Polar 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

Size: px
Start display at page:

Download "Polar 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"

Transcription

1 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. This mteril is not copyright free.

2 b r (+ sin ) Are (+ sin + sin )d + sin + cos d sin + cos d cos sin Use cos sin. Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free.

3 c r sin Are sin d (cos )d sin sin sin + + (+ ) 8 Use cos sin. Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free.

4 d r cos Are cosd sin sin () e r tn Are tn d lnsec ln () ln ln or Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free.

5 f r Are d () g r (+ cos ) ( cos cos )d + + [ ] + sin + sin + + () Are (9+ cos + cos )d ( + ) Use cos cos. Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free. 5

6 Are ( cos cos ) d p + pq + q q q cos cos d p + pq + + p + q q pq cos cos d + + p + q q sin sin + pq + p + q ( p + q ) + + () Use cos cos. Are cos d cos d ( cos )d + sin + () + Use cos cos. Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free.

7 In order to find the re enclosed by single loop, we clculte d ( 5sin ) d r + + ( sin + ) 5sin d 5 + sin ( cos ) d cos sin + ( + 5 ) nd >, we solve to obtin 9 In order to hve ( ) 5 First we find the intersection points of these two curves. sin sin sin sin + kor + k kor (+ k) k or (+ k) ( k Z). Since we re working in the rnge, we only cre bout the intersections occurring t, nd. In fct we only hve positive vlue for sin between nd so the loop is only defined in this rnge. Note tht sin when, which mens tht it intersects the origin. Now we clculte the re to be A ( sin ) d ( sin) d + + ( cos ) d ( cos8 ) sin sin d Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free. 7

8 The points of intersection re given by + sin sin So sin which mens 5 rcsin ( ) nd. So the polr coordintes of intersection re (, ) nd (, 5 ). Since there is symmetry bout the verticl xis, we my compute the re s Are ( sin) d+ ( + sin) d ( 9sin ) d + ( sin + ) + sin d 9 ( cos ) d + + sin ( cos ) d + 9 9sin sin + cos centred t (, ) with rdius 5. 7 The set of points A z: rgz { z: z + i 5} define segment of circle, Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free. 8

9 7 b The eqution for circle centred t (, ) with rdius 5, in Crtesin coordintes is ( x ) + ( y+ ) 5. To find the re of the region bounded bya, nd, we convert into polr coordintes, obtining ( rcos ) + ( rsin + ) 5. This simplifies to r 8cos sin when r. Now we clculte A (8cos sin ) d ( cos 9cos sin + sin )d ((+ cos ) 8sin + 8( cos ))d ( + cos 8( [ ( sin ) + + sin )) ] centred t (,5) with rdius. 8 The set of points A z: rgz { z: z+ 5 i } define region of circle, Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free. 9

10 8 b To find the re of the region bounded bya, r cos + sin to clculte ( cos sin ) A + d (57cos sin ) 8cos + sin d (88 (+ cos ) sin + 5 (cos ) ) d ( cos 5 ( ) 85. [ ( sin ) + ( sin )) ] nd, we use the polr form Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free.

11 9 In order to find the re of the shded region, we must find the re of the sector bounded by the curve nd the lineoa, then subtrct the re of the tringle OAB. The vlue of t the point A cn + be found by solving r + cos in order to get. We now find the re of the sector bounded by the curve nd the line OA. Asector ( + cos ) d ( + cos + cos ) d ( + cos + (cos + ) ) d [( sin ( sin )) ] Now we find the re of the tringle OAB by using the formul AreOAB bsinc, where is the length of OA, b is the length of OB nd C is the ngle between O A nd OB. AreOAB bsinc + + sin.77. Thus, the re of the shded region is found to be A Are Are sector (sf) OAB Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free.

12 Note tht there is symmetry bout the verticl xis nd so we my compute one side nd multiply by. In order to find the re of the shded region, we first must find the re of the sector of r + sin between nd the right hnd side intersection point of the two curves. This intersection point occurs when + sin sin i.e. when. So now we find the re of the sector of r + sin for. We will denote this re A. A ( + sin) d ( + sin + sin ) d ( + sin + (cos ) ) d [( cos + ( sin )) ] This integrl hs included smll extr region we do not wnt (the red region in the imge) We find the vlue of the unwnted regions re (which we will denote A ) by 9 A ( sin ) d ( 9sin ) 9 d (cos )d [( sin ) ] 9.8. So the right hnd side of the shded region hs re A right.97. Remember tht this is only hlf of the totl region we wnt to find, so tht mens we just need to double A right in order to find A totl.79 ( d.p.) Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free.

13 Chllenge The fr left hnd side of the shell is point which occurs when nd so we cn set r k. left Similrly for the fr right hnd side of the shell, which occurs t, giving r k. Thus the horizontl dimeter is d 7 k cm, so we conclude tht k 7 b In order to find the totl re of the cross section, we must ensure tht ll ngles re covered exctly once. So we choose to integrte between. A d d ( 8) right Person Eduction Ltd 8. Copying permitted for purchsing institution only. This mteril is not copyright free.

Polar Coordinates. July 30, 2014

Polar Coordinates. July 30, 2014 Polr Coordintes July 3, 4 Sometimes it is more helpful to look t point in the xy-plne not in terms of how fr it is horizontlly nd verticlly (this would men looking t the Crtesin, or rectngulr, coordintes

More information

10.4 AREAS AND LENGTHS IN POLAR COORDINATES

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 information

Section 10.2 Graphing Polar Equations

Section 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 information

9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations

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 information

Geometric quantities for polar curves

Geometric 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 information

Vocabulary Check. Section 10.8 Graphs of Polar Equations not collinear The points are collinear.

Vocabulary 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 information

Triangles and parallelograms of equal area in an ellipse

Triangles 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 information

c The scaffold pole EL is 8 m long. How far does it extend beyond the line JK?

c 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 information

Section 16.3 Double Integrals over General Regions

Section 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 information

FP2 POLAR COORDINATES: PAST QUESTIONS

FP2 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 information

Example. Check that the Jacobian of the transformation to spherical coordinates is

Example. 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 information

First Round Solutions Grades 4, 5, and 6

First 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 information

Vector Calculus. 1 Line Integrals

Vector 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

Section 17.2: Line Integrals. 1 Objectives. 2 Assignments. 3 Maple Commands. 1. Compute line integrals in IR 2 and IR Read Section 17.

Section 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 information

Lecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 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 information

NONCLASSICAL CONSTRUCTIONS II

NONCLASSICAL CONSTRUCTIONS II NONLSSIL ONSTRUTIONS II hristopher Ohrt UL Mthcircle - Nov. 22, 2015 Now we will try ourselves on oncelet-steiner constructions. You cn only use n (unmrked) stright-edge but you cn ssume tht somewhere

More information

Translate and Classify Conic Sections

Translate 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 information

LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY

LECTURE 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 information

REVIEW, pages

REVIEW, 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 information

INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS

INTRODUCTION 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 information

b = 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 )

b = 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 information

Theme: Don t get mad. Learn mod.

Theme: 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 information

SOLVING TRIANGLES USING THE SINE AND COSINE RULES

SOLVING 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 information

Exercise 1-1. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION

Exercise 1-1. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION Exercise 1-1 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 information

Math 116 Calculus II

Math 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 information

(1) Primary Trigonometric Ratios (SOH CAH TOA): Given a right triangle OPQ with acute angle, we have the following trig ratios: ADJ

(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 information

MATH 118 PROBLEM SET 6

MATH 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 information

Notes on Spherical Triangles

Notes 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 information

Domination and Independence on Square Chessboard

Domination 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 information

Student Book SERIES. Patterns and Algebra. Name

Student 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

Skills Practice Skills Practice for Lesson 4.1

Skills 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 information

Algebra Practice. Dr. Barbara Sandall, Ed.D., and Travis Olson, M.S.

Algebra 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 978-1-58037-754-6 Printing No. 404042-EB Mrk Twin Medi, Inc., Pulishers Distriuted y Crson-Dellos

More information

13.1 Double Integral over Rectangle. f(x ij,y ij ) i j I <ɛ. f(x, y)da.

13.1 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

Study 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 # 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 information

TIME: 1 hour 30 minutes

TIME: 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 information

MEASURE THE CHARACTERISTIC CURVES RELEVANT TO AN NPN TRANSISTOR

MEASURE 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 information

7KH4XLQFXQ; Earth/matriX SCIENCE IN ANCIENT ARTWORK. Charles William Johnson

7KH4XLQFXQ; Earth/matriX SCIENCE IN ANCIENT ARTWORK. Charles William Johnson Erth/mtriX SCIENCE IN ANCIENT ARTWORK 7KH4XLQFXQ; Chrles Willim Johnson Erth/mtriX P.O. Box 231126, New Orlens, Louisin, 70183-1126 2001 Copyrighted y Chrles Willim Johnson www.erthmtrix.om www.the-periodi-tle.om

More information

Section Thyristor converter driven DC motor drive

Section Thyristor converter driven DC motor drive Section.3 - Thyristor converter driven DC motor drive.3.1 Introduction Controllble AC-DC converters using thyristors re perhps the most efficient nd most robust power converters for use in DC motor drives.

More information

Application Note. Differential Amplifier

Application Note. Differential Amplifier Appliction Note AN367 Differentil Amplifier Author: Dve n Ess Associted Project: Yes Associted Prt Fmily: CY8C9x66, CY8C7x43, CY8C4x3A PSoC Designer ersion: 4. SP3 Abstrct For mny sensing pplictions, desirble

More information

Chapter 12 Vectors and the Geometry of Space 12.1 Three-dimensional Coordinate systems

Chapter 12 Vectors and the Geometry of Space 12.1 Three-dimensional Coordinate systems hpter 12 Vectors nd the Geometry of Spce 12.1 Three-dimensionl oordinte systems A. Three dimensionl Rectngulr oordinte Sydstem: The rtesin product where (x, y, z) isclled ordered triple. B. istnce: R 3

More information

Lecture 16. Double integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 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 information

MATHEMATICS Unit Pure Core 2

MATHEMATICS Unit Pure Core 2 General Certificate of Education January 2009 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Tuesday 1 January 2009 9.00 am to 10.0 am For this paper you must have: an 8-page answer

More information

Kirchhoff s Rules. Kirchhoff s Laws. Kirchhoff s Rules. Kirchhoff s Laws. Practice. Understanding SPH4UW. Kirchhoff s Voltage Rule (KVR):

Kirchhoff 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 information

Seven Sisters. Visit for video tutorials

Seven 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 information

STUDY GUIDE, CALCULUS III, 2017 SPRING

STUDY 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 information

SLOVAK UNIVERSITY OF TECHNOLOGY Faculty of Material Science and Technology in Trnava. ELECTRICAL ENGINEERING AND ELECTRONICS Laboratory exercises

SLOVAK 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 information

CHAPTER 2 LITERATURE STUDY

CHAPTER 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 information

SECOND EDITION STUDENT BOOK GRADE

SECOND 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 information

(b) ( 1, s3 ) and Figure 18 shows the resulting curve. Notice that this rose has 16 loops.

(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 information

Practice Problems: Calculus in Polar Coordinates

Practice Problems: Calculus in Polar Coordinates Practice Problems: Calculus in Polar Coordinates Answers. For these problems, I want to convert from polar form parametrized Cartesian form, then differentiate and take the ratio y over x to get the slope,

More information

Section 6.1 Law of Sines. Notes. Oblique Triangles - triangles that have no right angles. A c. A is acute. A is obtuse

Section 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 information

Shuli s Math Problem Solving Column

Shuli 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 information

Synchronous Machine Parameter Measurement

Synchronous 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 information

A Comparative Analysis of Algorithms for Determining the Peak Position of a Stripe to Sub-pixel Accuracy

A Comparative Analysis of Algorithms for Determining the Peak Position of a Stripe to Sub-pixel Accuracy A Comprtive Anlysis of Algorithms for Determining the Pek Position of Stripe to Sub-pixel Accurcy D.K.Nidu R.B.Fisher Deprtment of Artificil Intelligence, University of Edinburgh 5 Forrest Hill, Edinburgh

More information

Diffraction and Interference. 6.1 Diffraction. Diffraction grating. Diffraction grating. Question. Use of a diffraction grating in a spectrometer

Diffraction 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 information

MONOCHRONICLE STRAIGHT

MONOCHRONICLE STRAIGHT UPDATED 09-2010 HYDROCARBON Hydrocrbon is poncho-style cowl in bulky-weight 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 information

Performance Comparison between Network Coding in Space and Routing in Space

Performance 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 information

Design and Modeling of Substrate Integrated Waveguide based Antenna to Study the Effect of Different Dielectric Materials

Design and Modeling of Substrate Integrated Waveguide based Antenna to Study the Effect of Different Dielectric Materials Design nd Modeling of Substrte Integrted Wveguide bsed Antenn to Study the Effect of Different Dielectric Mterils Jgmeet Kour 1, Gurpdm Singh 1, Sndeep Ary 2 1Deprtment of Electronics nd Communiction Engineering,

More information

Magnetic monopole field exposed by electrons

Magnetic 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 information

Synchronous Generator Line Synchronization

Synchronous 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 information

Spherical Geometry. This is an article from my home page:

Spherical Geometry. This is an article from my home page: Spheril Geometry This is n rtile from my home pge: www.olewitthnsen.dk Ole Witt-Hnsen nov. 6 Contents. Geometry on sphere.... Spheril tringles...3. Polr tringles...4 3. The right-ngle spheril tringle...6

More information

MSC Studentenwettbewerb. Wintersemester 2012/13. Marc/Mentat 2012

MSC Studentenwettbewerb. Wintersemester 2012/13. Marc/Mentat 2012 MSC Studentenwettewer Wintersemester 2012/13 Mrc/Mentt 2012 Aufge Wie groß ist die mximle Verschieung in Y Richtung im nichtlineren Fll? Required File: tip_lod.mud. 2 TUTORIAL Prolem Description In this

More information

(CATALYST GROUP) B"sic Electric"l Engineering

(CATALYST GROUP) Bsic Electricl Engineering (CATALYST GROUP) B"sic Electric"l Engineering 1. Kirchhoff s current l"w st"tes th"t (") net current flow "t the junction is positive (b) Hebr"ic sum of the currents meeting "t the junction is zero (c)

More information

10.3 Polar Coordinates

10.3 Polar Coordinates .3 Polar Coordinates Plot the points whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r > and one with r

More information

Experiment 3: The research of Thevenin theorem

Experiment 3: The research of Thevenin theorem Experiment 3: The reserch of Thevenin theorem 1. Purpose ) Vlidte Thevenin theorem; ) Mster the methods to mesure the equivlent prmeters of liner twoterminl ctive. c) Study the conditions of the mximum

More information

Multivariable integration. Multivariable integration. Iterated integration

Multivariable 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

Student Book SERIES. Fractions. Name

Student 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 information

Electronic Circuits I - Tutorial 03 Diode Applications I

Electronic 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 reverse-bised, diode idelly ppers s short. F 3 A

More information

METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN. Inventor: Brian L. Baskin

METHOD 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 information

6ev O'Nly. i *Af. -0-St 4 THE REFLECTION OF RADIO WAVES FROM AN IRREGULAR IONOSPHERE MASSACHUSETTS INSTITUTE OF TECHNOLOGY M. L. V.

6ev O'Nly. i *Af. -0-St 4 THE REFLECTION OF RADIO WAVES FROM AN IRREGULAR IONOSPHERE MASSACHUSETTS INSTITUTE OF TECHNOLOGY M. L. V. i DOCU.EIT R~)6 R0 - RESEARCH LAB(Ai'RY CF EILCTROICS tkassachusetts 1It'S'IJE OF TCVlDDLOGY - -0-St 4 THE REFLECTION OF RADIO WAVES FROM AN IRREGULAR IONOSPHERE M. L. V. PITTEWAY TECHNICAL REPORT 382

More information

MAT01B1: Calculus with Polar coordinates

MAT01B1: 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 C-Ring 508

More information

Chapter 5 Analytic Trigonometry

Chapter 5 Analytic Trigonometry Section 5. Fundmentl Identities 03 Chter 5 Anlytic Trigonometry Section 5. Fundmentl Identities Exlortion. cos / sec, sec / cos, nd tn sin / cos. sin / csc nd tn / cot 3. csc / sin, cot / tn, nd cot cos

More information

Module 9. DC Machines. Version 2 EE IIT, Kharagpur

Module 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 (esson-40) 4 40.1 Gols

More information

Homework #1 due Monday at 6pm. White drop box in Student Lounge on the second floor of Cory. Tuesday labs cancelled next week

Homework #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 information

Unit 1: Chapter 4 Roots & Powers

Unit 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 information

Experiment 3: Non-Ideal Operational Amplifiers

Experiment 3: Non-Ideal Operational Amplifiers Experiment 3: Non-Idel Opertionl Amplifiers 9/11/06 Equivlent Circuits The bsic ssumptions for n idel opertionl mplifier re n infinite differentil gin ( d ), n infinite input resistnce (R i ), zero output

More information

Experiment 3: Non-Ideal Operational Amplifiers

Experiment 3: Non-Ideal Operational Amplifiers Experiment 3: Non-Idel Opertionl Amplifiers Fll 2009 Equivlent Circuits The bsic ssumptions for n idel opertionl mplifier re n infinite differentil gin ( d ), n infinite input resistnce (R i ), zero output

More information

HIGHER MATHEMATICS. Unit 2 Topic 3.2 Compound Angle Formula

HIGHER MATHEMATICS. Unit 2 Topic 3.2 Compound Angle Formula HIGHER MTHEMTICS Unit 2 Topic 3.2 Compound ngle Formula REMINDERS y P (y,) Let OP = r & POX = This gives the following sin = y r cos = r P(,y) r y O Now reflect OP in the line y = sin(90 - ) = r = cos

More information

A technical description of atmospheric sounding by GPS occultation

A technical description of atmospheric sounding by GPS occultation Journl of Atmospheric nd Solr-Terrestril Physics 64 (2002) 451 469 www.elsevier.com/locte/jstp A technicl description of tmospheric sounding by GPS occulttion G.A. Hjj, E.R. Kursinsi, L.J. Romns, W.I.

More information

ECE 274 Digital Logic. Digital Design. Datapath Components Shifters, Comparators, Counters, Multipliers Digital Design

ECE 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 information

Chapter 5 Analytic Trigonometry

Chapter 5 Analytic Trigonometry Section 5. Fundmentl Identities 03 Chter 5 Anlytic Trigonometry Section 5. Fundmentl Identities Exlortion. cos > sec, sec > cos, nd tn sin > cos. sin > csc nd tn > cot 3. csc > sin, cot > tn, nd cot cos

More information

Chapter 3, Part 1: Intro to the Trigonometric Functions

Chapter 3, Part 1: Intro to the Trigonometric Functions Haberman MTH 11 Section I: The Trigonometric Functions Chapter 3, Part 1: Intro to the Trigonometric Functions In Example 4 in Section I: Chapter, we observed that a circle rotating about its center (i.e.,

More information

Compared 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.

Compared 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 information

GCSE Mathematics Practice Tests: Set 3

GCSE Mathematics Practice Tests: Set 3 GCSE Mathematics Practice Tests: Set 3 Paper 1H (Non-calculator) Time: 1 hour 30 minutes You should have: Ruler graduated in centimetres and millimetres, protractor, pair of compasses, pen, HB pencil,

More information

EE Controls Lab #2: Implementing State-Transition Logic on a PLC

EE Controls Lab #2: Implementing State-Transition Logic on a PLC Objective: EE 44 - Controls Lb #2: Implementing Stte-rnsition 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 information

CHAPTER 3 EDGE DETECTION USING CLASICAL EDGE DETECTORS

CHAPTER 3 EDGE DETECTION USING CLASICAL EDGE DETECTORS CHAPTER 3 EDE DETECTION USIN CLASICAL EDE DETECTORS Edge detection is one o te most importnt opertions in imge nlsis. An edge is set o connected piels tt lie on te boundr between two regions. Te clssiiction

More information

CS 135: Computer Architecture I. Boolean Algebra. Basic Logic Gates

CS 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 information

& Y Connected resistors, Light emitting diode.

& 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 information

MA10103: Foundation Mathematics I. Lecture Notes Week 3

MA10103: Foundation Mathematics I. Lecture Notes Week 3 MA10103: Foundation Mathematics I Lecture Notes Week 3 Indices/Powers In an expression a n, a is called the base and n is called the index or power or exponent. Multiplication/Division of Powers a 3 a

More information

University of North Carolina-Charlotte Department of Electrical and Computer Engineering ECGR 4143/5195 Electrical Machinery Fall 2009

University of North Carolina-Charlotte Department of Electrical and Computer Engineering ECGR 4143/5195 Electrical Machinery Fall 2009 Problem 1: Using DC Mchine University o North Crolin-Chrlotte Deprtment o Electricl nd Computer Engineering ECGR 4143/5195 Electricl Mchinery Fll 2009 Problem Set 4 Due: Thursdy October 8 Suggested Reding:

More information

Misty. Sudnow Dot Songs

Misty. Sudnow Dot Songs Sudnow Dot Songs isty T The Dot Song is nottionl system tht depicts voiced chords in wy where the non-music reder cn find these firly redily. But the Dot Song is not intended be red, not s sight reder

More information

NUMBER THEORY Amin Witno

NUMBER THEORY Amin Witno WON Series in Discrete Mthemtics nd Modern Algebr Volume 2 NUMBER THEORY Amin Witno Prefce Written t Phildelphi University, Jordn for Mth 313, these notes 1 were used first time in the Fll 2005 semester.

More information

MAYWOOD. Hospitality & ConferenCe tables FURNITURE CORP. GSA Approved! Contract #GS-28F-0050W. Established 1918

MAYWOOD. Hospitality & ConferenCe tables FURNITURE CORP.   GSA Approved! Contract #GS-28F-0050W. Established 1918 MAYWOOD FURNITURE CORP. Mnufctured in the U.S.A. Estblished 1918 Hospitlity & ConferenCe tbles www.mywood.com GSA Approved! Contrct #GS-28F-0050W bout us Circ 1930 Mywood Furniture Corportion ws estblished

More information

(1) Non-linear system

(1) Non-linear system Liner vs. non-liner 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 information

http://my.nctm.org/eresources/view_article.asp?article_id=7655 Page 1 of 2 Advanced Search SIGN OFF MY NCTM MY MEMBERSHIP HELP HOME NCTM You are signed in as Jennifer Bergner. ON-Math 2006-2007 Volume

More information

CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates

CHAPTER 10 Conics, Parametric Equations, and Polar Coordinates CHAPTER Conics, Parametric Equations, and Polar Coordinates Section. Conics and Calculus.................... Section. Plane Curves and Parametric Equations.......... Section. Parametric Equations and Calculus............

More information

Synchronous Machine Parameter Measurement

Synchronous 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 information

Determine currents I 1 to I 3 in the circuit of Fig. P2.14. Solution: For the loop containing the 18-V source, I 1 = 0.

Determine currents I 1 to I 3 in the circuit of Fig. P2.14. Solution: For the loop containing the 18-V source, I 1 = 0. Prolem.14 Determine currents 1 to 3 in the circuit of Fig. P.14. 1 A 18 V Ω 3 A 1 8 Ω 1 Ω 7 Ω 4 Ω 3 Figure P.14: Circuit for Prolem.14. For the loop contining the 18-V source, Hence, 1 = 1.5 A. KCL t node

More information

Double Integrals over Rectangles

Double 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 single-vrible lulus, the definite integrl of

More information

Francis Gaspalou Second edition of February 10, 2012 (First edition on January 28, 2012) HOW MANY SQUARES ARE THERE, Mr TARRY?

Francis 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 information