Chapter 9. Arc Length and Surface Area


 Debra Simmons
 5 years ago
 Views:
Transcription
1 Chpter 9. Arc Length nd Surfce Are In which We ppl integrtion to stud the lengths of curves nd the re of surfces. 9. Arc Length (Tet ) P n P 2 P P 2 n b P i ( i, f( i )) P i ( i, f( i )) distnce from P i to P i ( i i ) 2 + ( f( i ) f( i ) ) 2
2 28 Chpter 9. Arc Length nd Surfce Are let i i be the sme for ll i n s P i P i i n i ( ) 2 + ( f( i ) f( i ) ) 2 n ( ) 2 f(i ) f( i ) + i n + f (c i ) 2 i for some i < c i < i b the M. V. Th. tke the limit s s b + ( f () ) 2 d notice tht for the limit (integrl) to eist we need some hpothesis on the integrnd continuit of f will do Emple 9.. Find the circumference of circle. 2 2
3 d d Arc Length (Tet ) 29 s ( 2 2 ) 2 d lim R [Arcsin ] R d d [Arcsin() Arcsin()] 4 π 2 2π Emple 9..2 Find the length long the curve from to 9. L 9 + d d 2 ( ) 2 d + d d this is not n es integrl d substitute t 2 4; 2t 4 d t L 6 t2 + t t t2 + now the substitution is t tn θ the integrl tht results is sec 3 θ dθ
4 L i 3 Chpter 9. Arc Length nd Surfce Are lternte method interchnge the roles of nd Find the length of 2 from to 3 L d d ( ) 2 d 3 d d + 42 d substitute 2 tn θ to give sec 3 θ dθ Emple 9..3 Find the length long the curve ln from to e. L d d e e + ( ) 2 d d d + e 2 d 2 + d substitute t to give t 2 t Arc Length of Prmetric Curves (Tet ) pproimte length b stright lines L ( i ) 2 + ( i ) 2 i i f(t i ) i f (c i ) t b MVTh. i g(t i ) i g (d i ) t b MVTh. (f (c i ) 2 + (g (d i ) 2 t
5 9.2 Arc Length of Prmetric Curves (Tet ) 3 L β α (d ) 2 + ( ) 2 d requires f (t), g (t) continuous for α t β the curve is trversed once gives old formul if prmeter is Emple 9.2. The circumference of circle. prmetric equtions of circle cos t, sin t; t 2π L 2π 2π 2π (d ) 2 ( ) 2 d + ( sin t)2 + ( cos t) 2 t 2π 2π Emple Find the length of the curve t 2 t; t 2 + t from t /2 to t 3/2. L 3/2 /2 (2t )2 + (2t + ) 2 3/2 8t /2 2 (2t)2 + /2 /2 to do this integrl substitute tn θ 2t recll sec θ ln sec θ + tn θ
6 32 Chpter 9. Arc Length nd Surfce Are Emple Find the length of the curve 4 cos 3 t; 4 sin 3 t. L 2π [2 cos 2 t( sin t)] 2 + [2 sin 2 t(cos t)] 2 2 2π cos t sin t ( sin 2 ) t π/ ! Wht hppens if we ignore the bsolute vlue signs? 9.3 Arc Length of Polr Curves (Tet ) length of sector of circle r r s compre length long the curve with length long the rc of circle ( s) 2 ( r) 2 + (r θ) 2 ( ( ) ) 2 r + r 2 ( θ) 2 θ s β α ( ) 2 dr + r dθ 2 dθ β α (f (θ) ) 2 + ( f(θ) ) 2 dθ
7 9.3 Arc Length of Polr Curves (Tet ) 33 formul cn lso be obtined from using ( ) 2 d + dθ r cos θ r sin θ ( ) 2 d dθ dθ Emple 9.3. Find the circumference of circle. circle, centre (, ), rdius is r circumference C 2π dθ 2π θ 2π dθ 2π Emple r sin 2θ. Find the rclength of one lef of review sketch (emple 8.7.4) one lef is swept out when θ π 2 s π 2 (2 cos 2θ)2 + (sin 2θ) 2 dθ this integrl is not es to do
8 34 Chpter 9. Arc Length nd Surfce Are 9.4 Surfce Are (Tet ) b pproimte the surfce using clinders rdius of the clinder is f(c i ) height of clinder is length long the curve for, length is + ( f (c i ) ) 2 s S 2πf(c i ) + ( f (c i ) ) 2 2πf(ci ) s S 2π b f() + ( f () ) b 2 d 2π f() ds
9 9.4 Surfce Are (Tet ) 35 rottion bout the is rdius of the clinder is length long the curve s bove S 2π b + ( f () ) b 2 d 2π ds Emple 9.4. Find the surfce re of sphere. 2 2 S 2π b 2 2π 4π + ( ) 2 d ( ) d 2 2 ( ) 2 2 d 2 2 4π d 4π 2
10 36 Chpter 9. Arc Length nd Surfce Are Emple Find the surfce re of cone. r 3'6" h 3'6" h r S 2π 2π r + + ( ) 2 h d r ( ) 2 [ h 2 r 2 ] r ( ) 2 [ ] h r 2 2π + r 2 πr r 2 + h 2 Emple Find the surfce re of the object formed when 3/2 from to 9 is revolved bout the is. S 2π 9 + ( ( ( 3/2 ) ) 2 d 2π ) ds 2π 2π ( ) /2 d d
11 9.4 Surfce Are (Tet ) 37 b first formul with S 2π 2/3 3/ ( ) 2 d d d ( 2π ) ds 2π 27 ( ) 2 2 2/3 + 3 /3 d Emple Find the volume nd surfce re of the infinitel long horn formed when is rotted bout the is from to. V π 2 d π ( ) 2 R d d π lim R 2 [ ] R [ ] π lim π lim R R R + π( + ) π Conclusion: The horn cn be filled with π units of pint.
12 38 Chpter 9. Arc Length nd Surfce Are S 2π 2π + ( ) 2 d + ( ) 2 2 d 4 + 2π 4 d this integrl need not be evluted notice for R d d lim R lim R ln R lim R ln R b comprison d Conclusion: The surfce cnnot be pinted! 9.5 Surfce Are nd Prmetric Curves ( ) S 2π ds for rottion bout the is for prmetric f(t), g(t), just showed (d ) 2 ds + needs f, g continuous nd g ( ) 2 d
13 9.5 Surfce Are nd Prmetric Curves ( ) 39 summr for rottion bout the is S β α (d ) 2 2π + ( ) 2 d for rottion bout the is S δ γ (d ) 2 2π + ( ) 2 d Emple 9.5. The surfce re of sphere. prmetric equtions of semicircle cos t, sin t; t π rotte bout the is to get sphere ds for the circle S π 2π ds 2π π 2π 2 ( cos t) π 4π 2 ( sin t) Emple The surfce re of e t t, 4e t/2, t, revolved bout the is. S 2π 4e t/2 (e t ) 2 + (2e t/2 ) 2 8π e t/2 e 2t 2e t + + 4e t 8π e t/2 (e t + ) 2 8π e 3t/2 + e t/2...
14 4 Chpter 9. Arc Length nd Surfce Are Emple The surfce re of e t t, 4e t/2, t, revolved bout the is. S 2π (e t t) (e t ) 2 + (2e t/2 ) 2 2π (e t t) e 2t 2e t + + 4e t 2π (e t t)(e t + ) 2π (e 2t te t + e t t)...
Total Score Maximum
Lst Nme: Mth 8: Honours Clculus II Dr. J. Bowmn 9: : April 5, 7 Finl Em First Nme: Student ID: Question 4 5 6 7 Totl Score Mimum 6 4 8 9 4 No clcultors or formul sheets. Check tht you hve 6 pges.. Find
More informationMath 231E, Lecture 33. Parametric Calculus
Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider
More informationThe problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.
ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion
More informationFinal Review, Math 1860 Thomas Calculus Early Transcendentals, 12 ed
Finl Review, Mth 860 Thoms Clculus Erly Trnscendentls, 2 ed 6. Applictions of Integrtion: 5.6 (Review Section 5.6) Are between curves y = f(x) nd y = g(x), x b is f(x) g(x) dx nd similrly for x = f(y)
More informationMATH1013 Tutorial 12. Indefinite Integrals
MATH Tutoril Indefinite Integrls The indefinite integrl f() d is to look for fmily of functions F () + C, where C is n rbitrry constnt, with the sme derivtive f(). Tble of Indefinite Integrls cf() d c
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationMultiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: olumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
More informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More informationUnit 5. Integration techniques
18.01 EXERCISES Unit 5. Integrtion techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A1 Evlute ) tn 1 3 b) sin 1 ( 3/) c) If θ = tn 1 5, then evlute sin θ, cos θ, cot θ, csc θ, nd
More information(b) Let S 1 : f(x, y, z) = (x a) 2 + (y b) 2 + (z c) 2 = 1, this is a level set in 3D, hence
Problem ( points) Find the vector eqution of the line tht joins points on the two lines L : r ( + t) i t j ( + t) k L : r t i + (t ) j ( + t) k nd is perpendiculr to both those lines. Find the set of ll
More informationMA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations
LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll
More informationNot for reproduction
AREA OF A SURFACE OF REVOLUTION cut h FIGURE FIGURE πr r r l h FIGURE A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundry of solid of revolution of the type
More informationExercise Qu. 12. a2 y 2 da = Qu. 18 The domain of integration: from y = x to y = x 1 3 from x = 0 to x = 1. = 1 y4 da.
MAH MAH Eercise. Eercise. Qu. 7 B smmetr ( + 5)dA + + + 5 (re of disk with rdius ) 5. he first two terms of the integrl equl to becuse is odd function in nd is odd function in. (see lso pge 5) Qu. b da
More informationAPPLICATIONS OF THE DEFINITE INTEGRAL
APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its crosssection in plne pssing through
More informationMultiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
More informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More information. Doubleangle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.
Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos(  1 2 ) = rcsin( 1 2 ) = rcsin(  1 2 ) = Cn you do similr problems? Review of Bsic Concepts
More information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More information5.2 Volumes: Disks and Washers
4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of crosssection or slice. In this section, we restrict
More informationAPPLICATIONS OF DEFINITE INTEGRALS
Chpter 6 APPICATIONS OF DEFINITE INTEGRAS OVERVIEW In Chpter 5 we discovered the connection etween Riemnn sums ssocited with prtition P of the finite closed intervl [, ] nd the process of integrtion. We
More information. Doubleangle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =
Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos(  1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin(  1 ) = π 2 6 2 6 Cn you do similr problems?
More informationES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
More informationImproper Integrals. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Improper Integrls MATH 2, Clculus II J. Robert Buchnn Deprtment of Mthemtics Spring 28 Definite Integrls Theorem (Fundmentl Theorem of Clculus (Prt I)) If f is continuous on [, b] then b f (x) dx = [F(x)]
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationApplications of Definite Integral
Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between
More informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
More informationIn Mathematics for Construction, we learnt that
III DOUBLE INTEGATION THE ANTIDEIVATIVE OF FUNCTIONS OF VAIABLES In Mthemtics or Construction, we lernt tht the indeinite integrl is the ntiderivtive o ( d ( Double Integrtion Pge Hence d d ( d ( The ntiderivtive
More informationSolutions to Problems Integration in IR 2 and IR 3
Solutions to Problems Integrtion in I nd I. For ec of te following, evlute te given double integrl witout using itertion. Insted, interpret te integrl s, for emple, n re or n verge vlue. ) dd were is te
More informationChapter 7: Applications of Integrals
Chpter 7: Applictions of Integrls 78 Chpter 7 Overview: Applictions of Integrls Clculus, like most mthemticl fields, egn with tring to solve everd prolems. The theor nd opertions were formlized lter. As
More informationSome Methods in the Calculus of Variations
CHAPTER 6 Some Methods in the Clculus of Vritions 6. If we use the vried function ( α, ) α sin( ) + () Then d α cos ( ) () d Thus, the totl length of the pth is d S + d d α cos ( ) + α cos ( ) d Setting
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 9 Prmetric Equtions nd Polr Coordintes Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition Section 91 Prmetric Curves
More informationUS01CMTH02 UNIT Curvature
Stu mteril of BSc(Semester  I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT 1 Curvture Let f : I R be sufficiently
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationApplications of Definite Integral
Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between
More informationTime : 3 hours 03  Mathematics  March 2007 Marks : 100 Pg  1 S E CT I O N  A
Time : hours 0  Mthemtics  Mrch 007 Mrks : 100 Pg  1 Instructions : 1. Answer ll questions.. Write your nswers ccording to the instructions given below with the questions.. Begin ech section on new
More informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
More information660 Chapter 10 Conic Sections and Polar Coordinates
Chpter Conic Sections nd Polr Coordintes 8. ( (b (c (d (e r r Ê r ; therefore cos Ê Ê ( ß is point of intersection ˆ ˆ Ê Ê Ê ß ß ˆ ß 9. ( r cos Ê cos ; r cos Ê r Š Ê r r Ê (r (b r Ê cos Ê cos Ê, Ê ß or
More informationSpace Curves. Recall the parametric equations of a curve in xyplane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xyplne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More informationA LEVEL TOPIC REVIEW. factor and remainder theorems
A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division
More informationIndefinite Integral. Chapter Integration  reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More information[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves
Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.
More informationMath 113 Exam 2 Practice
Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
More information(6.5) Length and area in polar coordinates
86 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS Totl mss 6 x ρ(x)dx + x 6 x dx + 9 kg dx + 6 x dx oment bout origin 6 xρ(x)dx x x dx + x + x + ln x ( ) + ln 6 kg m x dx + 6 6 x x dx Centre of mss +
More informationNORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y
LOCUS 50 Section  4 NORMALS Consider n ellipse. We need to find the eqution of the norml to this ellipse t given point P on it. In generl, we lso need to find wht condition must e stisfied if m c is to
More informationk ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.
Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 6 (First moments of an arc) A.J.Hobson
JUST THE MATHS UNIT NUMBER 13.6 INTEGRATION APPLICATIONS 6 (First moments of n rc) by A.J.Hobson 13.6.1 Introduction 13.6. First moment of n rc bout the yxis 13.6.3 First moment of n rc bout the xxis
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationGeometric and Mechanical Applications of Integrals
5 Geometric nd Mechnicl Applictions of Integrls 5.1 Computing Are 5.1.1 Using Crtesin Coordintes Suppose curve is given by n eqution y = f(x), x b, where f : [, b] R is continuous function such tht f(x)
More informationCHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS
CHAPTER 6 APPLICATIONS OF DEFINITE INTEGRALS 6. VOLUMES USING CROSSSECTIONS. A() ;, ; (digonl) ˆ Èˆ È V A() d d c d 6 (dimeter) c d c d c ˆ 6. A() ;, ; V A() d d. A() (edge) È Š È Š È ;, ; V A() d d 8
More informationImproper Integrals with Infinite Limits of Integration
6_88.qd // : PM Pge 578 578 CHAPTER 8 Integrtion Techniques, L Hôpitl s Rule, nd Improper Integrls Section 8.8 f() = d The unounded region hs n re of. Figure 8.7 Improper Integrls Evlute n improper integrl
More informationTrigonometric Functions
Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds
More informationn=0 ( 1)n /(n + 1) converges, but not n=100 1/n2, is at most 1/100.
Mth 07H Topics since the second exm Note: The finl exm will cover everything from the first two topics sheets, s well. Absolute convergence nd lternting series A series n converges bsolutely if n converges.
More informationMath 211/213 Calculus IIIIV. Directions. Kenneth Massey. September 17, 2018
Mth 211/213 Clculus V Kenneth Mssey CrsonNewmn University September 17, 2018 CN Mth 211  Mssey, 1 / 1 Directions You re t the origin nd giving directions to the point (4, 3). 1. n Mnhttn: go est 4
More informationReview Problem for Midterm #1
Review Problem for Midterm # Midterm I:  :5.m Fridy (Sep. ), Topics: 5.8 nd 6.6. Office Hours before the midterm I: W  pm (Sep 8), Th 5 pm (Sep 9) t UH 8B Solution to quiz cn be found t http://mth.utoledo.edu/
More informationMTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008
MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul
More informationr = cos θ + 1. dt ) dt. (1)
MTHE 7 Proble Set 5 Solutions (A Crdioid). Let C be the closed curve in R whose polr coordintes (r, θ) stisfy () Sketch the curve C. r = cos θ +. (b) Find pretriztion t (r(t), θ(t)), t [, b], of C in polr
More informationFurther integration. x n nx n 1 sinh x cosh x log x 1/x cosh x sinh x e x e x tan x sec 2 x sin x cos x tan 1 x 1/(1 + x 2 ) cos x sin x
Further integrtion Stndrd derivtives nd integrls The following cn be thought of s list of derivtives or eqully (red bckwrds) s list of integrls. Mke sure you know them! There ren t very mny. f(x) f (x)
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationPartial Differential Equations
Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for OneDimensionl Eqution The reen s function provides complete solution to boundry
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationMath 142: Final Exam Formulas to Know
Mth 4: Finl Exm Formuls to Know This ocument tells you every formul/strtegy tht you shoul know in orer to o well on your finl. Stuy it well! The helpful rules/formuls from the vrious review sheets my be
More informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
More informationdt. However, we might also be curious about dy
Section 0. The Clculus of Prmetric Curves Even though curve defined prmetricly my not be function, we cn still consider concepts such s rtes of chnge. However, the concepts will need specil tretment. For
More informationMATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 5 SOLUTIONS. cos t cos at dt + i
MATH 85: COMPLEX ANALYSIS FALL 9/ PROBLEM SET 5 SOLUTIONS. Let R nd z C. () Evlute the following integrls Solution. Since e it cos t nd For the first integrl, we hve e it cos t cos t cos t + i t + i. sin
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationChapter 8: Methods of Integration
Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln
More informationFirst Semester Review Calculus BC
First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewiseliner function f, for 4, is shown below.
More information1 Part II: Numerical Integration
Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble
More informationPROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
More informationMathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
More informationES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1  Thurs 28th Sept 17 Review of trigonometry and basic calculus
ES 111 Mthemticl Methods in the Erth Sciences Lecture Outline 1  Thurs 28th Sept 17 Review of trigonometry nd bsic clculus Trigonometry When is it useful? Everywhere! Anything involving coordinte systems
More informationChapter Direct Method of Interpolation More Examples Electrical Engineering
Chpter. Direct Method of Interpoltion More Emples Electricl Engineering Emple hermistors re used to mesure the temperture of bodies. hermistors re bsed on mterils chnge in resistnce with temperture. o
More informationJUST THE MATHS UNIT NUMBER INTEGRATION APPLICATIONS 8 (First moments of a volume) A.J.Hobson
JUST THE MATHS UNIT NUMBER 3.8 INTEGRATIN APPLICATINS 8 (First moments of volume) b A.J.Hobson 3.8. Introduction 3.8. First moment of volume of revolution bout plne through the origin, perpendiculr to
More informationFundamental Theorem of Calculus
Funmentl Teorem of Clculus Liming Png 1 Sttement of te Teorem Te funmentl Teorem of Clculus is one of te most importnt teorems in te istory of mtemtics, wic ws first iscovere by Newton n Leibniz inepenently.
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions
More information50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS
68 CHAPTE MULTIPLE INTEGALS 46. e da, 49. Evlute tn 3 4 da, where,. [Hint: Eploit the fct tht is the disk with center the origin nd rdius is smmetric with respect to both es.] 5. Use smmetr to evlute 3
More information( β ) touches the xaxis if = 1
Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I  Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without
More informationMath 113 Exam 1Review
Mth 113 Exm 1Review September 26, 2016 Exm 1 covers 6.17.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More informationShow that the curve with parametric equations x t cos t, ; Use a computer to graph the curve with the given vector
7 CHAPTER VECTOR FUNCTIONS. Eercises 2 Find the domin of the vector function.. 2. 3 4 Find the limit. 3. lim cos t, sin t, t ln t t l 4. lim t, e t l rctn 2t, ln t t 5 Mtch the prmetric equtions with the
More informationWhen e = 0 we obtain the case of a circle.
3.4 Conic sections Circles belong to specil clss of cures clle conic sections. Other such cures re the ellipse, prbol, n hyperbol. We will briefly escribe the stnr conics. These re chosen to he simple
More informationEigen Values and Eigen Vectors of a given matrix
Engineering Mthemtics 0 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE : Engineering Mthemtics I : 80/MA : Prolem Mteril : JM08AM00 (Scn the ove QR code for the direct downlod of this mteril) Nme
More informationCHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx
CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twicedifferentile function of x, then t
More informationYear 12 Mathematics Extension 2 HSC Trial Examination 2014
Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bordpproved clcultors my be used A tble of
More information13.4. Integration by Parts. Introduction. Prerequisites. Learning Outcomes
Integrtion by Prts 13.4 Introduction Integrtion by Prts is technique for integrting products of functions. In this Section you will lern to recognise when it is pproprite to use the technique nd hve the
More informationFinal Exam Review. [Top Bottom]dx =
Finl Exm Review Are Between Curves See 7.1 exmples 1, 2, 4, 5 nd exerises 133 (odd) The re of the region bounded by the urves y = f(x), y = g(x), nd the lines x = nd x = b, where f nd g re ontinuous nd
More informationl 2 p2 n 4n 2, the total surface area of the
Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n nsided regulr polygon of perimeter p n with vertices on C. Form cone
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationThe Fundamental Theorem of Calculus Part 2, The Evaluation Part
AP Clculus AB 6.4 Funmentl Theorem of Clculus The Funmentl Theorem of Clculus hs two prts. These two prts tie together the concept of integrtion n ifferentition n is regre by some to by the most importnt
More informationf(a+h) f(a) x a h 0. This is the rate at which
M408S Concept Inventory smple nswers These questions re openended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnkoutnnswer problems! (There re plenty of those in the
More informationNATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics Semester 1, 2002/2003 MA1505 Math I Suggested Solutions to T. 3
NATIONAL UNIVERSITY OF SINGAPORE Deprtment of Mthemtics Semester, /3 MA55 Mth I Suggested Solutions to T. 3. Using the substitution method, or otherwise, find the following integrls. Solution. ) b) x sin(x
More information