Volumes of Solids of Revolution via Summation Methods

Transcription

1 olums of Solds of Rvoluto va Summato Mthods Tlak d Alws talws@slu.du Dpartmt of Mathmats Southastr Lousaa Uvrsty Hammod, LA USA Astrat: I ths papr, w wll show how to alulat volums of rta solds of rvoluto wthout usg drt tgrato. Th tradtoal mthod of suh volum omputato uss dft tgrals as gv y Dsk Mthod or Shll Mthod a alulus ours. Howvr, stad of drt tgrato, w wll alulat ths volums as a lmt of a summato. Ev though somwhat logr th th tradtoal mthod gral, ths mthod mphaszs th fudamtal da hd a dft tgral,.. th dft tgral as th lmt of a sum. W wll also us th omputr algra systm Mathmata to faltat ad vrfy our alulatos.. Itroduto I ths sto, w wll rfly rvw th Dsk Mthod ad th Shll Mthod of fdg th volum of a sold of rvoluto (s [3]). Cosdr a ogatv otuous futo y f (x) dfd o a losd trval [ a, ] whr a ad ar ral umrs wth a <. Lt R th rgo oudd y th graphs of y f ( x), y 0, x a, ad x. Lt th volum of th sold otad y rotatg th rgo R aroud th x-axs. S th followg fgur: Y y fhxl O a x - x X Fgur. Th rgo R udr th graph of y f (x), ad th sold otad y rvolvg ths rgo aroud th x-axs

2 Aordg to th Dsk Mthod, th volum s gv y th followg dft tgral (s [3]): π [ f ( x)] dx (.) a Th aov formula (.) s asd o th volum of a yldr, whh s gv y π ( radus) ( hght). For xampl, suppos w dvd th trval [ a, ] to qual ps, whr s a atural umr, usg th partto a x < x < < x < x < < x Lt x th lgth of ay sutrval x, x ],,,...,. Th w hav th followg formulas: [ x ( a) / (.) x a + ( a) / (.3) W a ut th sold to sls prpdular to th x-axs, at th umrs x whr Eah sl s approxmatly a th yldr wth hght x ad radus x ),,...,. f (, so ts volum s approxmatly gv y π [ f ( x )] x. Thrfor, a approxmato for th volum of th sold s otad y addg all th smallr volums, as gv low: π [ f ( x )] x (.4) Th atual volum of th sold s gv y takg th lmt of th aov summato as : lm π [ f ( x )] x (.5) I th xt sto, w wll llustrat how to omput rta volums usg th aov formula (.5), wthout drtly valuatg th tgral gv y (.). Now to llustrat th Shll Mthod, lt W th volum of th sold otad y rotatg th aov dsrd rgo R aroud th y-axs. Aordg to th Shll Mthod formula W s gv y th followg tgral (s [3]): W π x f ( x) dx (.6) Th aov tgral s asd o th volum of a shll,.. th spa tw two yldrs, whh s gv y π ( radus of th shll)( hght of th shll)( thkss of th shll) (s [3]). Corrspodg to quato (.5), th summato vrso of th tgral (.6) s gv y th followg quato: a

3 W lm π x f ( x ) x (.7) I th xt stos w wll llustrat how to us formulas (.5) ad (.7) to alulat volums orrspodg to svral typs of futos f (x), stad of usg drt tgrato.. Th olum of a Sold Gratd y Rvolvg th Rgo Udr a Squar Root Futo Aroud th x-axs Cosdr th futo f ( x) k x ovr th losd trval [ 0, ], whr k ad ar ral postv ostats. Lt R th rgo oudd y th graphs of y f ( x), y 0, x 0, ad x. I ths sto, w wll omput th volum of th sold otad y rotatg th rgo R aroud th x-axs, usg quato (.5). S th followg fgur: Y y k è!!! x O X Fgur. Th rgo R udr th graph of f ( x) k x, ad th sold otad y rvolvg ths rgo aroud th x-axs By mployg th sam otato as gv sto, w a s that x /, ad x a + ( a) / /, for,,...,. Th usg quato (.5), o a ota th followg: π k lm π k lm (.) Th summato volvd quato (.) s a wll-kow quatty mathmats, ad s gv y th followg (s [] ad [3]): ( + ) (.) By usg (.) quato (.), w ar al to alulat th rqurd volum :

4 π k ( + ) lm. π k lm + π k O a of ours, vrfy th aov rsult y drt tgrato: Usg th Dsk Mthod formula gv y quato (.), π ( k x) dx π k [ x / ] π k just otad. 0 0 /, whh agrs wth th rsult I th xt sto, w wll osdr a slghtly mor hallgg prolm arsg from a quadrat futo. Ovously, as th futo f (x) oms mor omplatd, th orrspodg summato ad lmt alulatos om tdous. Howvr, th ours of th papr, th radr wll surprsd to osrv that o a stll us th summato mthod to fd th volums orrspodg to a wdr lass of futos, ludg polyomals, xpotal futos, logarthm futos, ad also s ad os futos. 3. Th olum of a Sold Gratd y Rvolvg th Rgo Udr a Quadrat Futo Aroud th x-axs Cosdr th futo f ( x) k x ovr th losd trval [ 0, ], whr k ad ar ral postv ostats. Lt R th rgo oudd y th graphs of y f ( x), y 0, x 0, ad x. I ths sto, w wll omput th volum of th sold otad y rotatg th rgo R aroud th x- axs, usg quato (.5). Usg th sam otato as sto, w a xprss th volum as th lmt of a summato: 5 π k 4 lm π k lm (3.) 5 Th summato 4 arsg quato (3.) rfrs to th sum of th fourth powrs of th frst postv tgrs, for whh a xprsso a foud most stadard mathmatal hadooks (s []). Altratvly, o a us a omputr algra systm (CAS) suh as Mathmata to valuat ths summato (s [] ad [5]). Th Mathmata ult futo Sum a alulat rasoal ft or ft sums. Spfally, th put l Sum [ ^ 4, {,, }] alulats th rqurd sum. Eah Mathmata ommad a xutd y prssg Shft-Etr at th d of th ommad l. Thus, w a ota th followg rsult: 4 ( + )( + )( ) (3.) Usg quato (3.) (3.), w ar a posto to fsh th omputato for :

5 π k lm ( + )( + )(3 π k + 3 ) π k lm O a of ours, vrfy th aov rsult drtly usg th Dsk Mthod formula (.). Th mthod outld ths sto a usd to fd th volums of solds orrspodg to ay polyomal futo f (x), ot just a quadrat futo. Howvr, as th dgr of th polyomal gts hghr, th had alulatos om mor tdous, ad thrfor, o a us a CAS for advatag. Th followg Mathmata program automats th produr for fdg th volum y summato mthod (s [] ad [5]): Program 3. Clar[f,a,,dltax,v] f[x_]: 3x^3 + x^ + x + a ; 3; dltax (-a)/; x [_]: a + (-a)/; v[_]: P * f[x[]]^*dltax vapprox Smplfy[Sum[v[], {,, }] ]; v Lmt[ vapprox, ->Ifty] I ths program, th usr a tr hs or hr ow puts for f[x_], a ad, ad th program alulats th volum of th sold gratd wh th orrspodg rgo s rotatd aout th x- 3 axs. I th aov, as a xampl, w hav hos f ( x) 3x + x + x +, a, ad 3. Th program a xutd y prssg Shft-Etr at aywhr th ommad ls. As th output, w ota th volum as 3608π / 7. O a also vrfy ths aswr drtly usg th Itgrat ommad of Mathmata, ad th Dsk Mthod formula (.), as show low: Iput: Itgrat[P(3x^3+x^+x+)^,{x,,3}] Prss Shft-Etr at th d of th ommad l to ota th output as 3608π / 7, ofrmg th prvous aswr. 4. Th olum of a Sold Gratd y Rvolvg th Rgo Udr a Expotal Futo Aroud th x-axs I ths sto, w would lk to hallg ourslvs y osdrg a xpotal futo for m x f (x). Cosdr th futo f ( x) k ovr th losd trval [ 0, ], whr k, m, ad ar ral ostats wth k, postv, ad m ozro. Lt R th rgo oudd y th graphs of y f ( x), y 0, x 0, ad x. W would lk to omput th volum of th sold otad y

6 rotatg th rgo R aroud th x-axs, usg th summato mthod. Th quato (.5) aga mpls th followg: m m π k lm π k lm (4.) Osrv that m s a gomtr srs wth th tal trm, ad th ommo rato ah qual to m /. Usg th fat that th sum of th frst trms of a gomtr srs wth tal trm a ad th ommo rato r s gv y a( r ) /( r) whr r, w ota th followg (s [3]): m / m ( ) m ( ) m / / Susttutg quato (4.) to (4.), w ar al to omput th volum, y makg us of L Hoptal s Rul (s [3]): (4.) lm π k ( m ) ( m / m / π k ( ) m ) lm ( m / ) π k ( π k ( m m / ) lm ( π k ) m m / ) ( m m π k ( ) m ) lm m / / ( m / ) (4.3) O a hk th auray of th aov alulato, usg Program 3.. By hoosg, mx f ( x) k, a 0,, th rsultg output s th sam as volum otad aov. Aothr way to vrfy th rsult drtly s, y usg th Dsk Mthod formula (.5), ad th Mathmata Itgrat ommad, as show low: Iput: Itgrat[P* (k*exp[m*x])^, {x,0,}] Prss Shft-Etr aywhr th ommad l to ota th sam aswr aov, vrfyg our alulatos. 5. Th olum of a Sold Gratd y Rvolvg th Rgo Udr a Logarthm Futo Aroud th x-axs Cosdr th futo f ( x) k l x ovr th losd trval [, ], whr k, ar ral ostats wth k postv, ad >. Lt R th rgo oudd y th graphs of y f ( x), y 0, x, ad x. W would lk to omput th volum of th sold otad y rotatg th rgo R aroud th x-axs, usg th summato mthod. Usg th sam otato

7 as sto, w s that x ( ) /, ad x + ( ) /. O th quato (.5) s usd to omput th volum, th dffulty s mmdatly lar, s w ar ow fad wth alulatg [ l( + ( ) / ]. It s vry dffult to smplfy ths summato v wth th hlp of a CAS, so w wll approah th prolm a drt way. Y y klhxl PH,L O X Fgur 5. Th rgos R ad I assoatd wth th graph of f ( x) k l x As gv th aov fgur, lt P (, ) th pot o th graph of f ( x) k l x / k orrspodg to x. Ths mas that k l, or quvaltly, W ow osdr th rgo I oudd y th graphs of y f ( x), y, y 0, ad x 0. Lt th volum of th sold gratd wh th rgo I s rotatd aroud th x-axs, ad lt th volum of th yldr gratd wh th rgo oudd y th graphs of y, y 0, x 0, ad x s rotatd aroud th x-axs. Clarly, π π k (l), ad th rqurd volum s just qual to. S th followg fgur: Fgur 5. Th sold otad y rvolvg rgo R aroud th x-axs, wth volum (o lft), ad th sold otad y rvolvg th rgo I aroud th x-axs, wth volum (o rght)

8 I ordr to fd th volum, w wll us th summato vrso of th Shll Mthod formula gv y quato (.7), wth x ad y varals trhagd. Dvd th trval [ 0, ] of th y-axs to qual ps usg th partto 0 y0 < y <... < y < y <... < y, whr s a atural umr. Lt y th lgth of ay sutrval [ y, y ],,,...,. It follows that y / ad y /. Wh usg th Shll Mthod as dsrd sto, ot that th radus of ah y k shll s qual to y, whl th hght s qual to /. Thus, th volum lmt of a typal shll y / k s approxmatly qual to π y y. By addg all ths volums, ad takg th lmts as, w a alulat as follows: y π k k lm π y k y lm π lm (5.) A srs suh as / ( k ) aov formula (5.) s alld a arthmto-gomtr srs, aus t s a hyrd of a arthmt srs ad a gomtr srs, ad a omputd y had wthout muh dffulty. Howvr, w opt to us th Sum ommad of Mathmata to omput ts sum, whr th rsult s gv low: k / k / k k / ( k ) / ( k ) (5.) ( ) ( ) By susttutg quato (5.) to (5.), w a prod wth th alulato for. I th / ( k ) followg, w usd th L Hoptal s Rul to ota lm/[ ( ] k /, smlar to th alulato of quato (4.3), ad th fat that k l. π π lm / ( k ) / k / ( ( k ) [ k ( ) + k l ] π k ( + l ) ) ( / k / ( k ) π ) ( / k k ) / k k (5.3) W ar fally a posto to alulat th rqurd volum as th dffr of th volums ad, as rmarkd arlr. π k (l) π k ( + l) π k [ (l) l + ( )] As do th prvous stos, o a of ours us th Itgrat ommad of Mathmata to vrfy th auray of th aov aswr.

9 6. Th olum of a Sold Gratd y Rvolvg th Rgo Udr a S or Cos Futo Aroud th x-axs As a fal llustrato, lt us osdr som trgoomtr futos. Cosdr th futo f ( x) k Sx ovr th losd trval [ 0, π ], whr k s a postv ral ostat. Lt R th rgo oudd y th graphs of y f ( x), y 0, x 0, ad x π. Lt us omput th volum of th sold otad y rotatg th rgo R aroud th x-axs, usg th summato mthod. Usg th sam otato as sto, w fd that x π / ad x π / for,,..., whr s a atural umr. Th quato (.5) ylds th followg xprsso for th volum : lm π k lm π π k S π π k lm π Cos π k lm π S π Cos (6.) Not that Cos( π / ) rfrs to a sum of th oss of a squ of agls arthmt progrsso, whh a omputd y th followg wll-kow formula, whr α ad β ar ral umrs wth S ( β / ) 0 (s [] ad [4]): Cos [ α + ( ) β ] Cos[ α + ( ) β / ] S( β / ) S( / ) β (6.) Usg th aov quato wth α β π /, w a asly s that th rqurd sum Cos( π / ) s qual to zro. Thus, th quato (6.) ylds th dsrd volum : π k lm π k (6.3) Th Itgrat ommad of Mathmata radly vrfs th aov rsult for th volum. Ev though ot ludd hr, o a us th sam mthod to fd th volum of th sold of rvoluto orrspodg to a os futo.

10 Coluso I ths papr, w showd how to alulat th volums of rta solds of rvoluto usg a summato mthod. Th tradtoal mthod drtly alulats ths volums va rta dft tgrals asd o thr Dsk Mthod or Shll Mthod, ad s grally fastr tha th proposd mthod. Howvr, th advatag of our mthod s that t mphaszs o of th most fudamtal aspts of dft tgral,.. th dft tgral as th lmt of a summato. Aothr mportat aspt of our mthod s that t gvs studts a xllt opportuty to dal wth two othr mportat aspts of alulus, amly summatos ad lmts. Also our mthod rats a w apprato for th dft tgral ad th Fudamtal Thorm of Calulus (FTC), as FTC provds a fftv short-ut for th summato mthod. Th papr also mphaszs th usag of a CAS, wthout sarfg th had alulatos altogthr. Th studt s ouragd to try th summato mthod dsrd ths papr to alulat volums of solds of rvolutos arsg from othr typs of futos. Rfrs [] Gradshty, I. S. ad Ryzhk, I. M. (980). Tal of Itgrals, Srs, ad Produts. Sa Dgo, CA : Aadm Prss [] Gray, T. ad Gly, J. (000). Th Bgr s Gud to Mathmata, rso 4. Camrdg, UK: Camrdg Uvrsty Prss. [3] Larso, H., Hosttlr, R., ad Edwards, B. (00). Calulus, 7th d. Bosto, MA: Houghto Mffl. [4] Loy, S. L. (96). Th Elmts of Coordat Gomtry. Lodo, UK: MaMlla. [5] Wolfram, S. (003). Mathmata Book, 5th d. Camrdg, UK: Camrdg Uvrsty Prss.