10.4 AREAS AND LENGTHS IN POLAR COORDINATES


 Preston Harrison
 5 years ago
 Views:
Transcription
1 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 formul for the re of sector of circle r A r FIGURE = r=f( ) = where, s in Figure, r is the rdius nd is the rdin mesure of the centrl ngle. Formul follows from the fct tht the re of sector is proportionl to its centrl ngle: A r r. (See lso Exercise 35 in Section 7.3.) Let e the region, illustrted in Figure, ounded y the polr curve r f nd y the rys nd, where f is positive continuous function nd where. We divide the intervl, into suintervls with endpoints,,,..., n nd equl width. The rys i then divide into n smller regions with centrl ngle i i. If we choose i* in the ith suintervl i, i, then the re A i of the ith region is pproximted y the re of the sector of circle with centrl ngle nd rdius f i*. (See Figure 3.) Thus from Formul we hve FIGURE A i f i* = FIGURE 3 f( i*) = = i = i Î nd so n pproximtion to the totl re A of is It ppers from Figure 3 tht the pproximtion in () improves s n l. But the sums in () re Riemnn sums for the function t f,so lim n l n i A n i f i* f i* y f d It therefore ppers plusile (nd cn in fct e proved) tht the formul for the re A of the polr region is 3 A y f d Formul 3 is often written s 4 A y r d with the understnding tht r f. Note the similrity etween Formuls nd 4. When we pply Formul 3 or 4, it is helpful to think of the re s eing swept out y rotting ry through tht strts with ngle nd ends with ngle. V EXAMPLE Find the re enclosed y one loop of the fourleved rose r cos. SLUTIN The curve r cos ws sketched in Exmple 8 in Section.3. Notice from Figure 4 tht the region enclosed y the right loop is swept out y ry tht rottes from
2 SECTIN.4 AREAS AND LENGTHS IN PLAR CRDINATES 65 r=cos = π 4 4 to 4 4 A y. Therefore Formul 4 gives 4 r d 4 y y 4 cos d 4 cos d y 4 [ cos 4 d 4 sin 4] 4 8 M =_ π 4 V EXAMPLE Find the re of the region tht lies inside the circle r 3 sin nd outside the crdioid r sin. FIGURE 4 r=3 sin SLUTIN The crdioid (see Exmple 7 in Section.3) nd the circle re sketched in Figure 5 nd the desired region is shded. The vlues of nd in Formul 4 re determined y finding the points of intersection of the two curves. They intersect when 3 sin sin,which gives sin,so, 56. The desired re cn e found y sutrcting the re inside the crdioid etween nd from the re inside the circle from 6 to 56. Thus = 5π 6 = π 6 A y 56 3sin d y 56 sin d 6 6 FIGURE 5 r=+sin Since the region is symmetric out the verticl xis, we cn write A y 9 sin d y sin sin d 6 6 y 6 8 sin sin d y cos sin d [ecuse sin cos ] 3 sin cos ]6 M = r=g( ) = FIGURE 6 r=f( ) Exmple illustrtes the procedure for finding the re of the region ounded y two polr curves. In generl, let e region, s illustrted in Figure 6, tht is ounded y curves with polr equtions r f, r t,, nd, where f t nd. The re A of is found y sutrcting the re inside r t from the re inside r f, so using Formul 3 we hve A y y f t f d y t d d CAUTIN The fct tht single point hs mny representtions in polr coordintes sometimes mkes it difficult to find ll the points of intersection of two polr curves. For instnce, it is ovious from Figure 5 tht the circle nd the crdioid hve three points of intersection; however, in Exmple we solved the equtions r 3sin nd r sin nd found only two such points, ( 3, 6) nd ( 3, 56). The origin is lso point of intersection, ut we cn t find it y solving the equtions of the curves ecuse the origin hs no single representtion in polr coordintes tht stisfies oth equtions. Notice tht, when represented s, or,, the origin stisfies r 3 sin nd so it lies on the circle; when represented s, 3, it stisfies r sin nd so it lies on the crdioid. Think of two points moving long the curves s the prmeter vlue increses from to. n one curve the origin is reched t nd ; on the
3 65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES r= FIGURE 7 π, 3 π, 6 r=cos other curve it is reched t. The points don t collide t the origin ecuse they rech the origin t different times, ut the curves intersect there nonetheless. Thus, to find ll points of intersection of two polr curves, it is recommended tht you drw the grphs of oth curves. It is especilly convenient to use grphing clcultor or computer to help with this tsk. EXAMPLE 3 Find ll points of intersection of the curves r cos nd r. SLUTIN If we solve the equtions r cos nd r, we get cos nd, therefore, 3, 5 3, 73, 3. Thus the vlues of etween nd tht stisfy oth equtions re 6, 5 6, 76, 6. We hve found four points of intersection: (, 6), (, 56), (, 76),nd(, 6). However, you cn see from Figure 7 tht the curves hve four other points of intersection nmely, (, 3), (, 3), (, 43), nd (, 53). These cn e found using symmetry or y noticing tht nother eqution of the circle is r nd then solving the equtions r cos nd r. M ARC LENGTH To find the length of polr curve r f,, we regrd s prmeter nd write the prmetric equtions of the curve s Using the Product Rule nd differentiting with respect to, we otin so, using cos sin, we hve dx dr cos r dr Assuming tht f is continuous, we cn use Theorem..6 to write the rc length s L y dx Therefore the length of curve with polr eqution r f,, is 5 dx dr cos r sin d d dy x r cos f cos d d 3 d dr dr r d L y d d d sin r dr d dy r dr y r sin f sin dy dr sin r cos d d d cos sin r sin d d d sin cos r cos V EXAMPLE 4 Find the length of the crdioid r sin. SLUTIN The crdioid is shown in Figure 8. (We sketched it in Exmple 7 in Section.3.) Its full length is given y the prmeter intervl,so
4 SECTIN.4 AREAS AND LENGTHS IN PLAR CRDINATES 653 FIGURE 8 r=+sin Formul 5 gives L y r dr y s sin d d d y We could evlute this integrl y multiplying nd dividing the integrnd y s sin,or we could use computer lger system. In ny event,we find tht the length of the crdioid is L 8. M s sin cos d.4 EXERCISES 4 Find the re of the region tht is ounded y the given curve nd lies in the specified sector.. r, 4. r e, 3. r sin, 3 4. r ssin, 5 8 Find the re of the shded region r=œ 9 4 Sketch the curve nd find the re tht it encloses.. r 4 cos r=4+3 sin r=+cos r=sin 9. r 3 cos. r 3 cos r sin 3. r cos 3 4. r cos ; 5 6 Grph the curve nd find the re tht it encloses. 7 Find the re of the region enclosed y one loop of the curve. 7. r sin 8. r 4 sin 3 5. r sin 6 6. r sin 3 sin 9 9. r 3 cos 5. r sin 6. r sin (inner loop). Find the re enclosed y the loop of the strophoid r cos sec. 3 8 Find the re of the region tht lies inside the first curve nd outside the second curve. 3. r cos, r 4. r sin, 5. r 8 cos, r 6. r sin, 7. r 3cos, 8. r 3sin, 9 34 Find the re of the region tht lies inside oth curves. 9. r s3 cos, 3. r cos, 3. r sin, 3. r 3 cos, 33. r sin, 34. r sin, r cos,, 35. Find the re inside the lrger loop nd outside the smller loop of the limçon r cos. 36. Find the re etween lrge loop nd the enclosed smll loop of the curve r cos Find ll points of intersection of the given curves. 37. r sin, 38. r cos, r cos r sin r sin r cos r cos r 3 sin r cos r 3 sin r sin 39. r sin, r 4. r cos 3, r r 3sin r sin 3 4. r sin, r sin 4. r sin, r cos
5 654 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES ; 43. The points of intersection of the crdioid r sin nd the spirl loop r,,cn t e found exctly. Use grphing device to find the pproximte vlues of t which they intersect. Then use these vlues to estimte the re tht lies inside oth curves. 44. When recording live performnces, sound engineers often use microphone with crdioid pickup pttern ecuse it suppresses noise from the udience. Suppose the microphone is plced 4 m from the front of the stge (s in the figure) nd the oundry of the optiml pickup region is given y the crdioid r 8 8 sin, where r is mesured in meters nd the microphone is t the pole. The musicins wnt to know the re they will hve on stge within the optiml pickup rnge of the microphone. Answer their question Find the exct length of the polr curve. 45. r 3 sin, r e, 47. r stge udience m 4 m microphone, 48. r, 49 5 Use clcultor to find the length of the curve correct to four deciml plces. 49. r 3 sin 5. r 4 sin 3 5. r sin 5. r cos3 ; Grph the curve nd find its length. 53. r cos r cos 55. () Use Formul..7 to show tht the re of the surfce generted y rotting the polr curve (where f is continuous nd ) out the polr xis is S y r f r sin r d dr () Use the formul in prt () to find the surfce re generted y rotting the lemniscte r cos out the polr xis. 56. () Find formul for the re of the surfce generted y rotting the polr curve r f, (where f is continuous nd ), out the line. () Find the surfce re generted y rotting the lemniscte r cos out the line. d.5 CNIC SECTINS In this section we give geometric definitions of prols, ellipses, nd hyperols nd derive their stndrd equtions. They re clled conic sections, or conics, ecuse they result from intersecting cone with plne s shown in Figure. ellipse prol hyperol FIGURE Conics
9.4. ; 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 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 informationGeometric quantities for polar curves
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
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 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 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 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 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 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 information(b) ( 1, s3 ) and Figure 18 shows the resulting curve. Notice that this rose has 16 loops.
SECTIN. PLAR CRDINATES 67 _ and so we require that 6n5 be an even multiple of. This will first occur when n 5. Therefore we will graph the entire curve if we specify that. Switching from to t, we have
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationDouble Integrals over Rectangles
Jim Lmbers MAT 8 Spring Semester 9 Leture Notes These notes orrespond to Setion. in Stewrt nd Setion 5. in Mrsden nd Tromb. Double Integrls over etngles In singlevrible lulus, the definite integrl of
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 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 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 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 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 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 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 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 informationTHE STUDY OF INFLUENCE CORE MATERIALS ON TECHNOLOGICAL PROPERTIES OF UNIVERSAL BENTONITE MOULDING MATERIALS. Matej BEZNÁK, Vladimír HANZEN, Ján VRABEC
THE STUDY OF INFLUENCE CORE MATERIALS ON TECHNOLOGICAL PROPERTIES OF UNIVERSAL BENTONITE MOULDING MATERIALS Mtej BEZNÁK, Vldimír HANZEN, Ján VRABEC Authors: Mtej Beznák, Assoc. Prof. PhD., Vldimír Hnzen,
More informationTIME: 1 hour 30 minutes
UNIVERSITY OF AKRON DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING 4400: 34 INTRODUCTION TO COMMUNICATION SYSTEMS  Spring 07 SAMPLE FINAL EXAM TIME: hour 30 minutes INSTRUCTIONS: () Write your nme
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 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 informationSynchronous Machine Parameter Measurement
Synchronous Mchine Prmeter Mesurement 1 Synchronous Mchine Prmeter Mesurement Introduction Wound field synchronous mchines re mostly used for power genertion but lso re well suited for motor pplictions
More informationMAT01B1: Calculus with Polar coordinates
MAT01B1: Calculus with Polar coordinates Dr Craig 23 October 2018 My details: acraig@uj.ac.za Consulting hours: Monday 14h40 15h25 Thursday 11h30 12h55 Friday (this week) 11h20 12h25 Office CRing 508
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 informationUnderstanding Basic Analog Ideal Op Amps
Appliction Report SLAA068A  April 2000 Understnding Bsic Anlog Idel Op Amps Ron Mncini Mixed Signl Products ABSTRACT This ppliction report develops the equtions for the idel opertionl mplifier (op mp).
More informationEnergy Harvesting TwoWay Channels With Decoding and Processing Costs
IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 3 Energy Hrvesting TwoWy Chnnels With Decoding nd Processing Costs Ahmed Arf, Student Member, IEEE, Abdulrhmn Bknin, Student
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 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 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 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 informationECE 274 Digital Logic. Digital Design. Datapath Components Shifters, Comparators, Counters, Multipliers Digital Design
ECE 27 Digitl Logic Shifters, Comprtors, Counters, Multipliers Digitl Design..7 Digitl Design Chpter : Slides to ccompny the textbook Digitl Design, First Edition, by Frnk Vhid, John Wiley nd Sons Publishers,
More informationFrancis Gaspalou Second edition of February 10, 2012 (First edition on January 28, 2012) HOW MANY SQUARES ARE THERE, Mr TARRY?
Frncis Gslou Second edition of Ferury 10, 2012 (First edition on Jnury 28, 2012) HOW MANY SQUARES ARE THERE, Mr TARRY? ABSTRACT In this er, I enumerte ll the 8x8 imgic sures given y the Trry s ttern. This
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 informationUnit 1: Chapter 4 Roots & Powers
Unit 1: Chpter 4 Roots & Powers Big Ides Any number tht cn be written s the frction mm, nn 0, where m nd n re integers, is nn rtionl. Eponents cn be used to represent roots nd reciprocls of rtionl numbers.
More informationFubini for continuous functions over intervals
Fuini for ontinuous funtions over intervls We first prove the following theorem for ontinuous funtions. Theorem. Let f(x) e ontinuous on ompt intervl =[, [,. Then [, [, [ [ f(x, y)(x, y) = f(x, y)y x =
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 informationTRIGONOMETRIC APPLICATIONS
HPTER TRIGONOMETRI PPLITIONS n ocen is vst expnse tt cn e lifetretening to person wo experiences disster wile oting. In order for elp to rrive on time, it is necessry tt te cost gurd or sip in te re e
More information10.1 Curves defined by parametric equations
Outline Section 1: Parametric Equations and Polar Coordinates 1.1 Curves defined by parametric equations 1.2 Calculus with Parametric Curves 1.3 Polar Coordinates 1.4 Areas and Lengths in Polar Coordinates
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 informationDESIGN OF CONTINUOUS LAG COMPENSATORS
DESIGN OF CONTINUOUS LAG COMPENSATORS J. Pulusová, L. Körösi, M. Dúbrvská Institute of Robotics nd Cybernetics, Slovk University of Technology, Fculty of Electricl Engineering nd Informtion Technology
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 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 informationAvailable online at ScienceDirect. 6th CIRP International Conference on High Performance Cutting, HPC2014
Aville online t www.sciencedirect.com ScienceDirect Procedi CIRP 4 ( 4 ) 76 8 6th CIRP Conference on High Performnce Cutting, HPC4 Investigting Eccentricity Effects in TurnMilling Opertions Emre Uysl,Umut
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 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 informationUniversity of North CarolinaCharlotte Department of Electrical and Computer Engineering ECGR 4143/5195 Electrical Machinery Fall 2009
Problem 1: Using DC Mchine University o North CrolinChrlotte Deprtment o Electricl nd Computer Engineering ECGR 4143/5195 Electricl Mchinery Fll 2009 Problem Set 4 Due: Thursdy October 8 Suggested Reding:
More informationSynchronous Generator Line Synchronization
Synchronous Genertor Line Synchroniztion 1 Synchronous Genertor Line Synchroniztion Introduction One issue in power genertion is synchronous genertor strting. Typiclly, synchronous genertor is connected
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 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 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 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 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 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 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 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 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 informationAreaTime Efficient DigitSerialSerial Two s Complement Multiplier
AreTime Efficient DigitSerilSeril Two s Complement Multiplier Essm Elsyed nd Htem M. ElBoghddi Computer Engineering Deprtment, Ciro University, Egypt Astrct  Multipliction is n importnt primitive
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 informationRobustness Analysis of Pulse Width Modulation Control of Motor Speed
Proceedings of the World Congress on Engineering nd Computer Science 2007 WCECS 2007, October 2426, 2007, Sn Frncisco, USA obustness Anlysis of Pulse Width Modultion Control of Motor Speed Wei Zhn Abstrct
More informationABB STOTZKONTAKT. ABB ibus EIB Current Module SM/S Intelligent Installation Systems. User Manual SM/S In = 16 A AC Un = 230 V AC
User Mnul ntelligent nstlltion Systems A B 1 2 3 4 5 6 7 8 30 ma 30 ma n = AC Un = 230 V AC 30 ma 9 10 11 12 C ABB STOTZKONTAKT Appliction Softwre Current Vlue Threshold/1 Contents Pge 1 Device Chrcteristics...
More informationSLOVAK UNIVERSITY OF TECHNOLOGY Faculty of Material Science and Technology in Trnava. ELECTRICAL ENGINEERING AND ELECTRONICS Laboratory exercises
SLOVAK UNIVERSITY OF TECHNOLOGY Fulty of Mteril Siene nd Tehnology in Trnv ELECTRICAL ENGINEERING AND ELECTRONICS Lbortory exerises Róbert Riedlmjer TRNAVA 00 ELECTRICAL ENGINEERING AND ELECTRONICS Lbortory
More informationSo Many Possibilities page 1 of 2
Otober Solving Problems Ativities & So Mny Possibilities pge of Use the blnk spe to solve eh problem. Show ll your work inluding numbers, words, or lbeled skethes. Write omplete sentene below your work
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 informationA Comparative Analysis of Algorithms for Determining the Peak Position of a Stripe to Subpixel Accuracy
A Comprtive Anlysis of Algorithms for Determining the Pek Position of Stripe to Subpixel Accurcy D.K.Nidu R.B.Fisher Deprtment of Artificil Intelligence, University of Edinburgh 5 Forrest Hill, Edinburgh
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 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 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 informationLATEST CALIBRATION OF GLONASS PCODE TIME RECEIVERS
LATEST CALIBRATION OF GLONASS PCODE TIME RECEIVERS A. Fos 1, J. Nwroci 2, nd W. Lewndowsi 3 1 Spce Reserch Centre of Polish Acdemy of Sciences, ul. Brtyc 18A, 00716 Wrsw, Polnd; Emil: fos@c.ww.pl; Tel.:
More informationTwolayer slottedwaveguide antenna array with broad reflection/gain bandwidth at millimetrewave frequencies
Twolyer slottedwveguide ntenn rry with rod reflection/gin ndwidth t millimetrewve frequencies S.S. Oh, J.W. Lee, M.S. Song nd Y.S. Kim Astrct: A 24 24 slottedwveguide rry ntenn is presented in
More informationSynchronous Machine Parameter Measurement
Synchronous Mchine Prmeter Mesurement 1 Synchronous Mchine Prmeter Mesurement Introduction Wound field synchronous mchines re mostly used for power genertion but lso re well suited for motor pplictions
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 informationJoanna Towler, Roading Engineer, Professional Services, NZTA National Office Dave Bates, Operations Manager, NZTA National Office
. TECHNICA MEMOANDM To Cc repred By Endorsed By NZTA Network Mngement Consultnts nd Contrctors NZTA egionl Opertions Mngers nd Are Mngers Dve Btes, Opertions Mnger, NZTA Ntionl Office Jonn Towler, oding
More informationShuli s Math Problem Solving Column
Shuli s Mth Problem Solvig Colum Volume, Issue Jue, 9 Edited d Authored by Shuli Sog Colordo Sprigs, Colordo shuli_sog@yhoocom Cotets Mth Trick: Metl Clcultio: b cd Mth Competitio Skill: Divisibility by
More informationTHE STUDY ON THE PLASMA GENERATOR THEORY FOR THIN DISC AND THIN RING CONFIGURATION
Interntionl Journl of Innovtive Reserch in Advnced Engineering (IJIRAE) ISSN: 249216 Issue 09, olume 4 (Septemer 2017) www.ijire.com THE STUDY ON THE PLASMA GENERATOR THEORY FOR THIN DISC AND THIN RING
More informationMONOCHRONICLE STRAIGHT
UPDATED 092010 HYDROCARBON Hydrocrbon is ponchostyle cowl in bulkyweight yrn, worked in the round. It ws designed to be s prcticl s it is stylish, with shping tht covers the neck nd shoulders nd the
More informationProbability and Statistics P(A) Mathletics Instant Workbooks. Copyright
Proility nd Sttistis Student Book  Series K P(A) Mthletis Instnt Workooks Copyright Student Book  Series K Contents Topis Topi  Review of simple proility Topi  Tree digrms Topi  Proility trees Topi
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 informationCompared to generators DC MOTORS. Back e.m.f. Back e.m.f. Example. Example. The construction of a d.c. motor is the same as a d.c. generator.
Compred to genertors DC MOTORS Prepred by Engr. JP Timol Reference: Electricl nd Electronic Principles nd Technology The construction of d.c. motor is the sme s d.c. genertor. the generted e.m.f. is less
More informationIntroduction to Planimetry of QuasiElliptic Plane
Originl scientific pper ccepted 4. 11. 2016.. Sliepčević, I. Božić Drgun: Introduction to Plnimetry of QusiElliptic Plne N SLIEPČEVIĆ IVN BOŽIĆ DRGUN Introduction to Plnimetry of QusiElliptic Plne Introduction
More information