# The Trapezoidal Rule

Size: px
Start display at page:

Transcription

1 _.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 Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson s Rule. The Trpezoidl Rule Some elementr functions simpl do not hve ntiderivtives tht re elementr functions. For emple, there is no elementr function tht hs n of the following functions s its derivtive., cos, If ou need to evlute definite integrl involving function whose ntiderivtive cnnot e found, the Fundmentl Theorem of Clculus cnnot e pplied, nd ou must resort to n pproimtion technique. Two such techniques re descried in this section. One w to pproimte definite integrl is to use n trpezoids, s shown in Figure.. In the development of this method, ssume tht f is continuous nd positive on the intervl,. So, the definite integrl f d represents the re of the region ounded the grph of f nd the -is, from to. First, prtition the intervl, into n suintervls, ech of width n, such tht < < <... < n. Then form trpezoid for ech suintervl (see Figure.). The re of the ith trpezoid is Are of i th trpezoid This implies tht the sum of the res of the n trpezoids is Are n f f... f n f n n n Letting n, ou cn tke the limit s n to otin lim n n f f... f n f n f f lim n n f i i f f lim lim n n n n f i i f d. f i f i n. f f f f... f n f n f f f... f n f n. The result is summrized in the following theorem. cos,, sin

2 _.qd // : PM Pge CHAPTER Integrtion THEOREM. The Trpezoidl Rule Let f e continuous on,. The Trpezoidl Rule for pproimting f d is given f d n f f f... f n f n. Moreover, s n, the right-hnd side pproches f d. NOTE Oserve tht the coefficients in the Trpezoidl Rule hve the following pttern.... EXAMPLE Approimtion with the Trpezoidl Rule = sin Use the Trpezoidl Rule to pproimte sin d. Compre the results for n nd n, s shown in Figure.. Four suintervls = sin Trpezoidl pproimtions Figure. 5 7 Eight suintervls Solution When n,, nd ou otin When n,, nd ou otin sin d sin d sin sin sin sin sin sin sin sin sin 5 sin 7 sin sin sin sin.97. For this prticulr integrl, ou could hve found n ntiderivtive nd determined tht the ect re of the region is..9. sin sin TECHNOLOGY Most grphing utilities nd computer lger sstems hve uilt-in progrms tht cn e used to pproimte the vlue of definite integrl. Tr using such progrm to pproimte the integrl in Emple. How close is our pproimtion? When ou use such progrm, ou need to e wre of its limittions. Often, ou re given no indiction of the degree of ccurc of the pproimtion. Other times, ou m e given n pproimtion tht is completel wrong. For instnce, tr using uilt-in numericl integrtion progrm to evlute d. Your clcultor should give n error messge. Does ours?

3 _.qd // : PM Pge SECTION. Numericl Integrtion It is interesting to compre the Trpezoidl Rule with the Midpoint Rule given in Section. (Eercises ). For the Trpezoidl Rule, ou verge the function vlues t the endpoints of the suintervls, ut for the Midpoint Rule ou tke the function vlues of the suintervl midpoints. f d n i f d n f i i i f i f i Midpoint Rule Trpezoidl Rule NOTE There re two importnt points tht should e mde concerning the Trpezoidl Rule (or the Midpoint Rule). First, the pproimtion tends to ecome more ccurte s n increses. For instnce, in Emple, if n, the Trpezoidl Rule ields n pproimtion of.99. Second, lthough ou could hve used the Fundmentl Theorem to evlute the integrl in Emple, this theorem cnnot e used to evlute n integrl s simple s ecuse sin sin d hs no elementr ntiderivtive. Yet, the Trpezoidl Rule cn e pplied esil to this integrl. Simpson s Rule One w to view the trpezoidl pproimtion of definite integrl is to s tht on ech suintervl ou pproimte f first-degree polnomil. In Simpson s Rule, nmed fter the English mthemticin Thoms Simpson (7 7), ou tke this procedure one step further nd pproimte f second-degree polnomils. Before presenting Simpson s Rule, we list theorem for evluting integrls of polnomils of degree (or less). THEOREM.7 Integrl of p A B C If p A B C, then p d p p p. Proof p d A B C d B epnsion nd collection of terms, the epression inside the rckets ecomes A B C A B C A B C p p p d p p p. nd ou cn write A A B C B C A B C p

4 _.qd // : PM Pge CHAPTER Integrtion p (, ) (, ) f To develop Simpson s Rule for pproimting definite integrl, ou gin prtition the intervl, into n suintervls, ech of width n. This time, however, n is required to e even, nd the suintervls re grouped in pirs such tht < < < < <... < n < n < n.,, n, n (, ) Figure. n p d f d On ech (doule) suintervl i, i, ou cn pproimte f polnomil p of degree less thn or equl to. (See Eercise 55.) For emple, on the suintervl,, choose the polnomil of lest degree pssing through the points,,,, nd,, s shown in Figure.. Now, using p s n pproimtion of f on this suintervl, ou hve, Theorem.7, f d p d p p n n p p p f f f. p Repeting this procedure on the entire intervl, produces the following theorem. THEOREM. Simpson s Rule (n is even) Let f e continuous on,. Simpson s Rule for pproimting f d is f d n f f f f... f n f n. Moreover, s n, the right-hnd side pproches f d. NOTE Oserve tht the coefficients in Simpson s Rule hve the following pttern.... In Emple, the Trpezoidl Rule ws used to estimte emple, Simpson s Rule is pplied to the sme integrl. sin d. In the net EXAMPLE Approimtion with Simpson s Rule NOTE In Emple, the Trpezoidl Rule with n pproimted sin d s.97. In Emple, Simpson s Rule with n gve n pproimtion of.. The ntiderivtive would produce the true vlue of. Use Simpson s Rule to pproimte Compre the results for n nd n. Solution When n, ou hve sin d. sin d sin sin sin When n, ou hve sin d.. sin sin.5.

5 _.qd // : PM Pge SECTION. Numericl Integrtion Error Anlsis If ou must use n pproimtion technique, it is importnt to know how ccurte ou cn epect the pproimtion to e. The following theorem, which is listed without proof, gives the formuls for estimting the errors involved in the use of Simpson s Rule nd the Trpezoidl Rule. THEOREM.9 Errors in the Trpezoidl Rule nd Simpson s Rule If f hs continuous second derivtive on,, then the error E in pproimting f d the Trpezoidl Rule is E n m f, Trpezoidl Rule Moreover, if f hs continuous fourth derivtive on,, then the error E in pproimting f d Simpson s Rule is. E 5 n m f,. Simpson s Rule TECHNOLOGY If ou hve ccess to computer lger sstem, use it to evlute the definite integrl in Emple. You should otin vlue of d ln.779. ( ln represents the nturl logrithmic function, which ou will stud in Section 5..) = +. d. Figure.5 n = Theorem.9 sttes tht the errors generted the Trpezoidl Rule nd Simpson s Rule hve upper ounds dependent on the etreme vlues of f nd f in the intervl,. Furthermore, these errors cn e mde ritrril smll incresing n, provided tht re continuous nd therefore ounded in,. EXAMPLE The Approimte Error in the Trpezoidl Rule Determine vlue of n such tht the Trpezoidl Rule will pproimte the vlue of d with n error tht is less thn.. Solution Begin letting f nd finding the second derivtive of f. f nd f The mimum vlue of f on the intervl, is f. So, Theorem.9, ou cn write E n f n n. To otin n error E tht is less thn., ou must choose n such tht n. n So, ou cn choose n (ecuse n must e greter thn or equl to.9) nd ppl the Trpezoidl Rule, s shown in Figure.5, to otin d.5. f nd f n.9 So, with n error no lrger thn., ou know tht. d..

6 _.qd // : PM Pge CHAPTER Integrtion In Eercises, use the Trpezoidl Rule nd Simpson s Rule to pproimte the vlue of the definite integrl for the given vlue of n. Round our nswer to four deciml plces nd compre the results with the ect vlue of the definite integrl.. d, n.. d, n. 5. d, n. 7. d, n. 9. d, n. In Eercises, pproimte the definite integrl using the Trpezoidl Rule nd Simpson s Rule with n. Compre these results with the pproimtion of the integrl using grphing utilit.. d.. d. 5. cos d. 7. sin d. 9.. Eercises for Section. 9 In Eercises, use the error formuls in Theorem.9 to estimte the error in pproimting the integrl, with n, using () the Trpezoidl Rule nd () Simpson s Rule.. tn d f d, f sin,,. d. > Writing Aout Concepts d, n d, n d, n d, n d, n d sin d tn d cos d. If the function f is concve upwrd on the intervl,, will the Trpezoidl Rule ield result greter thn or less thn f d? Eplin.. The Trpezoidl Rule nd Simpson s Rule ield pproimtions of definite integrl f d sed on polnomil pproimtions of f. Wht degree polnomil is used for ech? d 5.. d 7. cos d. In Eercises 9, use the error formuls in Theorem.9 to find n such tht the error in the pproimtion of the definite integrl is less thn. using () the Trpezoidl Rule nd () Simpson s Rule. 9.. d. d.. cos d. In Eercises 5, use computer lger sstem nd the error formuls to find n such tht the error in the pproimtion of the definite integrl is less thn. using () the Trpezoidl Rule nd () Simpson s Rule. 5. d. 7. tn d. 9. Approimte the re of the shded region using () the Trpezoidl Rule nd () Simpson s Rule with n. See for worked-out solutions to odd-numered eercises. 5 Figure for 9 Figure for. Approimte the re of the shded region using () the Trpezoidl Rule nd () Simpson s Rule with n.. Progrmming Write progrm for grphing utilit to pproimte definite integrl using the Trpezoidl Rule nd Simpson s Rule. Strt with the progrm written in Section., Eercises 59, nd note tht the Trpezoidl Rule cn e written s Tn Ln Rn nd Simpson s Rule cn e written s Sn Tn Mn. [Recll tht Ln, Mn, nd Rn represent the Riemnn sums using the left-hnd endpoints, midpoints, nd right-hnd endpoints of suintervls of equl width.] d sin d d d sin d d sin d

7 _.qd // : PM Pge 5 SECTION. Numericl Integrtion 5 Progrmming In Eercises, use the progrm in Eercise to pproimte the definite integrl nd complete the tle. n Ln. d. d. 5. Are Use Simpson s Rule with n to pproimte the re of the region ounded the grphs of cos,,, nd.. Circumference The elliptic integrl sin d gives the circumference of n ellipse. Use Simpson s Rule with n to pproimte the circumference. 7. Work To determine the size of the motor required to operte press, compn must know the mount of work done when the press moves n oject linerl 5 feet. The vrile force to move the oject is F 5 where F is given in pounds nd gives the position of the unit in feet. Use Simpson s Rule with n to pproimte the work W (in foot-pounds) done through one ccle if 5 W F d.. The tle lists severl mesurements gthered in n eperiment to pproimte n unknown continuous function f. () Approimte the integrl f d using the Trpezoidl Rule nd Simpson s Rule. Mn Rn.5 Tn Sn sin d () Use grphing utilit to find model of the form c d for the dt. Integrte the resulting polnomil over, nd compre the result with prt (). Approimtion of Pi In Eercises 9 nd 5, use Simpson s Rule with n to pproimte using the given eqution. (In Section 5.7, ou will e le to evlute the integrl using inverse trigonometric functions.) d d Are In Eercises 5 nd 5, use the Trpezoidl Rule to estimte the numer of squre meters of lnd in lot where nd re mesured in meters, s shown in the figures. The lnd is ounded strem nd two stright rods tht meet t right ngles Prove tht Simpson s Rule is ect when pproimting the integrl of cuic polnomil function, nd demonstrte the result for d, n. 5. Use Simpson s Rule with n nd computer lger sstem to pproimte t in the integrl eqution t sin d. 55. Prove tht ou cn find polnomil p A B C tht psses through n three points,,,, nd,, where the s re distinct. i 5 5 Rod Rod Strem Strem Rod Rod

### 4.6 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

### The 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

### Review 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

### 1 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

### LINEAR 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

### Math 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

### Goals: 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

### 2 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

### What 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

### APPROXIMATE INTEGRATION

APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose nti-derivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be

### Fundamental 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

### Arc 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

### Chapter 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

### Chapter 6 Notes, Larson/Hostetler 3e

Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

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

### Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED

Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type

### Chapter 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

### Topics Covered AP Calculus AB

Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

### Interpreting 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

### Time 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

### Calculus 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:

### 5.1 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

### Polynomial 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

### MA Lesson 21 Notes

MA 000 Lesson 1 Notes ( 5) How would person solve n eqution with vrible in n eponent, such s 9? (We cnnot re-write this eqution esil with the sme bse.) A nottion ws developed so tht equtions such s this

### x ) 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

### Math& 152 Section Integration by Parts

Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

### sec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5

Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to

### 5.1 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

### 7.8 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 (

### Section 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

### Suppose 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

### Improper 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

### y = 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

### 5: 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

### Continuous 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

### M344 - 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

### Distance And Velocity

Unit #8 - The Integrl Some problems nd solutions selected or dpted from Hughes-Hllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl

### Lab 11 Approximate Integration

Nme Student ID # Instructor L Period Dte Due L 11 Approximte Integrtion Ojectives 1. To ecome fmilir with the right endpoint rule, the trpezoidl rule, nd Simpson's rule. 2. To compre nd contrst the properties

### Section 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

### Calculus 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..

### Calculus - Activity 1 Rate of change of a function at a point.

Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus

### Bob Brown Math 251 Calculus 1 Chapter 5, Section 4 1 CCBC Dundalk

Bo Brown Mth Clculus Chpter, Section CCBC Dundlk The Fundmentl Theorem of Clculus Informlly, the Fundmentl Theorem of Clculus (FTC) sttes tht differentition nd definite integrtion re inverse opertions

### Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

### Chapter 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

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

### Ch 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,

### APPLICATIONS 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

### The 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

### Chapter 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

### 0.1 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.

### C 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

### 10.2 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

### Section 6.1 INTRO to LAPLACE TRANSFORMS

Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

### Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

### 1 Functions Defined in Terms of Integrals

November 5, 8 MAT86 Week 3 Justin Ko Functions Defined in Terms of Integrls Integrls llow us to define new functions in terms of the bsic functions introduced in Week. Given continuous function f(), consider

### 1 The fundamental theorems of calculus.

The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection

### Unit 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)

### AB 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

### Chapter 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

### INTRODUCTION 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

### ( ) 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

### ( ) 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

### Section 6: Area, Volume, and Average Value

Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

### Chapter 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

### METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS

Journl of Young Scientist Volume III 5 ISSN 44-8; ISSN CD-ROM 44-9; ISSN Online 44-5; ISSN-L 44 8 METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS An ALEXANDRU Scientific Coordintor: Assist

### Logarithmic Functions

Logrithmic Functions Definition: Let > 0,. Then log is the number to which you rise to get. Logrithms re in essence eponents. Their domins re powers of the bse nd their rnges re the eponents needed to

### ONLINE 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

### 3.1 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.

### The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

### Operations 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

### Math 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

### List 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

Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

### 1 The fundamental theorems of calculus.

The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous

### Midpoint Approximation

Midpoint Approximtion Sometimes, we need to pproximte n integrl of the form R b f (x)dx nd we cnnot find n ntiderivtive in order to evlute the integrl. Also we my need to evlute R b f (x)dx where we do

### The 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

### Z b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...

Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.

### Definite 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

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### MA 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

### Improper 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:

### Chapter 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

### The 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

### Thomas Whitham Sixth Form

Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

### Definite 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

### 13.3. 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

### = f (c) f (c) the height of the rectangle guaranteed by the MVT for integrals.

Get Rey: Given (t) = 8t n v() = 6, fin the isplcement n istnce of the oject from t= to t= If () = 4, fin the position of the prticle t t= I. Averge Vlue of Function Wht oes represent? Cn we rw rectngle

### 1 The Riemann Integral

The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

### Section 6.3 The Fundamental Theorem, Part I

Section 6.3 The Funmentl Theorem, Prt I (3//8) Overview: The Funmentl Theorem of Clculus shows tht ifferentition n integrtion re, in sense, inverse opertions. It is presente in two prts. We previewe Prt

### The Fundamental Theorem of Calculus, Particle Motion, and Average Value

The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt

### 10. AREAS BETWEEN CURVES

. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

### Mat 210 Updated on April 28, 2013

Mt Brief Clculus Mt Updted on April 8, Alger: m n / / m n m n / mn n m n m n n ( ) ( )( ) n terms n n n n n n ( )( ) Common denomintor: ( ) ( )( ) ( )( ) ( )( ) ( )( ) Prctice prolems: Simplify using common

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

### Chapter 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

### Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second

### Antiderivatives and Indefinite Integration

8 CHAPTER Integrtion Section EXPLORATION Finding Antiderivtives For ech derivtive, descrie the originl function F F F c F d F e F f F cos Wht strteg did ou use to find F? Antiderivtives nd Indefinite Integrtion

### HIGHER 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