56 CHAPTE 7 Applictions of Integtion Section 7 Volume: Te Disk Meto Fin te volume of soli of evolution using te isk meto Fin te volume of soli of evolution using te wse meto Fin te volume of soli wit known coss sections Te Disk Meto In Cpte we mentione tt e is onl one of te mn pplictions of te efinite integl Anote impotnt ppliction is its use in fining te volume of tee-imensionl soli In tis section ou will stu pticul tpe of teeimensionl soli one wose coss sections e simil Solis of evolution e use commonl in engineeing n mnufctuing Some emples e les, funnels, pills, ottles, n pistons, s sown in Figue 7 ectngle w Ais of evolution w Disk Solis of evolution Figue 7 Volume of isk: w Figue 7 If egion in te plne is evolve out line, te esulting soli is soli of evolution, n te line is clle te is of evolution Te simplest suc soli is igt cicul cline o isk, wic is fome evolving ectngle out n is jcent to one sie of te ectngle, s sown in Figue 7 Te volume of suc isk is Volume of isk e of iskwit of isk w wee is te ius of te isk n w is te wit To see ow to use te volume of isk to fin te volume of genel soli of evolution, consie soli of evolution fome evolving te plne egion in Figue 7 out te inicte is To etemine te volume of tis soli, consie epesenttive ectngle in te plne egion Wen tis ectngle is evolve out te is of evolution, it genetes epesenttive isk wose volume is V Appoimting te volume of te soli n suc isks of wit n ius i pouces Volume of soli n i i n i i
SECTION 7 Volume: Te Disk Meto 57 epesenttive ectngle Ais of evolution epesenttive isk Plne egion = = Disk meto Figue 7 Soli of evolution Appoimtion n isks Tis ppoimtion ppes to ecome ette n ette s n So, ou cn efine te volume of te soli s Volume of soli lim n i i Scemticll, te isk meto looks like tis Known Peclculus Fomul Volume of isk V w epesenttive Element V i New Integtion Fomul Soli of evolution V A simil fomul cn e eive if te is of evolution is veticl Te Disk Meto To fin te volume of soli of evolution wit te isk meto, use one of te following, s sown in Figue 75 Hoizontl Ais of evolution Volume V Veticl Ais of evolution Volume V c V = π [()] V = π c [()] NOTE In Figue 75, note tt ou cn etemine te vile of integtion plcing epesenttive ectngle in te plne egion pepenicul to te is of evolution If te wit of te ectngle is, integte wit espect to, n if te wit of te ectngle is, integte wit espect to () Hoizontl is of evolution Figue 75 c () Veticl is of evolution
58 CHAPTE 7 Applictions of Integtion Te simplest ppliction of te isk meto involves plne egion oune te gp of f n te -is If te is of evolution is te -is, te ius is simpl f EXAMPLE Using te Disk Meto f() = sin Fin te volume of te soli fome evolving te egion oune te gp of f sin n te -is out te -is π Plne egion π () Solution Fom te epesenttive ectngle in te uppe gp in Figue 76, ou cn see tt te ius of tis soli is f sin Figue 76 Soli of evolution π So, te volume of te soli of evolution is V cos sin sin Appl isk meto Simplif Integte T It Eplotion A Eplotion B EXAMPLE evolving Aout Line Tt Is Not Coointe Ais Plne egion f() = g() = Fin te volume of te soli fome evolving te egion oune f () n g out te line, s sown in Figue 77 Soli of evolution Ais of evolution Figue 77 f() g() Solution B equting f n g, ou cn etemine tt te two gps intesect wen ± To fin te ius, sutct g fom f f g Finll, integte etween n to fin te volume V 5 6 5 5 Appl isk meto Simplif Integte T It Eplotion A
SECTION 7 Volume: Te Disk Meto 59 w Te Wse Meto Ais of evolution Te isk meto cn e etene to cove solis of evolution wit oles eplcing te epesenttive isk wit epesenttive wse Te wse is fome evolving ectngle out n is, s sown in Figue 78 If n e te inne n oute ii of te wse n w is te wit of te wse, te volume is given Volume of wse w Disk w To see ow tis concept cn e use to fin te volume of soli of evolution, consie egion oune n oute ius n n inne ius, s sown in Figue 79 If te egion is evolve out its is of evolution, te volume of te esulting soli is given V Wse meto Figue 78 Soli of evolution Note tt te integl involving te inne ius epesents te volume of te ole n is sutcte fom te integl involving te oute ius () () Plne egion Soli of evolution wit ole = (, ) Figue 79 = (, ) = = Plne egion EXAMPLE Using te Wse Meto Fin te volume of te soli fome evolving te egion oune te gps of n out te -is, s sown in Figue 7 Solution In Figue 7, ou cn see tt te oute n inne ii e s follows Oute ius Inne ius Soli of evolution Figue 7 Soli of evolution Integting etween n pouces V 5 5 Appl wse meto Simplif Integte T It Eplotion A
6 CHAPTE 7 Applictions of Integtion In ec emple so f, te is of evolution s een oizontl n ou ve integte wit espect to In te net emple, te is of evolution is veticl n ou integte wit espect to In tis emple, ou nee two septe integls to compute te volume EXAMPLE Integting wit espect to, Two-Integl Cse Fin te volume of te soli fome evolving te egion oune te gps of,,, n out te -is, s sown in Figue 7 Fo : = = (, ) Soli of evolution Fo : = = Figue 7 Plne egion Solution Fo te egion sown in Figue 7, te oute ius is simpl Tee is, oweve, no convenient fomul tt epesents te inne ius Wen,, ut wen, is etemine te eqution, wic implies tt,, Using tis efinition of te inne ius, ou cn use two integls to fin te volume V Appl wse meto Simplif Integte Note tt te fist integl epesents te volume of igt cicul cline of ius n eigt Tis potion of te volume coul ve een etemine witout using clculus T It Eplotion A Eplotion B Figue 7 Genete Mtemtic TECHNOLOGY Some gping utilities ve te cpilit to genete (o ve uilt-in softwe cple of geneting) soli of evolution If ou ve ccess to suc utilit, use it to gp some of te solis of evolution escie in tis section Fo instnce, te soli in Emple migt ppe like tt sown in Figue 7
SECTION 7 Volume: Te Disk Meto 6 EXAMPLE 5 Mnufctuing () () = 5 = 5 () = 5 5 Plne egion () Figue 7 Soli of evolution = in 5 5 in A mnufctue ills ole toug te cente of metl spee of ius 5 inces, s sown in Figue 7() Te ole s ius of inces Wt is te volume of te esulting metl ing? Solution You cn imgine te ing to e genete segment of te cicle wose eqution is 5, s sown in Figue 7() Becuse te ius of te ole is inces, ou cn let n solve te eqution 5 to etemine tt te limits of integtion e ± So, te inne n oute ii e n 5 n te volume is given V 6 Solis wit Known Coss Sections 56 cuic inces Wit te isk meto, ou cn fin te volume of soli ving cicul coss section wose e is A Tis meto cn e genelize to solis of n spe, s long s ou know fomul fo te e of n it coss section Some common coss sections e sques, ectngles, tingles, semicicles, n tpezois 5 6 T It Eplotion A Open Eplotion Volumes of Solis wit Known Coss Sections Fo coss sections of e A tken pepenicul to te -is, Volume A See Figue 7() Fo coss sections of e A tken pepenicul to te -is, Volume A See Figue 7() c = = = c () Coss sections pepenicul to -is Figue 7 = () Coss sections pepenicul to -is
6 CHAPTE 7 Applictions of Integtion EXAMPLE 6 Tingul Coss Sections Fin te volume of te soli sown in Figue 75 Te se of te soli is te egion oune te lines Coss sections e equiltel tingles = g() Figue 76 f() = g() = + Tingul se in -plne Figue 75 Ae = A() Ae of se = B = = f() = ( ) n Te coss sections pepenicul to te -is e equiltel tingles Solution f, EXAMPLE 7 Te se n e of ec tingul coss section e s follows Bse Ae se A Becuse nges fom to, te volume of te soli is V A T It An Appliction to Geomet Lengt of se Ae of equiltel tingle Ae of coss section Pove tt te volume of pmi wit sque se is V B, wee is te eigt of te pmi n B is te e of te se Solution As sown in Figue 76, ou cn intesect te pmi wit plne pllel to te se t eigt to fom sque coss section wose sies e of lengt Using simil tingles, ou cn sow tt o wee is te lengt of te sies of te se of te pmi So, A Integting etween n pouces V A ) g, B Eplotion A B T It Eplotion A Eplotion B
SECTION 7 Volume: Te Disk Meto 6 Eecises fo Section 7 Te smol Click on Click on inictes n eecise in wic ou e instucte to use gping tecnolog o smolic compute lge sstem to view te complete solution of te eecise to pint n enlge cop of te gp In Eecises 6, set up n evlute te integl tt gives te volume of te soli fome evolving te egion out te -is 9 5, 6 In Eecises 7, set up n evlute te integl tt gives te volume of te soli fome evolving te egion out te -is 9 7 8 6, 5 In Eecises, fin te volume of te soli genete evolving te egion oune te gps of te equtions out te given lines,, () te -is () te -is (c) te line () te line 6,, () te -is () te -is (c) te line 8 () te line, () te -is () te line 6 6, 6 () te -is () te line In Eecises 5 8, fin te volume of te soli genete evolving te egion oune te gps of te equtions out te line 5,, 6,, 7 8,,, sec,, In Eecises 9, fin te volume of te soli genete evolving te egion oune te gps of te equtions out te line 6 9,,, 6 6,,,, 6,, 6, 6 In Eecises, fin te volume of te soli genete evolving te egion oune te gps of te equtions out te -is,,,,
6 CHAPTE 7 Applictions of Integtion 5 6 7 8 e, e,,,,, 9, 5,,,,, 8 In Eecises n, fin te volume of te soli genete evolving te egion oune te gps of te equtions out te -is,, 9,,, In Eecises 6, fin te volume of te soli genete evolving te egion oune te gps of te equtions out te -is Veif ou esults using te integtion cpilities of gping utilit,,,,,, 8 sin,,, cos,,, 5 e,,, 6 e e,,, In Eecises 7, use te integtion cpilities of gping utilit to ppoimte te volume of te soli genete evolving te egion oune te gps of te equtions out te -is 7 8 e, ln,,,,, 9 ctn,,, 5, Witing Aout Concepts In Eecises n, te integl epesents te volume of soli Descie te soli sin Tink Aout It In Eecises n, etemine wic vlue est ppoimtes te volume of te soli genete evolving te egion oune te gps of te equtions out te -is (Mke ou selection on te sis of sketc of te soli n not pefoming n clcultions) e,,, () () 5 (c) () 7 (e) ctn,,, () () (c) 5 () 6 (e) 5 Witing Aout Concepts (continue) 5 A egion oune te pol n te -is is evolve out te -is A secon egion oune te pol n te -is is evolve out te -is Witout integting, ow o te volumes of te two solis compe? Eplin 6 Te egion in te figue is evolve out te inicte es n line Oe te volumes of te esulting solis fom lest to getest Eplin ou esoning () -is () -is (c) 8 8 6 7 If te potion of te line ling in te fist qunt is evolve out te -is, cone is genete Fin te volume of te cone etening fom to 6 8 Use te isk meto to veif tt te volume of igt cicul cone is, wee is te ius of te se n is te eigt 9 Use te isk meto to veif tt te volume of spee is 5 A spee of ius is cut plne < units ove te equto Fin te volume of te soli (speicl segment) ove te plne 5 A cone of eigt H wit se of ius is cut plne pllel to n units ove te se Fin te volume of te soli (fustum of cone) elow te plne 5 Te egion oune,,, n is evolve out te -is () Fin te vlue of in te intevl, tt ivies te soli into two pts of equl volume () Fin te vlues of in te intevl, tt ivie te soli into tee pts of equl volume 5 Volume of Fuel Tnk A tnk on te wing of jet icft is fome evolving te egion oune te gp of 8 n te -is out te -is (see figue), wee n e mesue in metes Fin te tnk s volume 6 8 = 8
SECTION 7 Volume: Te Disk Meto 65 5 Volume of L Glss A glss contine cn e moele evolving te gp of 9, 95, out te -is, wee n e mesue in centimetes Use gping utilit to gp te function n fin te volume of te contine 55 Fin te volume of te soli genete if te uppe lf of te ellipse 9 5 5 is evolve out () te -is to fom polte speoi (spe like footll), n () te -is to fom n olte speoi (spe like lf of cn) Figue fo 55() 56 Minimum Volume Te c of Figue fo 55() on te intevl, is evolve out te line (see figue) () Fin te volume of te esulting soli s function of () Use gping utilit to gp te function in pt (), n use te gp to ppoimte te vlue of tt minimizes te volume of te soli (c) Use clculus to fin te vlue of tt minimizes te volume of te soli, n compe te esult wit te nswe to pt () 6 5 5 < 5 6 58 Moeling Dt A ftsmn is ske to etemine te mount of mteil equie to pouce mcine pt (see figue in fist column) Te imetes of te pt t equll spce points e liste in te tle Te mesuements e liste in centimetes () Use tese t wit Simpson s ule to ppoimte te volume of te pt () Use te egession cpilities of gping utilit to fin fout-egee polnomil toug te points epesenting te ius of te soli Plot te t n gp te moel (c) Use gping utilit to ppoimte te efinite integl ieling te volume of te pt Compe te esult wit te nswe to pt () 59 Tink Aout It Mtc ec integl wit te soli wose volume it epesents, n give te imensions of ec soli () igt cicul cline () Ellipsoi (c) Spee () igt cicul cone (e) Tous (i) (iii) (v) 5 8 7 5 57 6 7 8 9 58 5 9 6 (ii) (iv) 6 Cvliei s Teoem Pove tt if two solis ve equl ltitues n ll plne sections pllel to tei ses n t equl istnces fom tei ses ve equl es, ten te solis ve te sme volume (see figue) = Ae of e of Figue fo 56 Figue fo 58 57 Wte Dept in Tnk A tnk on wte towe is spee of ius 5 feet Detemine te epts of te wte wen te tnk is fille to one-fout n tee-fouts of its totl cpcit (Note: Use te zeo o oot fetue of gping utilit fte evluting te efinite integl) 6 Fin te volume of te soli wose se is oune te gps of n, wit te inicte coss sections tken pepenicul to te -is () Sques () ectngles of eigt
66 CHAPTE 7 Applictions of Integtion 6 Fin te volume of te soli wose se is oune te cicle wit te inicte coss sections tken pepenicul to te -is () Sques () Equiltel tingles In Eecises 67 7, fin te volume genete otting te given egion out te specifie line 5 = = 5 (c) Semicicles () Isosceles igt tingles 67 out 68 out 69 out 7 out 7 out 7 out 7 out 7 out 75 Te soli sown in te figue s coss sections oune te gp of, wee () Descie te coss section wen n () Descie poceue fo ppoimting te volume of te soli 6 Te se of soli is oune,, n Fin te volume of te soli fo ec of te following coss sections (tken pepenicul to te -is): () sques, () semicicles, (c) equiltel tingles, n () semiellipses wose eigts e twice te lengts of tei ses 6 Fin te volume of te soli of intesection (te soli common to ot) of te two igt cicul clines of ius wose es meet t igt ngles (see figue) Two intesecting clines Soli of intesection FO FUTHE INFOMATION Fo moe infomtion on tis polem, see te ticle Estimting te Volumes of Soli Figues wit Cuve Sufces Donl Coen in Mtemtics Tece MtAticle 65 A mnufctue ills ole toug te cente of metl spee of ius Te ole s ius Fin te volume of te esulting ing 66 Fo te metl spee in Eecise 65, let 5 Wt vlue of will pouce ing wose volume is ectl lf te volume of te spee? + = + = + = 76 Two plnes cut igt cicul cline to fom wege One plne is pepenicul to te is of te cline n te secon mkes n ngle of egees wit te fist (see figue) () Fin te volume of te wege if () Fin te volume of te wege fo n it ngle Assuming tt te cline s sufficient lengt, ow oes te volume of te wege cnge s inceses fom to 9? Figue fo 76 Figue fo 77 77 () Sow tt te volume of te tous sown is given te integl, wee > > 8 () Fin te volume of te tous θ 5