prove:

(a) cosh-1 x : ln(x + J? -11, x :- |. t /1+x\

(b) tanh-tx: lln (

- l, -l < "t < l.

I \t-^/61. Use Exercise 60 to obtain the derivative formulas for

cosh-lx and tanh-lx.62. Prove:

sech-lx:cosh-1(l/x), 0 < x < 1

coth-1 x : tanh-1 (1/x), lrl = 1

csch-1x :sinh-l(1/x), x *a63. Use Exercise 62 to express the integral

fduJr-d

entirely in terms of tanh-1.

dt

31.

39.

4't.

43.",,8

oo. l, {t4 r

4O4 Ghapter 6 I Applications of the Definite lntegral in Geometry, Science, and Engineering

64. Show that

c) ft.*n-'l*ll:@) flcscrr-I1x11:

1

xt/L + x2

(b) Find the length of the centerline to four decimal places.

(c) For what values of x is the height of the arch 100 ft?Round your answers to four decimal places.

(d) Approximate, to the nearest degree, the acute angle thatthe tangentline to the centedine makes with the ground

at the ends of the arch.

74. Suppose that a hollow tube rotates with a constant angular

velocity of ar rad/s about a horizontal axis at one end ofthetube, as shown in the accompanying flgure. Assume that an

object is free to slide without friction in the tube while the

tube is rotating. Let r be the distance from the object to the

pivot point at time / > 0, and assume that the object is at rest

and r : 0 when / : 0. It can be shown that if the tube is hor-

izontal at time f : 0 and rotating as shown in the figure, then

Ir : fillsinh(arr) - sin(ror)l

during the period that the object is in the hrbe. Assume that

r is in seconds and r is in meters, and use g : 9.8 m/s2 and

at:2rad/s.(a) Graph r versus r for0 < r < 1.

(b) Assuming that the tube has a length of 1 m, approxi-

mately how long does it take for the object to reach the

end of the tube?(c) Use the result of part (b) to approximate dr/dt atthe

instant that the object reaches the end of the tube.

-4": :/,

*''Y o\, _"4W* _l-w\*,1./ \,,\// 4 Figure Ex-74

75. The accompanying flgure (on the next page) shows a per-

son pulling a boat by holding a rope of length a attached

to the bow and walking along the edge of a dock. ff we

assume that the rope is always tafigent to the curve traced

by the bow of the boat, then this curve, which is called a

tractrix, has the properb/ that the segment of the tangent

line between the curve and the y-axis has a constant length

a. lt can be proved that the equation of this tractrix is

):asech-l t-Jo'z-7

(a) Show that to move rn" Ool, of the boat to a point (x, y),the person must walk a distance

from the origin.(b) If the rope has a length of 15 m, how far must the person

walk from the origin to bring the boat 10 m from the

dock? Round your answer to two decimal places.

(c) Find the distance traveled by the bow along the tractrixas it moves from its initial position to the point where

it is 5 m from the dock.

65. In eachpart, flnd the limit.

(a) rtrT-

(cosb-r x - tn x) (b) ,I!- TP66. Use the first and second derivatives to show that the graph

of y :1a4r-1 x is always increasing and has an inflectionpoint at the origin.

67. The integration formulas t* l/ nE17 in Theo,rem 6.9.6

are valid for u > a. Show that the following formula isvalid for u -< -a;

t #: - cosh-I (_i) *, or ul" + nE - ,*l+ c

68. Show that (sinhx * coshx)' : sinhnx * coshzx.

69. Showthat

f a 2 stnhat

J_oe'' dx

70. A cable is suspended between two poles as shown in Fig-we 6.9.2. Assume that the equation of the curve formed bythe cable is y : a cosh(x/ a), where a is apositive constant.

Suppose that the r-coordinates of the points of support are

x : -b and x : b,where b >- A.

(a) Show that the lenglh I of the cable is given by

L :2a sinh2

(b) Show that the sag S (the ,".ti"altirtunce between the

highest and lowest points on *re cable) is given bybS:acosh--a

"3|*VZ -flnese exercists reler lrl *ie iraxging c*ble desc*hed in

I:xetxisc'iA. "lt"tt

B 71. Assuming that the poles are 400 ft apart and the sag in the

cable is 30 ft, approximate the length of the cable by approx-

imating a. Express your flnal answer to the nearest tenth ofafoot. fHinr: Firstlet u:2A0/a.l

ffi 72. Assuming that the cable is 120 ft long and the poles me 100

ftapart, approximate the sag in the cable by approximating

a. Express your final answer to the nearest tenth of a foot.

lHint: Fkstlet u : 50/ a.l

B Z:. tne design of the Gateway Arch in St. Louis, Missouri, bymchitect Eero Saarinan was implemented using equations

provided by Dr. Hannskarl Badel. The equation used forthe centerline of the arch was

y :693.8597 - 68;7672cosh(0.0100333x) ftfor x between -299.2239 and 299.2239.(a) Use a graphing utiliqr to graph the centedine of the arch.

initiaipositio* dFigureEx-7S

Writ9xtg Suppr:sc that, b,v aflalog-y with thc trigo*omc,trie

1'unctions, we de.fix,t c+sFrl aiid si*hr ge*metricall,v usingEiigure *.9.3b:"For any real n*i:'lL'er f . sJe firle -r : eoslt, arrd r: : si*h r tobe the unique values of -r a*d v such that

Chapter 6 Review Exercises 485

(i) P(x, y) is on the right branch of the unit hyperbola-.2 -.2 -

1.L -

V -

t.

(ii) r and y Uur" it "

same sign (or are both 0);(iii) flre area of the region bounded by the x-axis, the right

branch of the unit hyperbola, and the segment fromthe origin ts P is Vl/Z;'

Discuss what properties would first need to be verffied inorder for this to be a legitimate definition.

lffrBti*'ag l*r.'*stigate rrhat prop*rties of coshr a*d sinhlc;r* be pr*vottr dilectl-v li'orn the g*ometric dcfraiticx inksrcise ?6. 1\ rit* a short descript!*x q:f ibe results of -yr-:r.:r

irlvestigation,

77.

76.

Mq*is*s{ flr*ffi(K erus\trsffis ffi.$

i.

:"

:\"

f,

r ta

Z

(^ -L-::-:*-*:::

e^+e

unit h;.per!:*la: -rl * J,2 .t1:4, sinh-r; coshxl -**el",:,r $. :ii.nl:rx * C; casir-t f C; t*i**sl::;; { C

,"-n------;'---i'- I

tr${&$}YKK $ RHW$H\St HH*ffieXSfiS

oo-qh.l- sinh -r tanh.r citlh :r -q*ch.r osrir .r

(-*. +*i 1-*, ili u (fi, +*) (--, 0) u iC. +*)

ll +ei i-r, l (-*, -1) Lj i 1. +*) (0. il (-*, it) \r (0. +*)

1. Describe the method of slicing for flnding volumes, and

use that method to derive an integral formula for flndingvolumes by the method of disks.

2. State an integral formula for flnding a volume by the methodof cylindrical shells, and use Riemann sums to derive the

formula.

3. State an integral formula for flnding the arc length of a

smooth curve y : f(x) over an interval [4, b], and use Rie-

mann sums to derive the formula.

4. State an integral formula for the work W done by a variableforce F(x) applied in the direction of motion to an objectmoving from x : a to x: b, and use Riemann sums toderive the formula.

5. State an integral formula for the fluid force F exerted on avertical flat surface immersed in a fluid of weight density p,and use Riemann sums to derive the formula.

6. Let R be the region in the first quadrant enclosed by ! : x2 ,

y :2 + x, and x : 0. In each part, set up, but do not eval-

uate, at integral or a sum of integrals that will solve the

problem.(a) Find the area of R by integrating withrespect to r.(b) Find the area of R by integrating with respect to y.(c) Find the volume of the solid generated by revolving R

about the x-axis by integrating with respect to x.(d) Find the volume of the solid generated by revolving rR

about the x-axis by integrating with respect to y.(e) Find the volume of the solid generated by revolving R

about the y-axis by integrating with respect to x.(f ) Find the volume of the solid generated by revolving R

about the y-axis by integrating with respect to y.(g) Find the volume of the solid generated by revolving R

about the line y : -3 by integrating with respect to x.(h) Find the volume of the solid generated by revolving R

about the line x : 5 by integrating with respect to r.7. (a) Set up a sum of deflnite integrals that represents the total

shaded area between the curves y : f (x) and y : g(x)in the accompanying figure on the nextpage. koe)