Area of a Region Between Two Curves
|
|
- Arron Gibson
- 6 years ago
- Views:
Transcription
1 6 CHAPTER 7 Applictions o Intertion Section 7 Are o Reion Between Two Curves Find the re o reion etween two curves usin intertion Find the re o reion etween intersectin curves usin intertion Descrie intertion s n ccumultion process Are o Reion Between Two Curves Reion etween two curves With ew modiictions ou cn etend the ppliction o deinite interls rom the re o reion under curve to the re o reion etween two curves Consider two unctions nd tht re continuous on the intervl, I, s in Fiure 7, the rphs o oth nd lie ove the -is, nd the rph o lies elow the rph o, ou cn eometricll interpret the re o the reion etween the rphs s the re o the reion under the rph o sutrcted rom the re o the reion under the rph o, s shown in Fiure 7 = = Fiure 7 Are o reion etween nd d Fiure 7 Are o reion under d Are o reion under d Animtion ( i ) Fiure 7 Representtive rectnle Heiht: ( i ) ( i ) Width: i ( i ) To veri the resonleness o the result shown in Fiure 7, ou cn prtition the intervl, into n suintervls, ech o width Then, s shown in Fiure 7, sketch representtive rectnle o width nd heiht i i, where i is in the ith intervl The re o this representtive rectnle is A i heihtwidth i i B ddin the res o the n rectnles nd tkin the limit s n, ou otin lim n n i i i Becuse nd re continuous on,, is lso continuous on, nd the limit eists So, the re o the iven reion is Are lim n n i i i d
2 SECTION 7 Are o Reion Between Two Curves 7 Are o Reion Between Two Curves I nd re continuous on, nd or ll in,, then the re o the reion ounded the rphs o nd nd the verticl lines nd is A d In Fiure 7, the rphs o nd re shown ove the -is This, however, is not necessr The sme internd cn e used s lon s nd re continuous nd or ll in the intervl, This result is summrized rphicll in Fiure 7 () () (, ()) () () (, ()) (, ()) (, ()) NOTE The heiht o representtive rectnle is rerdless o the reltive position o the -is, s shown in Fiure 7 Fiure 7 Representtive rectnles re used throuhout this chpter in vrious pplictions o intertion A verticl rectnle o width implies intertion with respect to, wheres horizontl rectnle o width implies intertion with respect to EXAMPLE Findin the Are o Reion Between Two Curves () = + (, ()) (, ()) () = Reion ounded the rph o, the rph o,, nd Fiure 7 Find the re o the reion ounded the rphs o,,, nd Solution Let nd Then or ll in,, s shown in Fiure 7 So, the re o the representtive rectnle is A nd the re o the reion is A d d 7 6 Editle Grph Tr It Eplortion A Eplortion B
3 8 CHAPTER 7 Applictions o Intertion Are o Reion Between Intersectin Curves In Emple, the rphs o nd do not intersect, nd the vlues o nd re iven eplicitl A more common prolem involves the re o reion ounded two intersectin rphs, where the vlues o nd must e clculted EXAMPLE A Reion Lin Between Two Intersectin Grphs Find the re o the reion ounded the rphs o nd (, ()) () = () = (, ()) Reion ounded the rph o nd the rph o Fiure 76 Solution In Fiure 76, notice tht the rphs o nd hve two points o intersection To ind the -coordintes o these points, set nd equl to ech other nd solve or Set equl to Write in enerl orm Fctor or Solve or So, nd Becuse or ll in the intervl,, the representtive rectnle hs n re o A nd the re o the reion is A d 9 Editle Grph Tr It EXAMPLE Eplortion A A Reion Lin Between Two Intersectin Grphs The sine nd cosine curves intersect ininitel mn times, oundin reions o equl res, s shown in Fiure 77 Find the re o one o these reions π (, ()) (, ()) π π () = cos () = sin One o the reions ounded the rphs o the sine nd cosine unctions Fiure 77 Solution sin cos sin cos Set equl to Divide ech side cos tn or, Trionometric identit Solve or So, nd Becuse sin cos or ll in the intervl,, the re o the reion is A sin cos d cos sin Editle Grph Tr It Eplortion A
4 SECTION 7 Are o Reion Between Two Curves 9 I two curves intersect t more thn two points, then to ind the re o the reion etween the curves, ou must ind ll points o intersection nd check to see which curve is ove the other in ech intervl determined these points EXAMPLE Curves Tht Intersect t More Thn Two Points Find the re o the reion etween the rphs o nd () () () () 6 (, ) (, ) 6 (, 8) 8 () = + () = On,,, nd on,, Fiure 78 Editle Grph Solution Bein settin nd equl to ech other nd solvin or This ields the -vlues t ech point o intersection o the two rphs,, 6 6 Set equl to Write in enerl orm Fctor Solve or So, the two rphs intersect when,, nd In Fiure 78, notice tht on the intervl, However, the two rphs switch t the oriin, nd on the intervl, So, ou need two interls one or the intervl, nd one or the intervl, A d d d d Tr It Eplortion A Open Eplortion NOTE In Emple, notice tht ou otin n incorrect result i ou interte rom to Such intertion produces d d I the rph o unction o is oundr o reion, it is oten convenient to use representtive rectnles tht re horizontl nd ind the re intertin with respect to In enerl, to determine the re etween two curves, ou cn use A top curve ottom curve d Verticl rectnles in vrile A riht curve let curve d Horizontl rectnles in vrile where, nd, re either djcent points o intersection o the two curves involved or points on the speciied oundr lines Technolo
5 CHAPTER 7 Applictions o Intertion EXAMPLE Horizontl Representtive Rectnles Find the re o the reion ounded the rphs o nd Solution Consider nd These two curves intersect when nd, s shown in Fiure 79 Becuse on this intervl, ou hve A So, the re is A d d 8 9 Tr It Eplortion A () = + (, ) = (, ) = (, ) () = Horizontl rectnles (intertion with respect to ) Fiure 79 Editle Grph (, ) = Verticl rectnles (intertion with respect to ) Fiure 7 In Emple, notice tht intertin with respect to ou need onl one interl I ou hd interted with respect to, ou would hve needed two interls ecuse the upper oundr would hve chned t, s shown in Fiure 7 A d d 6 d d 9
6 SECTION 7 Are o Reion Between Two Curves Intertion s n Accumultion Process In this section, the intertion ormul or the re etween two curves ws developed usin rectnle s the representtive element For ech new ppliction in the reminin sections o this chpter, n pproprite representtive element will e constructed usin preclculus ormuls ou lred know Ech intertion ormul will then e otined summin or ccumultin these representtive elements Known preclculus ormul Representtive element New intertion ormul For emple, in this section the re ormul ws developed s ollows A heihtwidth A A d EXAMPLE 6 Descriin Intertion s n Accumultion Process Find the re o the reion ounded the rph o nd the -is Descrie the intertion s n ccumultion process Solution The re o the reion is iven A d You cn think o the intertion s n ccumultion o the res o the rectnles ormed s the representtive rectnle slides rom to, s shown in Fiure 7 A d A d A d 6 A d 9 Fiure 7 A d Tr It Eplortion A
7 CHAPTER 7 Applictions o Intertion Eercises or Section 7 The smol Click on Click on indictes n eercise in which ou re instructed to use rphin technolo or smolic computer ler sstem to view the complete solution o the eercise to print n enlred cop o the rph In Eercises 6, set up the deinite interl tht ives the re o the reion In Eercises nd, ind the re o the reion intertin () with respect to nd () with respect to Think Aout It In Eercises nd 6, determine which vlue est pproimtes the re o the reion ounded the rphs o nd (Mke our selection on the sis o sketch o the reion nd not perormin n clcultions) In Eercises 7, the internd o the deinite interl is dierence o two unctions Sketch the rph o ech unction nd shde the reion whose re is represented the interl 7 8 d d sec d d d sec cos d, () () (c) (d) (e) 8 6, () () 6 (c) (d) (e) In Eercises 7, sketch the reion ounded the rphs o the leric unctions nd ind the re o the reion 7,,, 8 8 8,,, 8 9,,,,,,, 6, 7, 8, 9,,,,,,,, 6,,,,,
8 SECTION 7 Are o Reion Between Two Curves In Eercises, () use rphin utilit to rph the reion ounded the rphs o the equtions, () ind the re o the reion, nd (c) use the intertion cpilities o the rphin utilit to veri our results,,,, 6, 7, 8, 9, 6,,,,,, In Eercises 8, sketch the reion ounded the rphs o the unctions, nd ind the re o the reion sin, tn, sin, cos, 6 cos, cos, 6 In Eercises 9, () use rphin utilit to rph the reion ounded the rphs o the equtions, () ind the re o the reion, nd (c) use the intertion cpilities o the rphin utilit to veri our results In Eercises 6, () use rphin utilit to rph the reion ounded the rphs o the equtions, () eplin wh the re o the reion is diicult to ind hnd, nd (c) use the intertion cpilities o the rphin utilit to pproimte the re to our deciml plces sec tn 7 e,, 8, 9 sin sin,, sin cos,, < e,, ln,,,, e,,, 6,, cos,, In Eercises 7 6, ind the ccumultion unction F Then evlute F t ech vlue o the independent vrile nd rphicll show the re iven ech vlue o F 7 F t dt () F () F (c) F6 8 F t dt () F () F (c) F6 9 F cos () F () F (c) F d 6 F e d () F () F (c) F In Eercises 6 6, use intertion to ind the re o the iure hvin the iven vertices 6,,, 6, 6, 6,,,,, c 6 6,,,,,,,,,,,,,, 6 Numericl Intertion Estimte the surce re o the ol reen usin () the Trpezoidl Rule nd () Simpson s Rule t t 66 Numericl Intertion Estimte the surce re o the oil spill usin () the Trpezoidl Rule nd () Simpson s Rule mi t mi t mi t mi t In Eercises 67 7, set up nd evlute the deinite interl tht ives the re o the reion ounded the rph o the unction nd the tnent line to the rph t the iven point mi t mi t 6 t mi mi 6 t 67,, 68, 69 7,, Writin Aout Concepts,,, 7 The rphs o nd intersect t three points However, the re etween the curves cn e ound sinle interl Eplin wh this is so, nd write n interl or this re
9 CHAPTER 7 Applictions o Intertion Writin Aout Concepts (continued) 7 The re o the reion ounded the rphs o nd cnnot e ound the sinle interl d Eplin wh this is so Use smmetr to write sinle interl tht does represent the re 7 A collee rdute hs two jo oers The strtin slr or ech is $,, nd ter 8 ers o service ech will p $, The slr increse or ech oer is shown in the iure From strictl monetr viewpoint, which is the etter oer? Eplin Slr (in dollrs) 6,,,,,, S Fiure or 7 Fiure or 7 7 A stte leislture is detin two proposls or elimintin the nnul udet deicits the er The rte o decrese o the deicits or ech proposl is shown in the iure From the viewpoint o minimizin the cumultive stte deicit, which is the etter proposl? Eplin In Eercises 7 nd 76, ind such tht the line divides the reion ounded the rphs o the two equtions into two reions o equl re 7 9, 76 In Eercises 77 nd 78, ind such tht the line divides the reion ounded the rphs o the equtions into two reions o equl re 77,, 78, In Eercises 79 nd 8, evlute the limit nd sketch the rph o the reion whose re is represented the limit 79 lim i i, where i in nd n n i 8 lim i, where i in nd n n i Oer Oer 6 8 Yer t Deicit (in illions o dollrs) 9, Revenue In Eercises 8 nd 8, two models nd re iven or revenue (in illions o dollrs per er) or lre corportion The model R ives projected nnul revenues rom to, with t correspondin to, nd R ives projected revenues i there is decrese in the rte o rowth o corporte sles over the period Approimte the totl reduction in revenue i corporte sles re ctull closer to the model R 6 D Proposl R Proposl t 6 Yer R 8 R 8 R 7 6t t 7 8t R 7 t R 7 t t 8 Modelin Dt The tle shows the totl receipts R nd totl ependitures E or the Old-Ae nd Survivors Insurnce Trust Fund (Socil Securit Trust Fund) in illions o dollrs The time t is iven in ers, with t correspondin to 99 (Source: Socil Securit Administrtion) t R E t R E () Use rphin utilit to it n eponentil model to the dt or receipts Plot the dt nd rph the model () Use rphin utilit to it n eponentil model to the dt or ependitures Plot the dt nd rph the model (c) I the models re ssumed to e true or the ers throuh 7, use intertion to pproimte the surplus revenue enerted durin those ers (d) Will the models ound in prts () nd () intersect? Eplin Bsed on our nswer nd news reports out the und, will these models e ccurte or lon-term nlsis? 8 Lorenz Curve Economists use Lorenz curves to illustrte the distriution o income in countr A Lorenz curve,, represents the ctul income distriution in the countr In this model, represents percents o milies in the countr nd represents percents o totl income The model represents countr in which ech mil hs the sme income The re etween these two models, where, indictes countr s income inequlit The tle lists percents o income or selected percents o milies in countr () Use rphin utilit to ind qudrtic model or the Lorenz curve () Plot the dt nd rph the model (c) Grph the model How does this model compre with the model in prt ()? (d) Use the intertion cpilities o rphin utilit to pproimte the income inequlit
10 SECTION 7 Are o Reion Between Two Curves 8 Proit The chie inncil oicer o compn reports tht proits or the pst iscl er were $89, The oicer predicts tht proits or the net ers will row t continuous nnul rte somewhere etween % nd % Estimte the cumultive dierence in totl proit over the ers sed on the predicted rne o rowth rtes 86 Are The shded reion in the iure consists o ll points whose distnces rom the center o the squre re less thn their distnces rom the edes o the squre Find the re o the reion Fiure or 86 Fiure or Mechnicl Desin The surce o mchine prt is the reion etween the rphs o nd 8 k (see iure) () Find k i the prol is tnent to the rph o () Find the re o the surce o the mchine prt 88 Buildin Desin Concrete sections or new uildin hve the dimensions (in meters) nd shpe shown in the iure (, ) = 6 () Find the re o the ce o the section superimposed on the rectnulr coordinte sstem () Find the volume o concrete in one o the sections multiplin the re in prt () meters (c) One cuic meter o concrete weihs pounds Find the weiht o the section 89 Buildin Desin To decrese the weiht nd to id in the hrdenin process, the concrete sections in Eercise 88 oten re not solid Rework Eercise 88 to llow or clindricl openins such s those shown in the iure m 8 m + Rottle Grph m 6 = (, ) True or Flse? In Eercises 9 9, determine whether the sttement is true or lse I it is lse, eplin wh or ive n emple tht shows it is lse 9 I the re o the reion ounded the rphs o nd is, then the re o the reion ounded the rphs o h C nd k C is lso 9 I d A, then d A 9 I the rphs o nd intersect midw etween nd, then d 9 Are Find the re etween the rph o sin nd the line sements joinin the points, nd 7 s 6,, shown in the iure (, ) 6 π Fiure or 9 Fiure or 9 9 Are Let > nd > Show tht the re o the ellipse 7 6 π (, is (see iure) ( π + Putnm Em Chllene 9 The horizontl line c intersects the curve in the irst qudrnt s shown in the iure Find c so tht the res o the two shded reions re equl = c = This prolem ws composed the Committee on the Putnm Prize Competition The Mthemticl Assocition o Americ All rihts reserved = m 6 (, ) = + 6 = (, ) Rottle Grph
Area of a Region Between Two Curves
6 CHAPTER 7 Applictions of Integrtion Section 7 Are of Region Between Two Curves Find the re of region etween two curves using integrtion Find the re of region etween intersecting curves using integrtion
More informationThe Trapezoidal Rule
_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion
More information7Applications of. Integration
7Applictions of Integrtion The Atomium, locted in Belgium, represents n iron crstl molecule mgnified 65 illion times. The structure contins nine spheres connected with clindricl tues. The centrl sphere
More informationThe Trapezoidal Rule
SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion Approimte
More informationReview Exercises for Chapter 4
_R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises
More informationArc Length and Surfaces of Revolution. Find the arc length of a smooth curve. Find the area of a surface of revolution. <...
76 CHAPTER 7 Applictions of Integrtion The Dutch mthemticin Christin Hugens, who invented the pendulum clock, nd Jmes Gregor (6 675), Scottish mthemticin, oth mde erl contriutions to the prolem of finding
More information4.6 Numerical Integration
.6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationPaul s Notes. Chapter Planning Guide
Applictions of Integrtion. Are of Region Between Two Curves. Volume: The Disk nd Wsher Methods. Volume: The Shell Method. Arc Length nd Surfces of Revolution Roof Are (Eercise, p. ) Sturn (Section Project,
More informationChapter 6 Continuous Random Variables and Distributions
Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete
More informationChapter 3 Exponential and Logarithmic Functions Section 3.1
Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re non-lgebric functions. The re clled trnscendentl functions. The eponentil
More informationNAME: MR. WAIN FUNCTIONS
NAME: M. WAIN FUNCTIONS evision o Solving Polnomil Equtions i one term in Emples Solve: 7 7 7 0 0 7 b.9 c 7 7 7 7 ii more thn one term in Method: Get the right hnd side to equl zero = 0 Eliminte ll denomintors
More informationChapter 3 Single Random Variables and Probability Distributions (Part 2)
Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their
More informationTime in Seconds Speed in ft/sec (a) Sketch a possible graph for this function.
4. Are under Curve A cr is trveling so tht its speed is never decresing during 1-second intervl. The speed t vrious moments in time is listed in the tle elow. Time in Seconds 3 6 9 1 Speed in t/sec 3 37
More informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More information7.8 IMPROPER INTEGRALS
7.8 Improper Integrls 547 the grph of g psses through the points (, ), (, ), nd (, ); the grph of g psses through the points (, ), ( 3, 3 ), nd ( 4, 4 );... the grph of g n/ psses through the points (
More information5.1 Estimating with Finite Sums Calculus
5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during
More information3.1 Exponential Functions and Their Graphs
. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.
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 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 information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationPrecalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.
Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse
More information5.1 How do we Measure Distance Traveled given Velocity? Student Notes
. How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the x-xis & y-xis
More informationBelievethatyoucandoitandyouar. Mathematics. ngascannotdoonlynotyetbelieve thatyoucandoitandyouarehalfw. Algebra
Believethtoucndoitndour ehlfwtherethereisnosuchthi Mthemtics ngscnnotdoonlnotetbelieve thtoucndoitndourehlfw Alger therethereisnosuchthingsc nnotdoonlnotetbelievethto Stge 6 ucndoitndourehlfwther S Cooper
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 informationC Precalculus Review. C.1 Real Numbers and the Real Number Line. Real Numbers and the Real Number Line
C. Rel Numers nd the Rel Numer Line C C Preclculus Review C. Rel Numers nd the Rel Numer Line Represent nd clssif rel numers. Order rel numers nd use inequlities. Find the solute vlues of rel numers nd
More informationWhat Is Calculus? 42 CHAPTER 1 Limits and Their Properties
60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties The Mistress Fellows, Girton College, Cmridge Section. STUDY TIP As ou progress through this course, rememer tht lerning clculus is just one of
More information0.1 THE REAL NUMBER LINE AND ORDER
6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.
More information10.2 The Ellipse and the Hyperbola
CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point
More informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show
More informationTO: Next Year s AP Calculus Students
TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC
More informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in
More informationSection 7.1 Area of a Region Between Two Curves
Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More informationCalculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION
lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..
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 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 informationKEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a
KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider
More informationUnit 1 Exponentials and Logarithms
HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)
More informationM344 - ADVANCED ENGINEERING MATHEMATICS
M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More information(i) b b. (ii) (iii) (vi) b. P a g e Exponential Functions 1. Properties of Exponents: Ex1. Solve the following equation
P g e 30 4.2 Eponentil Functions 1. Properties of Eponents: (i) (iii) (iv) (v) (vi) 1 If 1, 0 1, nd 1, then E1. Solve the following eqution 4 3. 1 2 89 8(2 ) 7 Definition: The eponentil function with se
More information13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes
The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce
More informationGrade 10 Math Academic Levels (MPM2D) Unit 4 Quadratic Relations
Grde 10 Mth Acdemic Levels (MPMD) Unit Qudrtic Reltions Topics Homework Tet ook Worksheet D 1 Qudrtic Reltions in Verte Qudrtic Reltions in Verte Form (Trnsltions) Form (Trnsltions) D Qudrtic Reltions
More information7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
More informationPrerequisites CHAPTER P
CHAPTER P Prerequisites P. Rel Numers P.2 Crtesin Coordinte System P.3 Liner Equtions nd Inequlities P.4 Lines in the Plne P.5 Solving Equtions Grphiclly, Numericlly, nd Algericlly P.6 Comple Numers P.7
More informationR(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
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 informationChapter 2. Random Variables and Probability Distributions
Rndom Vriles nd Proilit Distriutions- 6 Chpter. Rndom Vriles nd Proilit Distriutions.. Introduction In the previous chpter, we introduced common topics of proilit. In this chpter, we trnslte those concepts
More informationMath 259 Winter Solutions to Homework #9
Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658-659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier
More informationMath 017. Materials With Exercises
Mth 07 Mterils With Eercises Jul 0 TABLE OF CONTENTS Lesson Vriles nd lgeric epressions; Evlution of lgeric epressions... Lesson Algeric epressions nd their evlutions; Order of opertions....... Lesson
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More information13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes
The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce
More information8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
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 piecewise-liner function f, for 4, is shown below.
More information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
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 informationy = f(x) This means that there must be a point, c, where the Figure 1
Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile
More informationCh AP Problems
Ch. 7.-7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,
More informationx dx does exist, what does the answer look like? What does the answer to
Review Guie or MAT Finl Em Prt II. Mony Decemer th 8:.m. 9:5.m. (or the 8:3.m. clss) :.m. :5.m. (or the :3.m. clss) Prt is worth 5% o your Finl Em gre. NO CALCULATORS re llowe on this portion o the Finl
More informationAPPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line
APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente
More informationChapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1
Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show
More informationInstantaneous Rate of Change of at a :
AP Clculus AB Formuls & Justiictions Averge Rte o Chnge o on [, ]:.r.c. = ( ) ( ) (lger slope o Deinition o the Derivtive: y ) (slope o secnt line) ( h) ( ) ( ) ( ) '( ) lim lim h0 h 0 3 ( ) ( ) '( ) lim
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
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 informationThe practical version
Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht
More informationAP Calculus AB Unit 5 (Ch. 6): The Definite Integral: Day 12 Chapter 6 Review
AP Clculus AB Unit 5 (Ch. 6): The Definite Integrl: Dy Nme o Are Approximtions Riemnn Sums: LRAM, MRAM, RRAM Chpter 6 Review Trpezoidl Rule: T = h ( y + y + y +!+ y + y 0 n n) **Know how to find rectngle
More informationIntroduction to Algebra - Part 2
Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite
More informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More informationONLINE PAGE PROOFS. Anti-differentiation and introduction to integral calculus
Anti-differentition nd introduction to integrl clculus. Kick off with CAS. Anti-derivtives. Anti-derivtive functions nd grphs. Applictions of nti-differentition.5 The definite integrl.6 Review . Kick off
More informationUnit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NON-CALCULATOR SECTION
Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NON-CALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)
More informationIntegration. 8.1 Kick off with CAS 8.2 The fundamental theorem of integral calculus 8.3 Areas under curves 8.4 Applications 8.
8 Integrtion 8. Kick off with CAS 8. The fundmentl theorem of integrl clculus 8. Ares under curves 8. Applictions 8.5 Review 8. Kick off with CAS Are under curves Using the grph ppliction on CAS, sketch
More informationONLINE PAGE PROOFS. Integration. 8.1 Kick off with CAS
8. Kick off with CAS Integrtion 8. The fundmentl theorem of integrl clculus 8. Ares under curves 8. Applictions 8.5 Review 8 8. Kick off with CAS Are under curves Using the grph ppliction on CAS, sketch
More informationCalculus AB. For a function f(x), the derivative would be f '(
lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:
More informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationNat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS
Nt USAP This ooklet contins : Questions on Topics covered in RHS USAP Em Tpe Questions Answers Sourced from PEGASYS USAP EF. Reducing n lgeric epression to its simplest form / where nd re of the form (
More informationMA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1
MA 15910, Lessons nd Introduction to Functions Alger: Sections 3.5 nd 7.4 Clculus: Sections 1. nd.1 Representing n Intervl Set of Numers Inequlity Symol Numer Line Grph Intervl Nottion ) (, ) ( (, ) ]
More informationSECTION 9-4 Translation of Axes
9-4 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationFUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y
FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = (- 1)(- ) The hperbol: p q The eponentil function: b p q Importnt fetures: -intercept : Let = 0 -intercept : Let = 0 Turning points (Where pplicble)
More informationIntroduction. Definition of Hyperbola
Section 10.4 Hperbols 751 10.4 HYPERBOLAS Wht ou should lern Write equtions of hperbols in stndrd form. Find smptotes of nd grph hperbols. Use properties of hperbols to solve rel-life problems. Clssif
More informationx ) dx dx x sec x over the interval (, ).
Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor
More informationThe semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer.
ALGEBRA B Semester Em Review The semester B emintion for Algebr will consist of two prts. Prt will be selected response. Prt will be short nswer. Students m use clcultor. If clcultor is used to find points
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
More informationDA 3: The Mean Value Theorem
Differentition pplictions 3: The Men Vlue Theorem 169 D 3: The Men Vlue Theorem Model 1: Pennslvni Turnpike You re trveling est on the Pennslvni Turnpike You note the time s ou pss the Lenon/Lncster Eit
More informationLinear Approximation and the Fundamental Theorem of Calculus
Mth 3A Discussion Session Week 9 Notes Mrch nd 3, 26 Liner Approimtion nd the Fundmentl Theorem of Clculus We hve three primry ols in tody s discussion of the fundmentl theorem of clculus. By the end of
More informationProceedings of the International Conference on Theory and Applications of Mathematics and Informatics ICTAMI 2003, Alba Iulia
Proceedings o the Interntionl Conerence on Theor nd Applictions o Mthemtics nd Inormtics ICTAMI 2003, Al Iuli CARACTERIZATIONS OF TE FUNCTIONS WIT BOUNDED VARIATION Dniel Lesnic Astrct. The present stud
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More information3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression.
SECTION. Eponents nd Rdicls 7 B 7 7 7 7 7 7 7 NOW TRY EXERCISES 89 AND 9 7. EXERCISES CONCEPTS. () Using eponentil nottion, we cn write the product s. In the epression 3 4,the numer 3 is clled the, nd
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 information