is needed and this can be established by multiplying A, obtained in step 3, by, resulting V = A x y =. = x, located in 1 st quadrant rotated about 2
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1 Ct Cllege f New Yk MATH (Calculus Ntes) Page 1 f 1 Essental Calculus, nd edtn (Stewat) Chapte 7 Sectn, and 6 auth: M. Pak Chapte 7 sectn : Vlume Suface f evlutn (Dsc methd) 1) Estalsh the tatn as and the nteval we need t calculate (ntesectn pnts f the functns). ) We need t get the aea etween ccles wth same cente (cente s the tatn as), see fgue. The aea f ccle s A = π, theefe we need t fgue the ad f the ccles (The adus s fm the tatn as t the functn). Ths wll gve us ad, call them ute ( ) and nne ( ) adus. Thus we have A = π and A = π. Hw t fnd adus: The heght f the functn s measued fm - -as t the functn, ut the adus (ethe ) s fm the tatn as t the functn. Yu ma need t add sutact t tan u adus. ) The aea f the shaded egn s A= A A. Ths s the functn that we wll ntegate (Daw the csssectn dagam). ememe that A s ethe a functn f. 4) A thn vlume V s needed and ths can e estalshed multplng A, taned n step,, esultng V = A V = A. 5) The fnal vlume s calculated V = lm Vj = lm Aj j. Snce we ae calculatng cntnuus n n functn, we wll calculate: V = Ad V = A d. a a Eample.1: Fnd the aea unded =, = Step 1: see fgue t the left = =, lcated n 1 st quadant tated aut = = 1= = = = 1 1 = ( ) =. = + = + Step : A = π ( + ) A = π ( + ) = π ( ) = π (4 4 ) = Step : A = A A = π (4 4 ) π(4 4 ) = π [ ] = π + 4 (4 4 )
2 Ct Cllege f New Yk MATH (Calculus Ntes) Page f 1 Essental Calculus, nd edtn (Stewat) Chapte 7 Sectn, and 6 auth: M. Pak 4 V = A = π (4 + 4 ) V = π(4 + 4 ) d= π Step 5: = π (1) + (1) (1) (1) [] 5 = π + = π π = π + = π = 1 Eample.: Same estctn as Eample.1, ut tated aut =. = Step 1: see fgue t the left and take same endpnts as Eample.1. = = Step : Ntce that n ths case we need t sutact the heght f the functn fm a cnstant value f t get u needed ad (unlke the Eample.1 whee we needed t add t the heght f the functn). Thus esultng wth u ad t e: = = ( ) π ( ) A = π A = = π + = + 4 (9 6 ) π (9 6 ) Step : A = A A = π (9 6 ) π(9 6 ) = π [ ] = π + 4 ( 6 6 ) 4 V = A = π ( ) 1 1 V = π( ) d= π Step 5: = π (1) (1) (1) + 4(1) [] 4 5 = π + = π π = π + = π =
3 Ct Cllege f New Yk MATH (Calculus Ntes) Page f 1 Essental Calculus, nd edtn (Stewat) Chapte 7 Sectn, and 6 auth: M. Pak Eample.: Calculate the vlume geneated tatng 1 st quadant f = 1. = and =, tated aut = = Step 1: see fgue t the left = = ( ) = = = = 1 Step : Ntce that unlke the pevus eamples, the ad s measued hzntall nt vetcall ecause we ae geneatng vlume tatng aut vetcal as. = 1+ = 1+ A = π + A = + (1 ) π(1 ) = π + + = π (1 4 4 ) (1 ) Step : A = A A = π + + π + + = π + + = π (1 4 4 ) (1 ) [ ] (4 ) = = + 4 V A π (4 ) V = π(4+ ) d = π () () () [] + = π Step 5: π = π 8 + = π + = π = Eample.4: Same estctn as Eample., ut tated aut = 5. Step 1: see fgue t the left and take same endpnts as Eample.. = = Step : Ntce that n ths case we need t sutact the heght f the functn fm a cnstant value f t get u needed ad (unlke the Eample. whee we needed t add 1 t the heght f the functn). Thus esultng wth u ad t e: = 5 = 5 ( 5 ) π ( 5 ) 4 (5 1 ) A = π A = = π + = π + (5 4 ) = 5
4 Ct Cllege f New Yk MATH (Calculus Ntes) Page 4 f 1 Essental Calculus, nd edtn (Stewat) Chapte 7 Sectn, and 6 auth: M. Pak Step : A = A A = π + π + = π + + = π + 4 ( 14 ) 4 4 (5 1 ) (5 4 ) [ ] = = + 4 V A π ( 14 ) V = π( 14 + ) d = π 1 () () 1() [] + = π Step 5: π = π + 4 = π + = π = Chapte 7 sectn : Vlume Suface f evlutn (Shell methd) 1) Ths methd s geneatng vlume wappng ectangula sheets n tp, lae afte lae. Imagne wappng alumnum fl n a all pnt pen; as we appl me fl, the veall damete gets thcke. Daw a css sectn dagam s we can estalsh the tatn as and the nteval we need t calculate (ntesectn pnts f the functns). ) We ae wappng ectangula sheets see fgue. What we need t get s the aea f ectangle (ata). Wdth s ease; just take the dffeence f the uppe and lwe functns (lke Sectn 6.1). The length s the tcke ne. Ths s the ccumfeence f the ccula pat that s wappng aund. The adus (thee s nl ne) s measued fm the tatng as t the pnt whee the functns ae epessed (emnde: adus s nt alwas = = ). Make sue t daw a css-sectn dagam s we can detemne f we need t take the sum the dffeence n epessng u adus. l = C = π [ ( ) ( )] w= f g f( ) uppe functn g ( ) lwe functn ) Detemne A = l w usng l and w fm pat. Then u n the tatng as. V = A V = A dependng 4) The fnal vlume s calculatng V V = A d V = A d. a a = lm n n j = 1 V j. Snce we ae calculatng cntnuus functns
5 Ct Cllege f New Yk MATH (Calculus Ntes) Page 5 f 1 Essental Calculus, nd edtn (Stewat) Chapte 7 Sectn, and 6 auth: M. Pak Eample.1: Calculate the vlume geneated tatng the aea f ntesectn pnts f =, and =, tated aut = 1. = and = Step 1: see fgue t the left = = = ( ) = = w Step : w= ( ) ( ) = = 1+ l = C = π = π (1 + ) = Step : A = l w= + π (1 )( ) = + π ( ) = + V = A π ( ) = 1 = π ( + ) V = π(+ ) d= π () () () [] + = π π = π + 18 = π + = π = Eample.: Same estctn as Eample.1, ut tated aut = 4. = Step 1: see fgue t the left and take same endpnts as Eample.1. w Step : The wdth s same as Eample.1. But ntce that n ths case we need t sutact fm a cnstant value f 4 t get u needed adus (unlke the Eample.1 whee we needed t add 1 t ). Thus esultng wth u adus t e: = 4 l = C = π = π (4 ) = = 4 Step : A = l w= π (4 )( ) = + π (1 4 ) = + π (1 7 ) π (1 7 ) V = A = +
6 Ct Cllege f New Yk MATH (Calculus Ntes) Page 6 f 1 Essental Calculus, nd edtn (Stewat) Chapte 7 Sectn, and 6 auth: M. Pak V = π(1 7 + ) d= π 6 6() () () [] + = π π = π = π 9+ = π + = π = Eample.: Calculate the vlume geneated tatng the aea f ntesectn pnts f =, tated aut =. = + and Step 1: see fgue t the left = + + = = w = ( + 1)( ) = = 1 = Step : w= ( ) ( + ) = + = + l = C = π = π ( + ) = Step : A = l w= + + π ( )( ) = π (4 ) = + V = A π (4 4 ) = π (4+ 4 ) V = π(4+ 4 ) d = π = π 4() + () () () 4( 1) + ( 1) ( 1) ( 1) = π = π = π 14 + = π 14 + = π π = π + = π =
7 Ct Cllege f New Yk MATH (Calculus Ntes) Page 7 f 1 Essental Calculus, nd edtn (Stewat) Chapte 7 Sectn, and 6 auth: M. Pak Eample.4: Same estctn as Eample., ut tated aut = 4. = 4 Step 1: see fgue t the left and take same endpnts as Eample.. = + w = Step : The wdth s same as Eample.. But ntce that n ths case we need t sutact fm a cnstant value f 4 t get u needed adus (unlke the Eample. whee we needed t add t ). Thus esultng wth u adus t e: = 4 l = C = π = π (4 ) Step : A = l w= + π (4 )( ) = + + π (8 4 4 ) = + + π (8 5 ) π (8 5 ) V = A = V = π( ) d = π = π 8() + () () + () 8( 1) ( 1) ( 1) ( 1) = π π = π = π 1 = π 1 15 = π 16 = π = π = Chapte 7 sectn 6: Wk (Lqud Pumpng methd) The set up f wk f pumpng lqud s smla t the set up f elated ates plems fm 1 st Calculus class. Theefe, we need t e ale t fmulate cectl the vlume equatn n sngle vaale. But nstead f a geneal vlume V, we need t fnd a thn vlume V. 1) Daw a css-sectn dagam (sde vew and tp vew). Tp vew s needed f us t fnd ut the gemetc shape f the vlume we ae lftng t calculate wk. Sde vew s needed t detemne the pumpng dstance and fmulatn f the aea, A, whch s the fgue taned fm the tp vew and eventuall geneate V. Make sue t fgue ut the nteval whee the lqud s eng pumped ( a ). ) Geneate the aea A (whch can e ectangula, ccula, tangula, etc.), then multpl and we wll get V. Ths s the thn vlume that we wll pump ut f the tank (thnk f lftng a eam f pape ut nstead f lftng ente eam nce, thnk f lftng a sheet at a tme untl the ente eam s lfted t a hghe lcatn). Then we can fmulate F (thee ae usuall cases f F that we need t w aut, metc fm and Englsh fm) shwn elw:
8 Ct Cllege f New Yk MATH (Calculus Ntes) Page 8 f 1 Essental Calculus, nd edtn (Stewat) Chapte 7 Sectn, and 6 auth: M. Pak V = A Metc [measuement dne usng metes (m)] Englsh [measuement dne wth feet (ft)] Get mass: m = ( denst)( vlume) Usuall denst f wate s used 1 kg / m Theefe: m= (1) V Get fce: F = m a, Usuall a s gavt whch s 9.8 m / sec F = (9.8) m = (9.8)(1) V = (98) V Ths case s smple ecause a cnstant nume vaale s gven such that fm vlume we can tan fce wthut calculatng the mass (call t δ ). F = δ ( vlume) F = δ V = δ A = (98) A Keep n mnd that usuall the methd usng metc measuement s lnge ecause fm vlume we need t get mass efe tanng the fce; whle the methd usng Englsh measuement s a t shte fm vlume we get the fce mmedatel. If we use sme the lqud nstead f wate n metc measuement, then the denst wll als change thus changng the cnstant n fce equatn t a dffeent value than the ne shwn ave. ) Wk fmula s W = F p, whee F = fce and p = pumpng dstance. Theefe, we get W = F p = p F = pa. Bth epessn p and A s a functn f ; the epessn pa shuld e dstuted and smplfed, n de f us t have a smple ntegatn. 4) Snce we ae dealng wth cntnuus functn: W = pad. a Eample 6.1: Cnsde a V shaped tank 1 metes wde, 5 metes at the tp, and metes hgh flled full f wate (see fgue t the ght). Attached t the tp s a ppe f 1 mete whee wate wll e pumped. Fnd the wk dne when 1 mete f wate (measued fm the tp) s pumped ut f the tank. 1 m 5m 1 m m Tp vew Sde vew 5m 1m p Step 1: See fgues t the left. Als fm sde vew we can calculate u pumpng dstance f p = 1+. Cnsdeng that the tp f the tank s =, we can cnclude that u ntegatng nteval s 1. m Step : The tp vew shws us that the aea we need t calculate s a ectangle. And the length f ths ectangle s. Usng tangula pptn, we get: = = 5 ( ) 5
9 Ct Cllege f New Yk MATH (Calculus Ntes) Page 9 f 1 Essental Calculus, nd edtn (Stewat) Chapte 7 Sectn, and 6 auth: M. Pak 5 A = 1 = (1) ( ) = 5( ) V = A = 5( ) m= ( denst) V = (1)(5( )) F = a m= (9.8)(1)(5( )) = (98)(5( )) = 45( ) Step : = = (1 + )(45( )) = 45( + ) W p F W = 45( + ) d= (1) (1) (1) [] + = = 45 + = 45 + = 45 = 15 = jules Eample 6.: Cnsde a cncal tank (see fgue t the ght), full f msteus lqud, wth pnted sde up; the ase adus s feet and heght s feet; attached t the tp s a ppe f 4 feet whee the lqud wll e pumped. The denst f lqud s 1 ls/ft. Fnd the wk dne when the ente msteus lqud s pumped fm the tank. 4 ft Tp vew ft Step 1: See fgues t the left. Als fm sde vew we can calculate u pumpng dstance f p = 4 +. Cnsdeng that the tp f the tank s =, we can cnclude that u ntegatng nteval s. ft Sde vew p 4 ft ft Step : The tp vew shws us that the aea we need t calculate s a ccle. And the adus f ths ccle s. Usng tangula pptn, we get: = = 4π A = π = π = 9 4π V = A = 9 Denst: δ = 1 ls/ft 4π 4π F = V = = 9 9 δ 1 ft ft Step : 4π 4π 9 9 W = p F = (4 + ) = (4 + )
10 Ct Cllege f New Yk MATH (Calculus Ntes) Page 1 f 1 Essental Calculus, nd edtn (Stewat) Chapte 7 Sectn, and 6 auth: M. Pak 4 4 4π 4π 4 1 4π 4 1 4π 81 W = (4 + ) d= () () [] 4(9) = + = π 9 4π = 4(9) + (9) = (9) 4 + = 4π + = 4π = 1 π(5) = 5π ft-punds Nw f u e up t the challenge, t ths fllwng eecse: Eecse 6.: Fnd the wk dne n pumpng the wate ve the m f a tank, whch s 5 feet lng and has a semccula end f adus 1 feet, f the tank s flled t a depth f 7 feet. [nte: Denst f wate s δ = 6.4 ].
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