Geometric quantities for polar curves


 Harriet Jones
 3 years ago
 Views:
Transcription
1 Roerto s Notes on Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10 Geometric quntities for polr curves Wht you need to know lredy: How to use integrls to compute res nd lengths of regions ounded y regulr nd prmetric curves Wht you cn lern here: How to use integrls to compute the sme quntities for region ounded y one or more polr curves In the pplictions we hve seen so fr, we hve used slices tht were verticl or horizontl Notice tht this mkes full use of the Crtesin coordintes we normlly use But wht if we re given informtion in polr coordintes? Polr coordintes use rdiclly different pproch to identify Therefore it should not surprise you tht the method needed to compute res of regions ounded y polr curves is sustntil vrition of those we hve used so fr To develop such method, we need to use, once gin, the four step process to construct n integrl, ut in novel wy Since in polr coordintes the independent vrile trces the polr curve rdilly, y rotting round the pole counter clockwise, let us ssume tht the region of interest is ounded y polr curve r r nd two rdii t nd, nd s shown here Let us now slice this region like piece of pie centered t the pole, through severl more rdii In this wy we cn pproximte the re of ech slice y using circulr sector whose ngle is smll nd whose rdius is the distnce from the pole to the curve t some point within the slice r r r r r r Notice tht ech pproximting slice looks like tringle, ut it isn t! Since we work in polr coordintes, we hve to rotte round the pole, so tht wht you re seeing is circulr sector Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10: Geometric quntities for polr curves Pge 1
2 We know (or should know!) tht the re of circulr sector of rdius r nd 1 ngle is given y r, so we use this formul to pproximte the corresponding re ounded y the curve: 1 Ai r i By dding the res of ll pproximting sectors we conclude tht: n 1 A r i i1 As we did in the rectngulr cse, we cn now go from this pproximtion to n exct vlue y tking the limit s the slicing is done into thinner nd more numerous slices This leds to the following method to e ddressed on cseycse The exmples nd prctice questions provide some illustrtion of how this cn e done nd how to hndle these difficulties Exmple: r 31 cos The grph of this polr curve shows n outer loop nd n inner loop: If Technicl fct r r is polr curve defined y function nd ounds, then the re of this ounded region is tht is continuous for rdillydefined region etween the hlflines nd given y: 1 A r d The more generl cse of region ounded y two polr curves, or y two different sections of the sme polr curve, cn e tckled s we hve done in other situtions, y descriing the region s the difference etween regions of the ove, sic type But tht is where is the key difficulty lies: identifying the curves, the limits of integrtion nd the potentil overlps in n pproprite wy These difficulties need To find the re ounded y the inner loop, we strt y noticing tht the curve crosses the pole when: cos 0 cos, 3 3 This mens tht the loop strts t the first vlue nd ends t the second Since this is exctly the simple sitution, the re is given y: 4 / 3 4 / A 91 cos d 1 4cos 4cos d / 3 / 3 We cn compute this integrl in the usul wy for trig integrls, thus otining: 9 4 A sin 4sin /3 / Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10: Geometric quntities for polr curves Pge
3 Exmple: r 31 cos Wht out the re of the region etween the outer nd inner loops of the sme polr curve? All we need to do is find the re ounded y the outer loop nd sutrct the re of the inner loop from it Let us compute the re of the region inside oth curves For the outer loop, we cn use symmetry, thus looking t twice the re of the top prt, or simply follow the rottion We see tht such loop strts t nd ends t 3 loop is: Aouter 3 1 Therefore the re ounded y the outer /3 /3 9 1cos d This cn e computed s efore nd then we need to sutrct the inner loop, leding to: Aouter Aouter Ainner /3 4 / cos d 91 cos d /3 /3 This region is symmetric, so we cn consider only the portion ove the x xis Exmple: 1 cos, r cos r By grphing these curves we cn see severl regions Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10: Geometric quntities for polr curves Pge 3
4 Moreover, y rotting counter clockwise, we notice tht this region consists of two smller regions: Technicl fct r1 cos r cos If r r, is finite polr curve, then its length is given y: dr L r d d In the first qudrnt, for 0, we use only the second curve, while in the second qudrnt, for, we use only the first curve Therefore the desired formul is: / 1 1 A cos d cos d 0 / / cos d cos d 0 / This is sic integrl whose computtion I leve to you Tht looks confusing! I m frid to sk wht hppens when we tckle lengths, surfce res nd volumes Well, cheer up: lthough surfce res nd volumes cn e tckled when deling with polr curves, they led to very messy formule tht re eyond the scope of our curse I ment course! So, we re left with rc length, which turns out to e surprisingly simple Proof We strt y writing the polr curve in prmetric form: x r cos x ' r 'cos r sin y r sin y ' r 'sin r cos Now we use the formul for the prmetric rc length: ' ' L x y d 'cos sin 'sin cos r r r r d By expnding the two squres, the two doule products cncel ech other nd the remining terms cn e grouped s follows: sin cos ' cos sin L r r d But wht we hve in rckets now is one side of the sic Pythgoren identity, whose other side is 1 Therefore, we cn write the formul s: Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10: Geometric quntities for polr curves Pge 4
5 s climed ' L r r d L / 0 sin 4cos d You my wnt to try nd compute this integrl Once gin, we get tough one, ut despir not: method for hndling these difficult definite integrls is coming Exmple: r sin Just s with this exmple, other exmples tend to e eqully chllenging, so the issue will e only to set up the integrl, which oils down to identifying the limits of integrtion The length of one petl of this rose is given y: Summry The formul for the re of region ounded y polr curve is otined y using the four step process to construct integrls nd is sed on the re of circulr sector, rther thn rectngle The min difficulty in setting up the required integrls is in identifying the polr region, or sometimes regions, involved The formul for rc length of polr curve is otined lgericlly from the prmetric version Generl formule for surfce res nd volumes tend to e complicted enough to e deferred to lter course Common errors to void Don t rush over the identifiction of the polr region: it is the key element nd it cn e tricky! Rememer tht you re deling with polr curves, so don t trce the curve left to right, ut counter clockwise Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10: Geometric quntities for polr curves Pge 5
6 Lerning questions for Section I 510 Review questions: 1 Explin how to set up the integrl representing the re of region ounded y polr curves Explin how to set up the integrl representing the length of polr curve Memory questions: 1 Wht geometricl shpes re used to construct the re formul in polr coordintes? Wht is the formul for the re ounded y polr curve of the form r f ( ),? 3 Wht is the integrl formul for the rc length of the grph of polr function r r? 4 When we nlyze polr region to identify the integrtion limits, how do we scn the region? Computtion questions: In questions 116, set up the integrl tht provides the re of the region descried there If possile, compute the integrl 1 The region ounded y the first loop of the polr curve polr xis r / e nd the The region elow the polr xis nd contined etween the two loops of the curve r 6sin 5 The region enclosed y one loop of the curve r 1 sin3 6 The region common to the circles r sin nd r cos 7 The figure8 region ounded y oth r 1 cos nd r 1 cos 3 The region ounded y y x nd r 3 sin 8 The finite region ounded y the loop in the conchoid r 4 sec 9 The region of intersection etween the two circles r sin nd r 1 4 The region inside the circle r 6cos nd outside the crdioid r cos 10 The finite region tht is inside oth the rose r sin 4 nd the circle r 1 Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10: Geometric quntities for polr curves Pge 6
7 11 The region inside oth r sin nd r 3 sin 14 One petl of the rose r 4cos3 1 The i determined y the polr curve r 1 sin 15 The region inside r 6cos nd outside r cos 13 The region ounded y the outermost lyer of the cochleoid shown here: r sin 16 The region outside r 6cos nd inside r cos nd the region common to them In questions 171,set up the integrl tht provides the length of the polr curve descried If possile, compute the integrl 17 The first loop of the spirl r / e 0 The limçon r 1 sin 18 One loop of the rose r cos3 1 r 3 sin 19 The rit ers r sin( cos ) Which function of n represents the length of the first n loops of the spirl r? I expect you to evlute ny integrls tht my e needed for the construction of such function Theory questions: 1 In the formul for the re ounded y polr curve, is it importnt tht the limits e used in incresing order? Is it possile for two polr regions to intersect t point tht does NOT corresponds to the sme vlue of for oth curves? Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10: Geometric quntities for polr curves Pge 7
8 3 Given the polr rc length formul, wht would e the formul for the surfce re otined y rotting polr curve round the x xis? 5 Which formul is used to simplify the expression in the formul for the rc length of polr curve? 4 In the formul for the re ounded y polr curve, which vlue goes in the ottom of the integrl nd which on top? Wht questions do you hve for your instructor? Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10: Geometric quntities for polr curves Pge 8
10.4 AREAS AND LENGTHS IN POLAR COORDINATES
65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the
More informationPolar Coordinates. July 30, 2014
Polr Coordintes July 3, 4 Sometimes it is more helpful to look t point in the xyplne not in terms of how fr it is horizontlly nd verticlly (this would men looking t the Crtesin, or rectngulr, coordintes
More informationSection 10.2 Graphing Polar Equations
Section 10.2 Grphing Polr Equtions OBJECTIVE 1: Sketching Equtions of the Form rcos, rsin, r cos r sin c nd Grphs of Polr Equtions of the Form rcos, rsin, r cos r sin c, nd where,, nd c re constnts. The
More information9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations
54 CHAPTER 9 PARAMETRIC EQUATINS AND PLAR CRDINATES 49. r, 5. r sin 3, 5 54 Find the points on the given curve where the tngent line is horizontl or verticl. 5. r 3 cos 5. r e 53. r cos 54. r sin 55. Show
More informationVocabulary Check. Section 10.8 Graphs of Polar Equations not collinear The points are collinear.
Section.8 Grphs of Polr Equtions 98 9. Points:,,,,.,... The points re colliner. 9. Points:.,,.,,.,... not colliner. Section.8 Grphs of Polr Equtions When grphing polr equtions:. Test for symmetry. () )
More informationPolar coordinates 5C. 1 a. a 4. π = 0 (0) is a circle centre, 0. and radius. The area of the semicircle is π =. π a
Polr coordintes 5C r cos Are cos d (cos + ) sin + () + 8 cos cos r cos is circle centre, nd rdius. The re of the semicircle is. 8 Person Eduction Ltd 8. Copying permitted for purchsing institution only.
More informationSection 16.3 Double Integrals over General Regions
Section 6.3 Double Integrls over Generl egions Not ever region is rectngle In the lst two sections we considered the problem of integrting function of two vribles over rectngle. This sitution however is
More informationTranslate and Classify Conic Sections
TEKS 9.6 A.5.A, A.5.B, A.5.D, A.5.E Trnslte nd Clssif Conic Sections Before You grphed nd wrote equtions of conic sections. Now You will trnslte conic sections. Wh? So ou cn model motion, s in E. 49. Ke
More informationExample. Check that the Jacobian of the transformation to spherical coordinates is
lss, given on Feb 3, 2, for Mth 3, Winter 2 Recll tht the fctor which ppers in chnge of vrible formul when integrting is the Jcobin, which is the determinnt of mtrix of first order prtil derivtives. Exmple.
More informationTriangles and parallelograms of equal area in an ellipse
1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA uonp0@studentsrownedu osler@rownedu Introduction In the pper
More informationSOLVING TRIANGLES USING THE SINE AND COSINE RULES
Mthemtics Revision Guides  Solving Generl Tringles  Sine nd Cosine Rules Pge 1 of 17 M.K. HOME TUITION Mthemtics Revision Guides Level: GCSE Higher Tier SOLVING TRIANGLES USING THE SINE AND COSINE RULES
More informationSection 17.2: Line Integrals. 1 Objectives. 2 Assignments. 3 Maple Commands. 1. Compute line integrals in IR 2 and IR Read Section 17.
Section 7.: Line Integrls Objectives. ompute line integrls in IR nd IR 3. Assignments. Red Section 7.. Problems:,5,9,,3,7,,4 3. hllenge: 6,3,37 4. Red Section 7.3 3 Mple ommnds Mple cn ctully evlute line
More informationMath Circles Finite Automata Question Sheet 3 (Solutions)
Mth Circles Finite Automt Question Sheet 3 (Solutions) Nickols Rollick nrollick@uwterloo.c Novemer 2, 28 Note: These solutions my give you the nswers to ll the prolems, ut they usully won t tell you how
More informationLecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.
Lecture 2 Intro to line integrls Dn Nichols nichols@mth.umss.edu MATH 233, Spring 218 University of Msschusetts April 12, 218 (2) onservtive vector fields We wnt to determine if F P (x, y), Q(x, y) is
More informationKirchhoff s Rules. Kirchhoff s Laws. Kirchhoff s Rules. Kirchhoff s Laws. Practice. Understanding SPH4UW. Kirchhoff s Voltage Rule (KVR):
SPH4UW Kirchhoff s ules Kirchhoff s oltge ule (K): Sum of voltge drops round loop is zero. Kirchhoff s Lws Kirchhoff s Current ule (KC): Current going in equls current coming out. Kirchhoff s ules etween
More informationc The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?
3 7. 7.2 Trigonometry in three dimensions Questions re trgeted t the grdes indicted The digrm shows the ck of truck used to crry scffold poles. L K G m J F C 0.8 m H E 3 m D 6.5 m Use Pythgors Theorem
More informationAlgebra Practice. Dr. Barbara Sandall, Ed.D., and Travis Olson, M.S.
By Dr. Brr Sndll, Ed.D., Dr. Melfried Olson, Ed.D., nd Trvis Olson, M.S. COPYRIGHT 2006 Mrk Twin Medi, Inc. ISBN 9781580377546 Printing No. 404042EB Mrk Twin Medi, Inc., Pulishers Distriuted y CrsonDellos
More informationStudent Book SERIES. Patterns and Algebra. Name
E Student Book 3 + 7 5 + 5 Nme Contents Series E Topic Ptterns nd functions (pp. ) identifying nd creting ptterns skip counting completing nd descriing ptterns predicting repeting ptterns predicting growing
More information& Y Connected resistors, Light emitting diode.
& Y Connected resistors, Light emitting diode. Experiment # 02 Ojectives: To get some hndson experience with the physicl instruments. To investigte the equivlent resistors, nd Y connected resistors, nd
More informationREVIEW, pages
REVIEW, pges 510 515 6.1 1. Point P(10, 4) is on the terminl rm of n ngle u in stndrd position. ) Determine the distnce of P from the origin. The distnce of P from the origin is r. r x 2 y 2 Substitute:
More information1 tray of toffee 1 bar of toffee. 10 In the decimal number, 0 7, the 7 refers to 7 tenths or
Chpter 3 Deciml Numers Do you know wht DECIMAL is? In chpter, we delt with units, s, 0 s nd 00 s. When you tke single unit nd divide it into (or 0 or 00) its, wht we then hve re deciml frctions of whole
More informationSection 6.1 Law of Sines. Notes. Oblique Triangles  triangles that have no right angles. A c. A is acute. A is obtuse
Setion 6.1 Lw of Sines Notes. Olique Tringles  tringles tht hve no right ngles h is ute h is otuse Lw of Sines  If is tringle with sides,, nd, then sin = sin = sin or sin = sin = sin The miguous se (SS)
More informationExercise 11. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION
Exercise 11 The Sine Wve EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the notion of sine wve nd how it cn be expressed s phsor rotting round the center of circle. You
More informationFP2 POLAR COORDINATES: PAST QUESTIONS
FP POLAR COORDINATES: PAST QUESTIONS. The curve C hs polr eqution r = cosθ, () Sketch the curve C. () (b) Find the polr coordintes of the points where tngents to C re prllel to the initil line. (6) (c)
More informationCHAPTER 2 LITERATURE STUDY
CHAPTER LITERATURE STUDY. Introduction Multipliction involves two bsic opertions: the genertion of the prtil products nd their ccumultion. Therefore, there re two possible wys to speed up the multipliction:
More informationSpiral Tilings with Ccurves
Spirl Tilings with curves Using ombintorics to Augment Trdition hris K. Plmer 19 North Albny Avenue hicgo, Illinois, 0 chris@shdowfolds.com www.shdowfolds.com Abstrct Spirl tilings used by rtisns through
More informationFirst Round Solutions Grades 4, 5, and 6
First Round Solutions Grdes 4, 5, nd 1) There re four bsic rectngles not mde up of smller ones There re three more rectngles mde up of two smller ones ech, two rectngles mde up of three smller ones ech,
More informationStudent Book SERIES. Fractions. Name
D Student Book Nme Series D Contents Topic Introducing frctions (pp. ) modelling frctions frctions of collection compring nd ordering frctions frction ingo pply Dte completed / / / / / / / / Topic Types
More informationNONCLASSICAL CONSTRUCTIONS II
NONLSSIL ONSTRUTIONS II hristopher Ohrt UL Mthcircle  Nov. 22, 2015 Now we will try ourselves on onceletsteiner constructions. You cn only use n (unmrked) strightedge but you cn ssume tht somewhere
More informationStudy Guide # Vectors in R 2 and R 3. (a) v = a, b, c = a i + b j + c k; vector addition and subtraction geometrically using parallelograms
Study Guide # 1 MA 26100  Fll 2018 1. Vectors in R 2 nd R 3 () v =, b, c = i + b j + c k; vector ddition nd subtrction geometriclly using prllelogrms spnned by u nd v; length or mgnitude of v =, b, c,
More informationb = and their properties: b 1 b 2 b 3 a b is perpendicular to both a and 1 b = x = x 0 + at y = y 0 + bt z = z 0 + ct ; y = y 0 )
***************** Disclimer ***************** This represents very brief outline of most of the topics covered MA261 *************************************************** I. Vectors, Lines nd Plnes 1. Vector
More informationMultibeam antennas in a broadband wireless access system
Multiem ntenns in rodnd wireless ccess system Ulrik Engström, Mrtin Johnsson, nders Derneryd nd jörn Johnnisson ntenn Reserch Center Ericsson Reserch Ericsson SE4 84 Mölndl Sweden Emil: ulrik.engstrom@ericsson.com,
More information(1) Primary Trigonometric Ratios (SOH CAH TOA): Given a right triangle OPQ with acute angle, we have the following trig ratios: ADJ
Tringles nd Trigonometry Prepred y: S diyy Hendrikson Nme: Dte: Suppose we were sked to solve the following tringles: Notie tht eh tringle hs missing informtion, whih inludes side lengths nd ngles. When
More informationSequential Logic (2) Synchronous vs Asynchronous Sequential Circuit. Clock Signal. Synchronous Sequential Circuits. FSM Overview 9/10/12
9//2 Sequentil (2) ENGG5 st Semester, 22 Dr. Hden So Deprtment of Electricl nd Electronic Engineering http://www.eee.hku.hk/~engg5 Snchronous vs Asnchronous Sequentil Circuit This Course snchronous Sequentil
More informationAlgorithms for Memory Hierarchies Lecture 14
Algorithms for emory Hierrchies Lecture 4 Lecturer: Nodri Sitchinv Scribe: ichel Hmnn Prllelism nd Cche Obliviousness The combintion of prllelism nd cche obliviousness is n ongoing topic of reserch, in
More informationSolutions to exercise 1 in ETS052 Computer Communication
Solutions to exercise in TS52 Computer Communiction 23 Septemer, 23 If it occupies millisecond = 3 seconds, then second is occupied y 3 = 3 its = kps. kps If it occupies 2 microseconds = 2 6 seconds, then
More informationVector Calculus. 1 Line Integrals
Vector lculus 1 Line Integrls Mss problem. Find the mss M of very thin wire whose liner density function (the mss per unit length) is known. We model the wire by smooth curve between two points P nd Q
More informationMAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNELSHAPED NODES
MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNELSHAPED NODES Romn V. Tyshchuk Informtion Systems Deprtment, AMI corportion, Donetsk, Ukrine Emil: rt_science@hotmil.com 1 INTRODUCTION During the considertion
More informationMixed CMOS PTL Adders
Anis do XXVI Congresso d SBC WCOMPA l I Workshop de Computção e Aplicções 14 20 de julho de 2006 Cmpo Grnde, MS Mixed CMOS PTL Adders Déor Mott, Reginldo d N. Tvres Engenhri em Sistems Digitis Universidde
More informationAquauno Select MINUTES. (duration) FREQUENCY LED. OFF 8h AQUAUNO SELECT 5 MIN FREQUENCY. the timer is being programmed;
Aquuno Select Pg. INSTALLATION. Attch the timer to cold wter tp, following these simple instructions. Do not instll the timer in pit or vlve ox, elow ground level or indoors. Do not use the timer with
More informationCS 135: Computer Architecture I. Boolean Algebra. Basic Logic Gates
Bsic Logic Gtes : Computer Architecture I Boolen Algebr Instructor: Prof. Bhgi Nrhri Dept. of Computer Science Course URL: www.ses.gwu.edu/~bhgiweb/cs35/ Digitl Logic Circuits We sw how we cn build the
More informationDirect Current Circuits. Chapter Outline Electromotive Force 28.2 Resistors in Series and in Parallel 28.3 Kirchhoff s Rules 28.
P U Z Z L E R If ll these pplinces were operting t one time, circuit reker would proly e tripped, preventing potentilly dngerous sitution. Wht cuses circuit reker to trip when too mny electricl devices
More informationRegular languages can be expressed as regular expressions.
Regulr lnguges cn e expressed s regulr expressions. A generl nondeterministic finite utomton (GNFA) is kind of NFA such tht: There is unique strt stte nd is unique ccept stte. Every pir of nodes re connected
More informationLECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY
LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY 1. Bsic roerties of qudrtic residues We now investigte residues with secil roerties of lgebric tye. Definition 1.1. (i) When (, m) 1 nd
More informationSTUDY GUIDE, CALCULUS III, 2017 SPRING
TUY GUIE, ALULU III, 2017 PING ontents hpter 13. Functions of severl vribles 1 13.1. Plnes nd surfces 2 13.2. Grphs nd level curves 2 13.3. Limit of function of two vribles 2 13.4. Prtil derivtives 2 13.5.
More informationPRO LIGNO Vol. 11 N pp
THE INFLUENCE OF THE TOOL POINT ANGLE AND FEED RATE ON THE DELAMINATION AT DRILLING OF PRELAMINATED PARTICLEBOARD Mihi ISPAS Prof.dr.eng. Trnsilvni University of Brsov Fculty of Wood Engineering Address:
More informationCHAPTER 3 AMPLIFIER DESIGN TECHNIQUES
CHAPTER 3 AMPLIFIER DEIGN TECHNIQUE 3.0 Introduction olidstte microwve mplifiers ply n importnt role in communiction where it hs different pplictions, including low noise, high gin, nd high power mplifiers.
More informationPatterns and Relationships
Series Techer Ptterns nd Reltionships opyright 009 3P Lerning. All rights reserved. First edition printed 009 in Austrli. A ctlogue record for this ook is ville from 3P Lerning Ltd. ISBN 97819186034
More informationDataflow Language Model. DataFlow Models. Applications of Dataflow. Dataflow Languages. Kahn process networks. A Kahn Process (1)
The slides contin revisited mterils from: Peter Mrwedel, TU Dortmund Lothr Thiele, ETH Zurich Frnk Vhid, University of liforni, Riverside Dtflow Lnguge Model Drsticlly different wy of looking t computtion:
More informationMultivariable integration. Multivariable integration. Iterated integration
Multivrible integrtion Multivrible integrtion Integrtion is ment to nswer the question how muh, depending on the problem nd how we set up the integrl we n be finding how muh volume, how muh surfe re, how
More information(1) Nonlinear system
Liner vs. nonliner systems in impednce mesurements I INTRODUCTION Electrochemicl Impednce Spectroscopy (EIS) is n interesting tool devoted to the study of liner systems. However, electrochemicl systems
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Prctice Skills Prctice for Lesson.1 Nme Dte Tiling Bthroom Wll Simplifying Squre Root Expressions Vocbulry Mtch ech definition to its corresponding term. 1. n expression tht involves root. rdicnd
More informationMagnetic monopole field exposed by electrons
Mgnetic monopole field exposed y electrons A. Béché, R. Vn Boxem, G. Vn Tendeloo, nd J. Vereeck EMAT, University of Antwerp, Groenenorgerln 171, 22 Antwerp, Belgium Opticl xis Opticl xis Needle Smple Needle
More informationABOUT THIS MANUAL ABOUT THIS MANUAL
ABOUT THIS MANUAL ABOUT THIS MANUAL This mnul provides detils on IQ Designer, which is ville with the upgrde. Mke sure tht the mchine hs een upgrded to the most recent version. When you find this icon
More informationLecture 16. Double integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.
Leture 16 Double integrls Dn Nihols nihols@mth.umss.edu MATH 233, Spring 218 University of Msshusetts Mrh 27, 218 (2) iemnn sums for funtions of one vrible Let f(x) on [, b]. We n estimte the re under
More informationMath 116 Calculus II
Mth 6 Clculus II Contents 7 Additionl topics in Integrtion 7. Integrtion by prts..................................... 7.4 Numericl Integrtion.................................... 7 7.5 Improper Integrl......................................
More informationOperation Manual. Addendum. Embroidery Machine. Product Code: 884T13
Emroidery Mchine Opertion Mnul Addendum Product Code: 884T13 Be sure to red this document efore using the mchine. We recommend tht you keep this document nery for future reference. ABOUT THIS MANUAL ABOUT
More informationINTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS
CHAPTER 8 INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: sine of A = sin A = side opposite
More informationSeven Sisters. Visit for video tutorials
Seven Sisters This imge is from www.quiltstudy.org. Plese visit this website for more informtion on Seven Sisters quilt ptterns. Visit www.blocloc.com for video tutorils 1 The Seven Sisters design cn be
More informationChapter 12 Vectors and the Geometry of Space 12.1 Threedimensional Coordinate systems
hpter 12 Vectors nd the Geometry of Spce 12.1 Threedimensionl oordinte systems A. Three dimensionl Rectngulr oordinte Sydstem: The rtesin product where (x, y, z) isclled ordered triple. B. istnce: R 3
More informationMEASURE THE CHARACTERISTIC CURVES RELEVANT TO AN NPN TRANSISTOR
Electricity Electronics Bipolr Trnsistors MEASURE THE HARATERISTI URVES RELEVANT TO AN NPN TRANSISTOR Mesure the input chrcteristic, i.e. the bse current IB s function of the bse emitter voltge UBE. Mesure
More informationAlternatingCurrent Circuits
chpter 33 AlterntingCurrent Circuits 33.1 AC Sources 33.2 esistors in n AC Circuit 33.3 Inductors in n AC Circuit 33.4 Cpcitors in n AC Circuit 33.5 The LC Series Circuit 33.6 Power in n AC Circuit 33.7
More informationNetwork Theorems. Objectives 9.1 INTRODUCTION 9.2 SUPERPOSITION THEOREM
M09_BOYL3605_13_S_C09.indd Pge 359 24/11/14 1:59 PM f403 /204/PH01893/9780133923605_BOYLSTAD/BOYLSTAD_NTRO_CRCUT_ANALYSS13_S_978013... Network Theorems Ojectives Become fmilir with the superposition theorem
More informationDomination and Independence on Square Chessboard
Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 A.A. Omrn Deprtment of Mthemtics, College of Eduction for Pure Science, University of bylon, bylon, Irq pure.hmed.omrn@uobby lon.edu.iq Domintion
More informationElectronic Circuits I  Tutorial 03 Diode Applications I
Electronic Circuits I  Tutoril 03 Diode Applictions I 1 / 9  T & F # Question 1 A diode cn conduct current in two directions with equl ese. F 2 When reversebised, diode idelly ppers s short. F 3 A
More informationCrime Scene Documentation. Crime Scene Documentation. Taking the C.S. What should my notes include. Note Taking 9/26/2013
Crime Scene Documenttion Crime Scene Documenttion Most importnt step in C.S. processing Purpose: permnently record the condition of C.S. & physicl evidence Time consuming Documenter must be orgnized nd
More information13.1 Double Integral over Rectangle. f(x ij,y ij ) i j I <ɛ. f(x, y)da.
CHAPTE 3, MULTIPLE INTEGALS Definition. 3. Double Integrl over ectngle A function f(x, y) is integrble on rectngle [, b] [c, d] if there is number I such tht for ny given ɛ>0thereisδ>0 such tht, fir ny
More informationET 51 EXTERIOR ROOF DRIP SIDE FINISH MOULDING INSTALLATION
EXTERIOR ROOF DRIP SIDE FINISH MOULDING 51 INSTALLATION The procedure descried elow is for the LH side. Use the sme procedure for oth the RH nd LH sides, unless otherwise specified. 1. INSTALL ROOF SIDE
More informationNotes on Spherical Triangles
Notes on Spheril Tringles In order to undertke lultions on the elestil sphere, whether for the purposes of stronomy, nvigtion or designing sundils, some understnding of spheril tringles is essentil. The
More informationDigital Design. Sequential Logic Design  Controllers. Copyright 2007 Frank Vahid
Digitl Design Sequentil Logic Design  Controllers Slides to ccompny the tetook Digitl Design, First Edition, y, John Wiley nd Sons Pulishers, 27. http://www.ddvhid.com Copyright 27 Instructors of courses
More informationCS2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2005
CS2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2005 EXPERIMENT 1 FUNDAMENTALS 1. GOALS : Lern how to develop cr lrm digitl circuit during which the following re introduced : CS2204 l fundmentls, nd
More informationHomework #1 due Monday at 6pm. White drop box in Student Lounge on the second floor of Cory. Tuesday labs cancelled next week
Announcements Homework #1 due Mondy t 6pm White drop ox in Student Lounge on the second floor of Cory Tuesdy ls cncelled next week Attend your other l slot Books on reserve in Bechtel Hmley, 2 nd nd 3
More informationModule 9. DC Machines. Version 2 EE IIT, Kharagpur
Module 9 DC Mchines Version EE IIT, Khrgpur esson 40 osses, Efficiency nd Testing of D.C. Mchines Version EE IIT, Khrgpur Contents 40 osses, efficiency nd testing of D.C. mchines (esson40) 4 40.1 Gols
More informationASSEMBLY INSTRUCTIONS
ASSEMBLY INSTRUCTIONS Multi Line 6 x8 255x193x203cm / 100 1 /2 x76 x80 PolyTex, Inc. PO Box 458 27725 Dnville Avenue Cstle Rock, MN 55010 We Site: www.polytex.com English  69717 Hoy Greenhouse Service
More informationThe Math Learning Center PO Box 12929, Salem, Oregon Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 300Q Model the concept of ddition for sums to 10. (QT N 36) Model the concept of sutrction using numers less thn or equl to 10. (QT N 37) Write ddition
More informationSECOND EDITION STUDENT BOOK GRADE
SECOND EDITION STUDENT BOOK GRADE 5 Bridges in Mthemtics Second Edition Grde 5 Student Book Volumes 1 & 2 The Bridges in Mthemtics Grde 5 pckge consists of: Bridges in Mthemtics Grde 5 Techers Guide Units
More informationTUR DOORS SHOWER DOORS
TUR DOORS SHOWER DOORS INSTALLATION INSTRUCTIONS TUB DOORS: LBTDB6062 SHOWER DOORS: LBSDB4876 LBSDB6076 VERSION: 3.2 PREPARATION FOR INSTALLATION TUB DOORS SHOWER DOORS PREPARATION FOR INSTALLATION READ
More informationmac profile Configuration Guide Adobe Photoshop CS/CC Sawgrass Virtuoso SG400/SG800 Macintosh v
mc profile Mcintosh 10.510.10 Configurtion Guide Adoe Photoshop CS/CC Swgrss Virtuoso SG400/SG800 v20150427 Configurtion Guide  Photoshop CS/CC Swgrss SG400/800 Before proceeding, ensure the correct
More information* * 33/3549A CS 33/3549A. 2Point Latch includes these additional parts. 1Point Latch (LBL) Customer Service. Installation Instructions
*23970726* 23970726 33/3549A CS 33/3549A Instlltion Instructions This instruction covers new instlltion of the 33/3549A conceled verticl device for luminum nd hollow metl doors. RF Also covered is the
More informationProceedings of Meetings on Acoustics
Proceedings of Meetings on Acoustics Volume 19, 2013 http://cousticlsociety.org/ ICA 2013 Montrel Montrel, Cnd 27 June 2013 Signl Processing in Acoustics Session 4SP: Sensor Arry Bemforming nd Its Applictions
More informationITEC2620 Introduction to Data Structures
/5/20 ITEC220 Introdution to Dt Strutures Leture 0 Gme Trees TwoPlyer Gmes Rules for gme define the sttespe Nodes re gme sttes Links re possile moves Build serh tree y rute fore Exmple I Exmple II A Our
More informationMATH 118 PROBLEM SET 6
MATH 118 PROBLEM SET 6 WASEEM LUTFI, GABRIEL MATSON, AND AMY PIRCHER Section 1 #16: Show tht if is qudrtic residue modulo m, nd b 1 (mod m, then b is lso qudrtic residue Then rove tht the roduct of the
More informationMisty. Sudnow Dot Songs
Sudnow Dot Songs isty T The Dot Song is nottionl system tht depicts voiced chords in wy where the nonmusic reder cn find these firly redily. But the Dot Song is not intended be red, not s sight reder
More informationThe Discussion of this exercise covers the following points:
Exercise 4 Bttery Chrging Methods EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the different chrging methods nd chrgecontrol techniques commonly used when chrging NiMI
More informationChapter 2 Literature Review
Chpter 2 Literture Review 2.1 ADDER TOPOLOGIES Mny different dder rchitectures hve een proposed for inry ddition since 1950 s to improve vrious spects of speed, re nd power. Ripple Crry Adder hve the simplest
More informationMacroscopic and Microscopic Springs Procedure
Mrosopi nd Mirosopi Springs Proedure OBJECTIVE Purpose In this l you will: investigte the springlike properties of stright wire, disover the strethiness of mteril, independent of the size nd shpe of n
More informationDiffraction and Interference. 6.1 Diffraction. Diffraction grating. Diffraction grating. Question. Use of a diffraction grating in a spectrometer
irction nd Intererence 6.1 irction irction grting Circulr dirction irction nd intererence re similr phenomen. Intererence is the eect o superposition o 2 coherent wves. irction is the superposition o mny
More informationMake Your Math Super Powered
Mke Your Mth Super Powered: Use Gmes, Chllenges, nd Puzzles Where s the fun? Lern Mth Workshop model by prticipting in one nd explore fun nocost/lowcost gmes nd puzzles tht you cn esily bring into your
More informationThreePhase Synchronous Machines The synchronous machine can be used to operate as: 1. Synchronous motors 2. Synchronous generators (Alternator)
ThreePhse Synchronous Mchines The synchronous mchine cn be used to operte s: 1. Synchronous motors 2. Synchronous genertors (Alterntor) Synchronous genertor is lso referred to s lterntor since it genertes
More informationMETHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN. Inventor: Brian L. Baskin
METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN Inventor: Brin L. Bskin 1 ABSTRACT The present invention encompsses method of loction comprising: using plurlity of signl trnsceivers to receive one or
More informationASY P.O. BOX 729 TERRELL, TEXAS / PAGE 1 OF 13 SAM
203 Madix Inc., ll rights reserved ommon Parts 2 MXI GRI WIRE GRI SHELF WITH (GPWGS) MXI GRI FIXTURE PNEL (GPWFP) FIXTURE PNELS RE USE S EN SUPPORT. SHELF N E USE NYWHERE. MXI GRI REINFORMENT R 3 (GPR)
More informationAnalysis of circuits containing active elements by using modified T  graphs
Anlsis of circuits contining ctive elements using modified T  grphs DALBO BOLEK *) nd EA BOLKOA**) Deprtment of Telecommunictions *) dioelectronics **) Brno Universit of Technolog Purknov 8, 6 Brno CECH
More informationPerformance Comparison between Network Coding in Space and Routing in Space
Performnce omprison etween Network oding in Spce nd Routing in Spce Yunqing Ye, Xin Hung, Ting Wen, Jiqing Hung nd lfred Uwitonze eprtment of lectronics nd Informtion ngineering, Huzhong University of
More informationEnglish Printed in Taiwan XG
COVER 14 C M Y K English Printed in Tiwn XG0091001 Congrtultions on choosing our product! Thnk you very much for purchsing our product. To otin the est performnce from this device nd to ensure sfe nd
More informationEE Controls Lab #2: Implementing StateTransition Logic on a PLC
Objective: EE 44  Controls Lb #2: Implementing Stternsition Logic on PLC ssuming tht speed is not of essence, PLC's cn be used to implement stte trnsition logic. he dvntge of using PLC over using hrdwre
More informationCalculation of OffCore Inductance in DualCircuit Model of Transformer
Clcultion of OffCore Inductnce in DulCircuit Model of Trnsformer As Lotfi NTNU Trondheim, Norwy s.lotfi@ntnu.no Hns Kr. Hoidlen NTNU Trondheim, Norwy hns.hoidlen@ntnu.no Nicol Chies Sttoil Trondheim,
More informationSPECIAL EDITION. Spring 2012 Ezine. where crafty is contagious
Spring 2012 Ezine SPECIAL EDITION www.clubchiccircle.com where crfty is contgious shmrock sttionery It is so esy to mke these festive homemde crds! Mke homemde stmp from the end of n pencil erser. With
More informationSeries. Teacher. Numbers
Series B Techer Copyright 2009 3P Lerning. All rights reserved. First edition printed 2009 in Austrli. A ctlogue record for this book is vilble from 3P Lerning Ltd. ISBN 9781921860171 Ownership of
More informationTheme: Don t get mad. Learn mod.
FERURY When 1 is divided by 5, the reminder is. nother wy to sy this is opyright 015 The Ntionl ouncil of Techers of Mthemtics, Inc. www.nctm.org. ll rights reserved. This mteril my not be copied or distributed
More informationPatterns and Algebra
Student Book Series D Mthletis Instnt Workooks Copyright Series D Contents Topi Ptterns nd funtions identifying nd reting ptterns skip ounting ompleting nd desriing ptterns numer ptterns in tles growing
More information