The Disk Method. 456 CHAPTER 7 Applications of Integration
|
|
- Dulcie Holmes
- 5 years ago
- Views:
Transcription
1 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
2 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
3 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 Appl isk meto Simplif Integte T It Eplotion A
4 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
5 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
6 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
7 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
8 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 , 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,,,,
9 6 CHAPTE 7 Applictions of Integtion 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) 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
10 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 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 () < 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) (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 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
11 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 () Sow tt te volume of te tous sown is given te integl, wee > > 8 () Fin te volume of te tous θ 5
Essential Question How can you find the surface area and the volume of a cone? 3 in. π
11.7 Suface Aeas and Volumes of Cones Essential Question How can you find te suface aea and te volume of a cone? Finding te Suface Aea of a Cone Wok wit a patne. Constuct a cicle wit a adius of 3 inces.
More information5.10. Area and Perimeter INSERT
5.10 Aea and Peimete INSERT A Peimete We egin tis section y eviewing te definition of a polygon, and te definition of peimete. Definition A polygon is a closed geometic figue, wit at least tee sides, in
More informationTotal surface area: the area of the lateral faces combined with the area of both bases
Capte 9: Measuement and te Metic System Section 9.: Volume and Suface Aea Total Suface Aea of a Cylinde and a Pism Lateal aea: te aea of te egions bounded by te lateal faces of a pism Total suface aea:
More informationGeometry Supplement for Math 60 Perimeter, Area, and Volume
Geomety Suppement fo Mat 60 Peimete, Aea, and Voume Geomety comes fom te Geek wods geo meaning eat and meton meaning measue. Today we wi ean about ow to measue cetain featues of two- and tee-dimensiona
More informationPerimeter, Area, and Circumference
51 CHAPTER 9 Geomety B C E A 5x x M x D F To contuct a golden ectangle, one in wic te atio of te lengt to te widt i equal to te atio of te lengt plu te widt to te lengt, egin wit a quae ABCD. Wit te point
More information20.1 Heights and distances
20 Heigts an istances INTROUTION In tis capter, we will learn te practical use of trigonometry in our ay-to-ay life. We will see ow trigonometry is use for fining te eigts an istances of various objects,
More informationLAND COVER CLASSIFICATION USING E-SAR POLARIMETRIC DATA
LAND COVR CLAIFICATION UING -AR POLARIMTRIC DATA V. Ktnssi *, M. Dbboo Lbotoy of Remote ensing, cool of Rul nd uveying ngineeing, Ntionl Tecnicl Univesity of Atens, Heoon Polytecniou 9, Zogpos, 15780,
More informationEssential Question How can you prove the Pythagorean Theorem?
9.1 Te Pytgoren Teorem Essentil Question How n you prove te Pytgoren Teorem? Proving te Pytgoren Teorem witout Words Work wit prtner.. Drw nd ut out rigt tringle wit legs nd, nd ypotenuse.. Mke tree opies
More information16.1 Volume of Prisms and Cylinders
Name Class Date 16.1 Volume of Prisms and Cylinders Essential Question: How do te formulas for te volume of a prism and cylinder relate to area formulas tat you already know? Explore G.11.D Apply te formulas
More informationArea CHAPTER. Overview. The Mathematics. Area 421A
CHAPTER 8 Are Overview In Lessons 8.1, 8., 8.4, nd 8.5, students discover formuls for finding te res of rectngles, prllelogrms, tringles, trpezoids, kites, regulr polygons, nd circles y cutting prt nd
More informationFinding Minimum Values of Distances and Distance Ratios in ATS Spectrum Management
MP93 MITRE PRODUCT Fining Minimm Vales of Distances an Distance Ratios in ATS Spectm Management D. Leone C. Monticone D. Rica E. Snow Fank Bo Decembe 8 Te contents of tis mateial eflect te views of te
More information234 The National Strategies Secondary Mathematics exemplification: Y7
234 Te National Strategies Secondary Matematics exemplification: Y7 Pupils sould learn to: Deduce and use formulae to calculate lengts, perimeters, areas and volumes in 2-D and 3-D sapes As outcomes, Year
More informationMathematics, Music, and Means Math Teacher s Circle Tuesday, October 4, 2016
Wrm-up Mtemtics, Music, nd Mens Mt Tecer s ircle Tuesdy, Octoer 4, 2016 If 0 < xx re te coordintes of points on te rel numer line, solve ec of te followin equtions for xx in terms of nd. 0 xx 1. xx xx
More informationTo find the volume of a pyramid and of a cone
- Volumes of Pyramids and Cones Common Core State Standards G-GMD.A. Use volume formulas for... pyramids, cones... to solve problems. G-MG.A. Use geometric sapes, teir measures, and teir properties to
More informationDesign Aspects of Coils for EDS Levitation Systems
Design Aspects of Coils fo EDS Levittion Sstes M. Andiollo Diptiento di Ingegnei Elettic Univesità degli Studi di Pdov Pdov Itl T. Betoncelli. Cstelli De D. osti Diptiento di Elettotecnic Politecnico di
More informationPhysics Engineering PC 1431 Experiment P2 Heat Engine. Section B: Brief Theory (condensed from Serway & Jewett)
Pysics Engineering PC 1431 Experiment P2 Heat Engine Section A: Introduction Te invention of steam engine played a very significant role in te Industrial Revolution from te late 1700s to early 1800s. Te
More informationMath GPS. 2. Art projects include structures made with straws this week.
Number of Plants Mat GPS. List te measurements in order from greatest to least., inces,5 feet mile 75 yards Greatest. Art projects include structures made wit straws tis week. Number of Projects, p Total
More informationApplications. 38 Looking for Pythagoras. Find the missing length(s).
Applications. A rigt triangle as legs of lengt inces and inces. a. Find te area of a square drawn on te ypotenuse of te triangle. b. Wat is te lengt of te ypotenuse?. Use te Pytagorean Teorem to find te
More information4.2 Using Similar Shapes
LESSON 4.2 Using Similar Sapes Proportionalit 7.5. Generalize te critical attributes of similarit, including ratios witin and between similar sapes. ESSENTIL QUESTION How can ou use similar sapes to find
More informationMath Practice Use a Formula
9.4 Volumes of Prisms How can you find te volume of a prism? ACTIVITY: Pearls in a Treasure Cest Work wit a partner. A treasure cest is filled wit valuable pearls. Eac pearl is about centimeter in diameter
More informationDistributed Forces: Centroids and Centers of Gravity
5 H P T E R Distributed Forces: entroids nd enters of Grvit 219 pter 5 Distributed Forces: entroids nd enters of Grvit 5.1 Introduction 5.2 enter of Grvit of Two- Dimensionl od 5.3 entroids of res nd Lines
More information1 CLEANING THE FILTER.
Dul Select Pg. 1 CLENING THE FILTER. Te tier is protected y wire es filter tt ensures long-lsting nd troule-free opertion. Te filter sould e inspected periodiclly nd wsed t te eginning of every seson.
More informationVolumes of Pyramids. Essential Question How can you find the volume of a pyramid?
11.6 Volumes of Pyramids Essential Question How can you find te volume of a pyramid? Finding te Volume of a Pyramid Work wit a partner. Te pyramid and te prism ave te same eigt and te same square base.
More informationRevision Topic 12: Area and Volume Area of simple shapes
Revision Topic : Area and Volume Area of simple sapes You need to learn ALL of te following area formulae: Rectangle Triangle W L b Area = lengt widt Area = base eigt = ½ b Parallelogram Trapezium a b
More informationOverall stability of multi-span portal sheds at right-angles to the portal spans
Overall stability of multi-span portal seds at rigt-angles to te portal spans SCI s Senior Manager for Standards, Carles M King, explains te approac for design of long-span portal seds. 1. Introduction
More information青藜苑教育 Example : Find te area of te following trapezium. 7cm 4.5cm cm To find te area, you add te parallel sides 7
青藜苑教育 www.tetopedu.com 00-6895997 3095457 Area of simple sapes Revision Topic : Area and Volume You need to learn ALL of te following area formulae: Rectangle Triangle W L Area = lengt widt Area = b base
More information6.2 Pressure driven Microsyringe
. Pessue diven Miosyinge Te deposition system developed t Cento Intediptimentle di ie E. Piggio t te Univesity of Pis onsists of stinless steel syinge wit mion glss pilly needle (fig.) [, ]. Te needle
More informationApplying Trigonometric Functions. ENTERTAINMENT The circus has arrived and the roustabouts must put
5-4 OJETIVE Use trigonometry to find te measures of te sides of rigt triangles. pplying Trigonometric Functions ENTERTINMENT Te circus as arrived and te roustabouts must put up te main tent in a field
More informationKINEMATICS OF TWO SIMPLE PLANETARY GEARS
ARPN Journl of Engineering nd Applied Sciences 2006-2017 Asin Reserc Pulising Network (ARPN). All rigts reserved. www.rpnjournls.com KINEMATICS OF TWO SIMPLE PLANETARY GEARS V. I. Nekrsov 1, O. O. Gorskov
More informationarxiv: v1 [q-bio.pe] 14 Sep 2016
Feuay 7, 8 :4 Poceedings Tim Size: 9in x 6in axiv:69.434v [q-io.pe] 4 Sep 6 A EPIDEIOLOGICAL ODEL OF VIRAL IFECTIOS I A VARROA-IFESTED EE COLOY: THE CASE OF A EE-DEPEDET ITE POPULATIO SIZE SARA ERARDI,
More informationPlanck + Einstein on -Day Fundamental Physical Constants in a Relativistic Perspective and The Design of a Black Hole Gun!
Plank + Einstein on -Day Fundamental Pysial Constants in a Relatiisti Pesetie and Te Design of a Blak Hole Gun! Esen Gaade Haug Nowegian Uniesity of Life Sienes Ma 7, 06 Abstat Tis ae sow ow we an maniulate
More informationp x The revenue function is 5. What is the maximum vertical distance between the line
SETION 4.7 OTIMIZTION ROLEMS 331 and Ris called te revenue function. Te derivative R of te revenue function is called te marginal revenue function and is te rate of cange of revenue wit respect to te numer
More information10. Consider the following problem: A box with an open top is to. 11. A farmer wants to fence an area of 1.5 million square feet in a
8 HTER 4 LITIONS OF DIFFERENTITION 4.7 EXERISES 1. onsider te following prolem: Find two numers wose sum is and wose product is a maximum. (a) Make a tale of values, like te following one, so tat te sum
More informationPassive Cooling of Glazing Surfaces Using Solar Chimneys
Intenational Confeence on Renewable Enegies and Powe Quality (ICREPQ 16 Madid (Spain, 4 t to 6 t May, 2016 Renewable Enegy and Powe Quality Jounal (RE&PQJ ISSN 2172-038 X, No.14 May 2016 Passive Cooling
More informationReview & Summary. Questions
QUESTIONS 405 Review & Summry Density Te ensity r of ny mteril is efine s te mteril s mss per unit volume: r m (14-1) V. Usully, were mteril smple is muc lrger tn tomic imensions, we cn write Eq. 14-1
More informationCOMPARATIVE STUDY OF SPECIFIC HEAT CAPACITIES OF SOME VEGETABLE OILS OBTAINED BY DSC AND MICROWAVE OVEN
Jounal of Temal Analysis and Caloimety, Vol. 79 (2005) 283 287 COMPARATIVE TUDY OF PECIFIC HEAT CAPACITIE OF OME VEGETABLE OIL OBTAINED BY DC AND MICROWAVE OVEN J. C. O. antos 1*, M. G. O. antos 2, J.
More informationDesigning an Optimal Lightning Protection Scheme for Substations Using Shielding Wires
Engineering, Tecnolog & Applied Science Reserc Vol. 7, No. 3, 217, 1595-1599 1595 Designing n Optiml Ligtning Protection Sceme for Substtions Using Sielding Wires Abolfzl Kodddi Electricl Engineering Deprtment
More informationOptimization Model of Oil-Volume Marking with Tilted Oil Tank
Open Journal of Optimization 1 1 - ttp://.doi.org/1.36/ojop.1.1 Publised Online December 1 (ttp://www.scirp.org/journal/ojop) Optimization Model of Oil-olume Marking wit Tilted Oil Tank Wei Xie 1 Xiaojing
More informationBalanced Binary Trees
Balanced Binary Trees 1 Binary searc trees provide O(log N) searc times provided tat te nodes are distributed in a reasonably balanced manner. Unfortunately, tat is not always te case and performing a
More informationOD DVOSTRUKO ZASTAKLJENOG PROZORA DO DVOSTRUKE FASADE INDIKATORI PRENOSA TOPLOTE STACIONARNOG STANJA
OD DVOSTRUKO ZASTAKLJENOG PROZORA DO DVOSTRUKE FASADE INDIKATORI PRENOSA TOPLOTE STACIONARNOG STANJA FROM DOUBLE-GLAZED WINDOW TO DOUBLE-SKIN FACADE STEADY STATE HEAT TRANSFER INDICATORS Gabriel NĂSTASE
More informationCOMPONENTS OF THE MECHANICAL DISPLACEMENTS OF THE PIEZOCERAMIC RING POINTS UDC
FACTA UNIVERITATI eies: Woing nd Living Envionmentl Potetion Vol. 5 N o 8 pp. 9-57 COMPONENT OF THE MECHANICAL IPLACEMENT OF THE PIEZOCERAMIC RING POINT UC 5. Miln Đ. Rdmnović gn. Mnčić Zon M. Petušić
More informationDIRECT AND INDIRECT LOW TEMPERATURE GEOTHERMAL DISTRICT HEATING SYSTEMS
DIRECT AND INDIRECT LOW TEMPERATURE GEOTHERMAL DISTRICT HEATING SYSTEMS Cuansan Dai and Jun Liang Tianjin Geoteal Reseac and Taining Cente, Tianjin Univesity, 30007, Tianjin, P.R.Cina Key Wods: low teeatue
More informationCO-ROTATING FULLY INTERMESHING TWIN-SCREW COMPOUNDING: ADVANCEMENTS FOR IMPROVED PERFORMANCE AND PRODUCTIVITY
CO-ROTATING FULLY INTERMESHING TWIN-SCREW COMPOUNDING: ADVANCEMENTS FOR IMPROVED PERFORMANCE AND PRODUCTIVITY Paul G. Andersen, Coperion Corporation, Ramsey, NJ Frank Lecner, Coperion GmbH, Stuttgart,
More information1/1 FULL SIZE 3/4 QUARTER SIZE 1/2 HALF SIZE EXTRA LARGE SIZE EXTRA LONG SIZE
STERILE CONTAINER SYSTEMS BIO-BARRIER 1/1 FULL SIZE 3/4 QUARTER SIZE 1/2 HALF SIZE EXTRA LARGE SIZE EXTRA LONG SIZE Aygün Bio-Barrier model sterilization container systems are designed wit mecanical valves
More informationEECS 556, 2003, Noll Exam #1: 4
EECS 556 003 o Exa #: 4 3. Fite esig: a. Si tae te oute ouct *. b. Use δ [ δ ]'*[ δ ]. c. Hee we foow te fo of fequec saig escibe i sectio 4.3. of te text. We sae at te x 7 x7 ocatios a ivese DFT to get
More information2011 ORGANIC SOYBEAN VARIETY TRIAL MATERIALS AND METHODS
2011 ORGANIC SOYBEAN VARIETY TRIAL In 2011, the University of Vermont Extension conducted soyben vriety tril in Alburgh, VT. The purpose of this tril ws to provide yield comprisons of food- nd feed-grde
More informationPrediction of steel plate deformation due to triangle heating using the inherent strain method
J Mar Sci Tecnol (005) 10:11 16 DOI 10.1007/s00773-005-00-5 Prediction of steel plate deformation due to triangle eating using te inerent strain metod Cang Doo Jang 1, Tae Hoon Kim, Dae Eun Ko 3, Tomas
More information1/1 FULL SIZE 3/4 QUARTER SIZE 1/2 HALF SIZE EXTRA LARGE SIZE EXTRA LONG SIZE
BIO-BARRIER 1/1 FULL SIZE 3/4 QUARTER SIZE 1/2 HALF SIZE EXTRA LARGE SIZE EXTRA LONG SIZE Aygün Bio-Barrier model sterilization container systems are designed wit mecanical valves bot in bottom and lid
More information2 2D 2F. 1pc for each 20 m of wire. h (min. 45) h (min. 45) 3AC. see details J, E
TEXTILE AIR DIFFUSERS - - INSTRUCTIONS AIR FOR MOUNTING 1 1D 1F see details A, B, C 2 2D 2F see details A, B, C (min. 32) min. every 0 mm 1pc for eac 20 m of wire min. every 0 mm 1pc for eac 20 m of wire
More informationWOOD GRILLS FOR BBS & PTS SERIES BACK BARS MODEL: DZS60
4 2 3 1 WOO GRILLS FOR S & PTS SERIES K RS 1 SINGLE ONTROL GRILL UL ONTROL GRILL SEE SHEETS 2 THRU 4 FOR SSEMLY ETILS PSS THROUGH RER GRILL SEE SHEETS 5 THRU 7 FOR SSEMLY ETILS SEE SHEETS 8 THRU 10 FOR
More informationBribery and Favoritism by Auctioneers in Sealed-Bid Auctions
Bribery an Favoritism by Auctioneers in Seale-Bi Auctions Roberto Burguet an Martin K. Perry Abstract We consier a moel of bribery in an asymmetric procurement auction. In return for a bribe from te isonest
More informationNumerical Simulation of Stresses in Thin-rimmed Spur Gears with Keyway B. Brůžek, E. Leidich
Numerical Simulation of Stresses in Tin-rimmed Spur Gears wit Keyway B. Brůžek, E. Leidic Tis paper contains an investigation of te key on a stress distribution in a tin-rimmed spur gear. A stress analysis
More informationPh.D. Thesis - Abstract
Inveteşte în oameni! FONDUL SOCIAL UROPAN Pogamul Opeaţional Sectoial Dezvoltaea Reuelo Umane 7 1 ng. Maiu-Citian SUCIU P.D. Tei - Atact CONTRIBUTIONS TO TH DVLOPMNT OF A NW ACTIV PARALLL ROBOT FOR MINIMALLY
More informationPostharvest Application of 1-Methylcyclopropene (1-MCP) Extends Shelf Life of Kiwifruit
Posthrvest Appliction of 1-Methylcyclopropene (1-MCP) Extends Shelf Life of Kiwifruit C.M. Cntin 1, D. Holcroft 2 nd C.H. Crisosto 1 1 University of Cliforni, Dvis, Deprtment of Plnt Sciences, Dvis, CA
More informationInfluence of the mass flow ratio water-air on the volumetric mass transfer coefficient in a cooling tower
International Journal of CemTec Researc CODEN (UA): IJCRGG, IN: 974-49, IN(Online):455-9555 Vol.11 No.1, pp 167-173, 18 Influence of te mass flow ratio water-air on te volumetric mass transfer coefficient
More informationFixation effects: do they exist in design problem solving?
Environment and Planning B: Planning and Design, 1993, volume 20, pages 333-345 Fixation effects: do tey exist in design problem solving? A T Purcell, P Williams, J S Gero, B Colbron Department of Arcitectural
More informationRed Green Black Trees: Extension to Red Black Trees
Red Green Black Trees: Extension to Red Black Trees Seyfeddine Zouana*, Djamel Eddine Zegour Laboratoire de la Communication dans les Systèmes Informatiques, Ecole nationale Supérieure d'informatique,
More informationInvestigation of a hydraulic impact. a technology in rock breaking
Investigation of a ydaulic imact a tecnology in ock beaking Matin Genet * Laboatoie de Mécanique et de Tecnologie, 61 av. du Pésident Wilson, 94235 Cacan Cedex, Fance D. Wenyi Yan Senio Lectue Deatment
More informationPoint Pollution Sources Dimensioning
Point Pollution Sources Diensioning Georgeta CUCULEANU 1 ABSTRACT In tis paper a etod for deterining te ain pysical caracteristics of te point pollution sources is presented. It can be used to find te
More informationPaddy McGinty's Goat
2 Paddy McGinty's Goat Taditional olk song a. Leslie Helett 5 q 15 4 J â ä he, sue o goat's milk, I mean to have me ill he ound it as a Bill. 9 A & # # no live all oh They â 1 Mis - te Pa - tick - ty,
More informationStrathprints Institutional Repository
Strtprints Institutionl Repository Fernndez, J. nd Blnco, E. nd Prrondo, J. nd Sticklnd, M.T. nd Scnlon, T.J. (001) Ironwork of Teixois-Trmundi (Asturis) Spin. In: Interntionl Symposium on History of Mcines
More informationStatus, Caste, and the Time Allocation of Women in Rural India
Status, Caste, and te Time Allocation of Women in Rual India By Mukes Esaan, Baat Ramasami, and Wilima Wada * June 9 ABSTRACT We ague tat omen may be disinclined to paticipate in maket ok in te ual aeas
More informationBribery and Favoritism by Auctioneers in Sealed-Bid Auctions
Bribery an Favoritism by Auctioneers in Seale-Bi Auctions Roberto Burguet Institute for Economic Analysis (CSIC) an CREA Campus UAB Bellaterra, Barcelona, Spain 08193 Martin K. Perry * Department of Economics
More informationANALYSIS OF WORK ROLL THERMAL BEHAVIOR FOR 1450MM HOT STRIP MILL WITH GENETIC ALGORITHM
Journal of Teoretical and Applied Information Tecnology 3 t September 2. Vol. 43 No.2 5-2 JATIT & LLS. All rigts reserved. ANALYSIS OF WORK ROLL THERMAL BEHAVIOR FOR 45MM HOT STRIP MILL WITH GENETIC ALGORITHM
More informationGoal: Measure the pump curve(s)
Pump Performance Tes;ng Goal: Measure te pump curve(s) Head versus flow rate: Efficiency versus flow rate: Maximum ead at = 0 Maximum flow rate at = 0 η η = 0 wen = 0 and = 0 Maximum efficiency Pump tes;ng
More informationStudy of Steam Export Transients in a Combined Cycle Power Plant
Study of Steam Export Transients in a Combined Cycle ower lant Alfonso Junquera Delgado Departamento Mecánico, Empresarios Agrupados c\ Magallanes 3 Madrid 8003 ajd@empre.es Almudena Travesí de los Santos
More informationQuality and pricing of delivery services in the e-commerce sector 1
Quality and icing of delivey sevices in te e-commece secto Claie Bosenbege 2, Helmut Ceme 3, Pilie De Donde 4 and Denis Joam 5 INTRODUCTION In most uoean counties, te national univesal ostal sevice ovides
More informationAnalysing the energy consumption of air handling units by Hungarian and international methods
Analysing te energy consumption of air andling units by Hungarian and international metods László Kajtár 1, Miklós Kassai 2,* 1,2 Budapest University of Tecnology and Economics (BUTE), Budapest, Hungary
More informationEugene O'Neill
Euge O'Neill 1888-1953 te ead squ for Buo ull mo i n te trades: Te old ooke I lay te bow sprit facing tern wit te water foam me: Te mts wit every sail wite in te mo ligt towering ig above me: I became
More informationBIOLOGICALLY INSPIRED MULTIFUNCTIONAL COMPOSITE PANEL WITH INTEGRATED CIRCULATORY SYSTEM FOR THERMAL CONTROL
BIOLOGICALLY INSPIRED MULTIFUNCTIONAL COMPOSITE PANEL WITH INTEGRATED CIRCULATORY SYSTEM FOR THERMAL CONTROL A. D. Williams, M. E. Lyall, L. E. Underwood, and B. J. Arritt Air Force Researc Laboratory,
More informationAn experimental study on the design method of a real-sized Mobile Bridge for a moving vehicle
Mobile and Rapidly ssembled Structures I 93 n experimental study on te design metod of a real-sized Mobile ridge for a moving veicle Y. ikairo, I. rio, M. Nakazawa, S. Ono 3, J. olnicki-szulc 4, P. Pawlowski
More informationCalculation of Theoretical Torque and Displacement in an Internal Gear Pump
TECHNICAL REPORT Calculation of Teoretical Torque and Displacement in an Internal Gear Pump Y. INAGUMA Tis paper describes numerical determination of teoretical torque (ideal torque) and teoretical stroke
More informationRussell James Department of Scientific and Industrial Research Taupo-ldairakei, New Zealand
MEASUREMENT OF STEAM-TJATER FLOWS FOR THE TOTAL FLOW TURBIlJE Russell James Department of Scientific and Industrial Researc Taupo-ldairakei, New Zealand Hot water geotermal fields discarge steam-water
More informationCalculation Methodology of Translucent Construction Elements in Buildings and Other Structures
MATEC Web of Conferences 96, 005 (08) ttps://doi.org/0.05/matecconf/0896005 XXVII R-S-P Seminar 08, Teoretical Foundation of Civil Engineering Calculation Metodology of Translucent Construction Elements
More informationMeasured Adiabatic Effectiveness and Heat Transfer for Blowing From the Tip of a Turbine Blade
See discussions, stats, and autor profiles for tis publication at: ttps://www.researcgate.net/publication/4554895 Measured Adiabatic Effectiveness and Heat Transfer for Blowing From te Tip of a Turbine
More informationReflections on the drinking bowl 'Balance'
Supplement to Fun wit oueold object and centre of ma, originally publied in te German journal Pyik in unerer Zeit, 9-96 eflection on te drinking bowl Balance I te bowl made of maive tainle teel? No: ma
More informationP/N: U77-3GC1638-X071 SHOWN
EN/E-R ESRIPTION TE PROPOSL UG, PPROVE MTERIL (RoHS OMPLINT): GE: OPPER LLOY LIGHT PIPE KIT (P/N: U--00P) (IF ORER SEPRTELY) PLTING : 00 u" [.um] MIN. NIKEL PLTING : 00 u" [.um] MIN. MTTE TIN LIGHT PIPE:
More informationInstallation the DELTABEAM Frame
Tese installation instructions are intended to be used togeter wit te project s erection metod statement were te instructions may be complemented. If tere are differences between te erection metod statement
More informationFIRST COMPARISON OF REMOTE CESIUM FOUNTAINS
2001 EEE nternational Frequency Control Symposium and PDA Exibition FRST COMPARSON OF REMOTE CESUM FOUNTANS T. Parker*, P. Hetzel', S. Jefferts*, S. Weyers', L. Nelson", A. Bauc', J. Levine* *National
More informationS^STT*" NEGOCIATIONS AU TITRE DE L'ARTICLE XXVIII:k LISTE XXIV - FINLANDE. Addendum
Voir dans cette série le volume relié de ^JMS-JJAJ concernant ce document. /rdf DECRET ACCORD GENERAL SUR LES TARIFS DOUANIERS ET LE COMMERCE No. S^STT*" Original: anglais NEGOCIATIONS AU TITRE DE L'ARTICLE
More informationGas Flow into Rotary Valve Intake and Exhaust Mechanism in Internal Combustion Engine
World Academy of Science, Engineering and Tecnology Vol:7, No:4, 13 Gas Flow into Rotary Valve Intake and Exaust Mecanism in Internal Combustion Engine R. Usubamatov, Z. A. Rasid International Science
More informationMECHANISMS OF ENVIRONMENTAL INCENTIVE REGULATION: WHY ECOLOGICAL POLICIES IN TRANSITION AND DEVELOPING COUNTRIES ARE NOT EFFECTIVE?
MECHANISMS OF ENIRONMENTAL INCENTIE REGULATION: HY ECOLOGICAL POLICIES IN TRANSITION AND DEELOPING COUNTRIES ARE NOT EFFECTIE? Autor laimir D Mateenko National Reearc Unierity Higer Scool of Economic Ruia
More informationDeveloping a building damage function using SAR images and post-event data after the Typhoon Haiyan in The Philippines
Developing a building damage function using SAR images and post-event data after te Typoon Haiyan in Te Pilippines Bruno ADRIANO 1, Erick MAS 2 and Sunici KOSHIMURA 3 1 Member of JSCE, Graduate Student,
More informationConcentric Dual-tubing Steam Injection: A New Model for Predicting Steam Pressure in the Annulus
Jounal of Industial Intelligent Infomation Vol., No. 4, ecembe 014 Concentic ual-g Steam Injection: A New Model fo Pedictg Steam Pessue te Hao Gu, Lsong Ceng, Sijun Huang, Suang Ai, Saolei Wei epatment
More informationA snapshot of SO 2 concentrations in Australian wines
A snapshot of SO 2 concentations in Austalian ines Sulfu dioxide (SO 2 ) is a pesevative idely used in foods and beveages. In inemaking, SO 2 is a vey common additive, due to its antioxidant and antimicobial
More informationQuestions. denotes answer available in Student Solutions Manual/Study Guide; O denotes objective question
Questions 407 Questions denotes answer available in Student Solutions Manual/Study Guide; O denotes objective question 1. O Figure Q14.1 sows aerial views from directly above two dams. Bot dams are equally
More informationBundling in Vertically Differentiated Communication Markets
unding in Verticy Differentited Communiction Mrkets Tierno Dio strct We ook t te cometition nd te wefre effects of unding in te context of verticy differentited communiction services Teevision, Teeone
More informationEMPIRICAL MATHEMATICAL MODELS OF THE DEPENDENCE OF THE SPECIFIC CUTTING FORCE ON THICKNESS OF CUT IN TURNING
. Sten VELCHEV,. Ivn KOLEV, 3. Ksii IVANOV EMPIRICAL MATHEMATICAL MODELS OF THE DEPENDENCE OF THE SPECIFIC CUTTING FORCE ON THICKNESS OF CUT IN TURNING -3. DEPARTMENT OF MACHINE TOOLS & MANUFACTURING,
More information2. The differential pressure across control valves must not vary too much
2. Te differential pressre across control valves mst not vary too mc Common roblems roblems, typical indicating tat condition nmber two is not met: - Continos oscillation of room temperatre. - Room temperatres
More informationRegression Analysis and Indoor Air Temperature Model of Greenhouse in Northern Dry and Cold Regions
Regression Anlysis nd Indoor Air Temperture Model of Greenouse in Nortern Dry nd Cold Regions Ting Zo, Heru Xue College of computer & informtion Engineering, Inner Mongoli Agriculturl Uniersity, Hoot,
More informationTHE ANNALS OF "DUNAREA DE JOS" UNIVERSITY OF GALATI FASCICLE III, 2003 ISSN X ELECTROTECHNICS, ELECTRONICS, AUTOMATIC CONTROL, INFORMATICS
FASCICLE III, 003 ISSN -454X ELECTROTECHNICS, ELECTRONICS, AUTOMATIC CONTROL, INFORMATICS A NEW METHOD OF GENE CODING FOR A GENETIC ALGORITHM DESIGNED FOR PARAMETRIC OPTIMIZATION Rau Belea * an Liviu Beliman
More informationEXPERIMENTAL AND NUMERICAL ANALYSIS OF HEAT TRANSFER IN THE CAVITIES OF HOLLOW BLOCKS
EXPERIMENTAL AND NUMERICAL ANALYSIS OF HEAT TRANSFER IN THE CAVITIES OF HOLLOW BLOCKS Pietro Stefanizzi, Antonio Lippolis, Stefania Liuzzi Politecnico i Bari, Via Orabona 4, 715 Bari SUMMARY Given te importance
More informationHigh Directivity Microstrip Coupler Using Variable Width Structure
IOS Jounal of Eletonis and Communiation Engineeing IOS-JECE e-issn: 78-834,p- ISSN: 78-8735.Volume 9, Issue 4, Ve. V Jul - Aug. 04, PP 4-8.iosjounals.og Hig Dietivity Miostip Couple Using Vaiable Widt
More informationGround Improvement Using Preloading with Prefabricated Vertical Drains
DISCUSSION of: Ground Improvement Using Preloading wit Prefabricated Vertical Drains Full Reference: Dar, A.S., Siddique, A., Ameen, S.F., (211). Ground Improvement using Pre-loading wit Prefabricated
More informationVariance Estimation of the Design Effect
JSM 013 - Survey Researc Metods Section Variance Estimation of te Design Effect Alberto Padilla Banco de México Abstract Sample size determination is a crucial part of te planning process of a survey and
More informationCalculation of the Age of the Universe
www.pagtec.com/pysiqe/ Calclation of te Age of te Univese Clade Mecie eng., Apil 9 t, Rev. Octobe 7 t, 5 clade.mecie@gctda.com og is obsevations, Edwin Powell Hbble discoveed in 99 tat te nivese is expanding.
More informationThe coin-through-the-rubber trick: an elastically stabilized invagination
Te oin-toug-te-ubbe tik: an elastially stabilized invagination Fanlong Meng Institute of Teoetial Pysis, Cinese Aademy of Sienes Pone: 86-15001345869 flm@itp.a.n Masao Doi Cente fo Soft Matte Pysis and
More informationThe Violin Bow: Taper, Camber and Flexibility
Te Violin Bow: Taper, Camber and lexibility Colin Goug Scool of Pysics and Astronomy, University of Birmingam, B13 9SN,UK a) (Dated: 11/22/1) An analytic, small-deflection, simplified model of te modern
More informationMode I Concrete Characterization using Rebar Induction Heating
Mode I Concrete Crcteriztion using Rebr Induction Heting Fokw Didier #, F. Ngpgue *, T. Nsongo * M. Mpess #4, T. Tmo Ttietse #4 # GRMS, Deprtment of Civil Engineering, ENSET, University of Doul PO Box
More informationPreview. Chapter 3. Labor Productivity and Comparative Advantage: The Ricardian Model
hpter 3 Lbor roductivity nd omprtive Advntge: The Ricrdin Model review Opportunity costs nd comprtive dvntge roduction possibilities Reltive supply, reltive demnd & reltive prices Trde possibilities nd
More information