LECTURE 19. Numerical Integration. Z b. is generally thought of as representing the area under the graph of fèxè between the points x = a and


 Vanessa Gregory
 3 years ago
 Views:
Transcription
1 LECTURE 9 Numericl Integrtion Recll from Clculus I tht denite integrl is generlly thought of s representing the re under the grph of fèxè between the points x = nd x = b, even though this is ctully only true when fèxè is nonnegtive onë; bë. However, the geometric interprettion of the derivtive t lest mkes its the forml denition lim N! NX fè~x i èx ; x = b, N ; ~x i ë + ix; +èi + èxë little more pltble: for one thinks of the sum on the right hnd side s sum over the res of set of innitesiml rectngles tht just bout cover the grph of fèxè. Of course, in prctice one never uses the limit denition to compute n integrl. Rther, one relies on the Fundmentl Theorem of Clculus, which sys tht = F èbè, F èè where F èxè isny function such tht df = fèxè. Thus, the problem of crrying out direct integrl is dx reduced to nding n ntiderivtive of the integrnd. However, there re mny exmples of functions f for which there is no closed formul for ntiderivtive of f. A fmous exmple is fèxè =e x : Therefore, numericl integrtion is often the only mens for scribing vlues to expressions like Z 0 e x dx : Now just s in the cse of the derivtive, the forml denition leds nturlly to simple numericl procedure. We simply choose lrge vlue for N, set nd clculte x = b, N ~x i = +xi + x ; the midpoint of the intervl ëx i ;x i+ ë NX fè~x i èx : However, while it's esy to write this computtionl lgorithm down, it's not so obvious how to gure out how ccurte the resulting clcultion will be for ny given vlue of N.
2 9. NUMERICAL INTEGRATION In order to obtin relible estimte of the error we will replce the integrnd fèxè by polynomil interpoltion P f èxè. The reson for doing this is twofold. First of ll, since P f èxè is polynomil we cn compute its integrl exctly: è A i x i! dx = èi +èa i èb i+, i+ è Secondly, wehve n exct expression for the error tht's introduced when we replce fèxè bynn th degree polynomil interpoltion: From this we cn deduce, fèxè, P f èxè = P f èxè = èn + è! f èn+è èç x è èn + è! f èn+è èç x è è! Just s in the cse of polynomil interpoltion we would like tochoose the interpoltion nodes x i in such wy tht the fctor W = è! dx ç!èxèdx is minimized. Recll tht in the cse of polynomil interpoltion we discovered tht the optiml choice of n + nodes x i, t lest on the intervl ë,; ë would be the roots of the Chebyshev polynomil T n+ èxè. We hve similr sitution here; however, insted of minimizing the mximl vlue of the product of the 's we must insted try to minimize the integrl of such product. This leds us to using, insted of the roots of the ordinry Chebyshev polynomils, the roots of nother specil set of polynomils, the Chebyshev polynomils of the second kind. These polynomils re dened s follows: U n èxè = sin, èn + è cos, èxè sin ècos, èxèè nd hve the following properties U n èxè =,n x n +lower order terms,n, U n+ èxè =èx, x 0 èèx, x èèx, x è èx, x n è where ç ç èi +èç x i = cos ; i =0; ; ;:::;n n + If x 0 ;:::;x n re the roots of U n+ èxè then Z, dx =,èn+è If f~x i j i =0; ;:::;ng is ny other collection of n + point inë,; ë, then Z, èx, ~x i è Thus, one optiml technique for computing n integrl Z, ç,èn+è would be to clculte n n + point interpoltion P f èxè on the intervl ë,; ë, using the points ç ç èi +èç x i = cos ; i =0; ; ;:::;n n +
3 . OTHER QUADRATURES 3 s interpoltion nodes nd then setting Z, Z, P f èxèdx : To hndle n integrl over more generl intervl, sy ë; bë we'd use the the sme trick tht we used for nding optiml nodes for ordinry interpoltion; we just mp the intervl ë,; ë linerly onto the intervl ë; bë nd look to see where the the nodes of U n èxè lnd. Thus, we set x i = + b + b, cos ç èi +èç n + ç ; i =0; ; ;:::;n determine the interpoltion polynomil P f èxè corresponding to this set of nodes, nd nlly set Problem 9.. Suppose = P f èxèdx is clculted numericlly by interpolting the function fèxè t the points ç ç x i = + b + b, èi +èç cos ; i =0; ; ;:::;n n + nd then integrting the interpoltion polynomil between nd b. Express the mximl error in terms of derivtive of f, n, nd the end points of integrtion nd b. èhint: Write down chnge of vribles formul reduces the integrl over ë; bë to n integrl over ë,; ë.è. Other Qudrtures Now ctully the progrm outlined bove cn be bit too computer intensive. Certinly we cn use up lot of computer time nding the Newton form for the interpoltion polynomil for lrge set of nodes, converting the Newton form of the interpoltion polynomil to stndrd form, nd then integrting term by term. Often wht suces is modest improvement over the nive lgorithm coming from the forml denition of the Riemnn integrl. Here's the bsic ide. The formul è9.è fèxè ç NX fè~x i èx ; x = b, N ; ~x i ë + ix; +èi + èxë ws bsed on the ide tht the re under the grph of fèxè cn be pproximted by forming prtitioning the intervl ë; bë into N subintervls ëx i ;x i+ ë of width x, pproximting the re A i under the grph of fèxè under between x i nd x i+ by è9.è The ide we'll pursue now ishow to improve the pproxi nd then summing over the contributions A i. mtions è9.è. A i ç f è~x i èx ; ~x i ëx i ;x i+ ë Let's rst note tht the pproximtion è9.è is bout the worst possible. For eectively, we're replcing the function fèxè bypoint polynomil interpoltion of f on the intervl ëx i ;x i+ ë. No doubt our ccurcy would improve if we insted used three point polynomil interpoltion insted.
4 . OTHER QUADRATURES 4 For nottionl clrity, let's replce x i nd x i+ by nd x +h, nd consider the Lgrnge form of the three point interpoltion of fèxè on the intervl ë; +hë using the points ; + h; +h. We then hve è9.3è fèxè ç fèè èx,, hèèx,, hè è,hèè,hè + f è + hè èx, èèx,, hè èhèè,hè èx, èèx,, hè + fè +hè èhèèhè The expresion on the right hnd side is second order polynomil in x. Ifwe expnd it in powers of x nd integrte it between nd +h we obtin è9.4è Z +h h ç ëfèè+4fè + hè+fè +hèë 3 This formul is equivlent to Simpson's Rule è rule tht is often presented in elementry clculus courses.è We cn immeditely pply this formul to get new nd improved version of è9.è. Setting we hve = x i ; h = x nd so, setting we hve Z xi+x x i fèxè ç çèx i è= x = ëfèè+4fè +x=è + fè +xèë x = b, x i = + ix i= Z xi x i, i= ç èx i,èx Of course, there's nothing stopping us from interpolting the subintervls ëx i ;x i+ ë t four, ve or more points to obtin even more ccurte qudrture formule. However, the lgebr between the nlogs of equtions è9.3è nd è9.4è becomes pretty strenuous. There is n esy wy of guessing the correct qudture formul. To demonstrte this technique let me rework derivtion of eqution è9.4è. The key ide is tht polynomil interpoltion t the points x 0, x, :::; x n will lwys produce formul of the form è9.5è Z è n X f èx i è `ièxè! dx = f èx i è where the `ièxè re the crdinl functions for the nodes x 0 ;x ;:::;x n : If we dene then è??è tkes the form è9.6è `ièxè = A i ç j=0 j6=i èx, x j è èx i, x j è `ièxèdx f èx i è A i `ièxèdx
5 . OTHER QUADRATURES 5 no mtter wht the function fèxè is. Moreover, if fèxè is polynomil of degree less thn or equl to n, then fèxè is identicl to its polynomil interpoltion èrecll tht the error term is proprotionl to f èn+è èçè which would be zero if f were polynomil of degree ç nè. By tking fèxè =;x;x ;:::;x n we rrive t series of n + equtions for the n + constnts A i : b, =, b, =, b n+, n+ = n +. dx = xdx = x n dx = Solving this system of equtions for the constnts A i will give us qudture formul è9.6è tht cn be used for ny function fèxè. Let me now demonstrte this technique for the cse where we do three point interpoltion to clculte Z +h : A i x i A i x n i A i Set x 0 =, x = + h, x = +h. The interpoltion is exct when fèxè =: So A 0 + A + A = The interpoltion is lso exct when fèxè = x. So A 0 èè+a è + hè+a è +hè = Z +h,h And the interpoltion is exct when fèxè =x.thus, A 0 èè + A è + hè + A è +hè = We thus rrive t the following system of equtions The solution of this system nd so which is identicl to è9.4è. + h +h è + hè è +hè Z +h 0 A Z +h xdx = Z A 0 A A A 0 = 3 h A = 4 3 h A = 3 h Problem 9.. Find qudrture formul for the integrl dx =h, è +hè, èè =h, h x dx = 3 0 A è +hè 3, èè 3 = h è +hè, h +4h h3 ç 3 hfèè +4 3 hfè + hè+ hfè +hè 3 Z +3h fèxè A
6 . OTHER QUADRATURES 6 corresponding to the cse where the function fèxè is interpolted t four points: x 0 =, x = + h, x = +h, x 3 = +3h;
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationProperties 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
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
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 informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
More information3.4 Numerical integration
3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationLecture 14: Quadrature
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 relvlues nd smooth The pproximtion of n integrl by numericl
More information1 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
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics  A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationLecture 20: Numerical Integration III
cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed
More informationNumerical Integration. 1 Introduction. 2 Midpoint Rule, Trapezoid Rule, Simpson Rule. AMSC/CMSC 460/466 T. von Petersdorff 1
AMSC/CMSC 46/466 T. von Petersdorff 1 umericl Integrtion 1 Introduction We wnt to pproximte the integrl I := f xdx where we re given, b nd the function f s subroutine. We evlute f t points x 1,...,x n
More informationIII. Lecture on Numerical Integration. File faclib/dattab/lecturenotes/numericalinter03.tex /by EC, 3/14/2008 at 15:11, version 9
III Lecture on Numericl Integrtion File fclib/dttb/lecturenotes/numericalinter03.tex /by EC, 3/14/008 t 15:11, version 9 1 Sttement of the Numericl Integrtion Problem In this lecture we consider the
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More informationNumerical Integration
Numericl Integrtion Wouter J. Den Hn London School of Economics c 2011 by Wouter J. Den Hn June 3, 2011 Qudrture techniques I = f (x)dx n n w i f (x i ) = w i f i i=1 i=1 Nodes: x i Weights: w i Qudrture
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 20172018 Tble of contents 1 Antiderivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Antiderivtive Function Definition Let f : I R be function
More informationOrthogonal Polynomials
Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils
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 informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationNUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose ntiderivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More information1. GaussJacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. GussJcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
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 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 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 informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More informationDOING PHYSICS WITH MATLAB MATHEMATICAL ROUTINES
DOIG PHYSICS WITH MATLAB MATHEMATICAL ROUTIES COMPUTATIO OF OEDIMESIOAL ITEGRALS In Cooper School of Physics, University of Sydney in.cooper@sydney.edu.u DOWLOAD DIRECTORY FOR MATLAB SCRIPTS mth_integrtion_1d.m
More information1 The Lagrange interpolation formula
Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x
More informationNumerical Integration
Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationNumerical Analysis. 10th ed. R L Burden, J D Faires, and A M Burden
Numericl Anlysis 10th ed R L Burden, J D Fires, nd A M Burden Bemer Presenttion Slides Prepred by Dr. Annette M. Burden Youngstown Stte University July 9, 2015 Chpter 4.1: Numericl Differentition 1 ThreePoint
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationMath& 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
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/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
More informationChapter 5. Numerical Integration
Chpter 5. Numericl Integrtion These re just summries of the lecture notes, nd few detils re included. Most of wht we include here is to be found in more detil in Anton. 5. Remrk. There re two topics with
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationBig idea in Calculus: approximation
Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More information1 Error Analysis of Simple Rules for Numerical Integration
cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion
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 informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationConstruction of Gauss Quadrature Rules
Jim Lmbers MAT 772 Fll Semester 201011 Lecture 15 Notes These notes correspond to Sections 10.2 nd 10.3 in the text. Construction of Guss Qudrture Rules Previously, we lerned tht NewtonCotes qudrture
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationLecture 12: Numerical Quadrature
Lecture 12: Numericl Qudrture J.K. Ryn@tudelft.nl WI3097TU Delft Institute of Applied Mthemtics Delft University of Technology 5 December 2012 () Numericl Qudrture 5 December 2012 1 / 46 Outline 1 Review
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationLecture 1: Introduction to integration theory and bounded variation
Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You
More informationCOT4501 Spring Homework VII
COT451 Spring 1 Homework VII The ssignment is due in clss on Thursdy, April 19, 1. There re five regulr problems nd one computer problem (using MATLAB). For written problems, you need to show your work
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationCOSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III)  Gauss Quadrature and Adaptive Quadrature
COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III  Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting
More informationJDEP 384H: Numerical Methods in Business
BT 3.4: Solving Nonliner Systems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Methods Instructor: Thoms Shores Deprtment of Mthemtics Lecture 20, Februry 29, 2007 110 Kufmnn Center Instructor:
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationLecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at UrbanaChampaign. March 20, 2014
Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t UrbnChmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method
More information6.5 Numerical Approximations of Definite Integrals
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 6.5 Numericl Approximtions of Definite Integrls Sometimes the integrl of function cnnot be expressed with elementry functions, i.e., polynomil,
More information5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship
5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is
More informationDiscrete Leastsquares Approximations
Discrete Lestsqures Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
More informationThe 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
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixedpoint itertion to converge when solving the eqution
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationIntegrals  Motivation
Integrls  Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is nonliner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
More informationTHE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrdlindeloftheorem/ This document is proof of the existenceuniqueness theorem
More informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationF (x) dx = F (x)+c = u + C = du,
35. The Substitution Rule An indefinite integrl of the derivtive F (x) is the function F (x) itself. Let u = F (x), where u is new vrible defined s differentible function of x. Consider the differentil
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
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 informationZ 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.
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
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 informationTaylor Polynomial Inequalities
Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil
More informationThe Fundamental Theorem of Calculus
The Fundmentl Theorem of Clculus MATH 151 Clculus for Mngement J. Robert Buchnn Deprtment of Mthemtics Fll 2018 Objectives Define nd evlute definite integrls using the concept of re. Evlute definite integrls
More informationThe Evaluation Theorem
These notes closely follow the presenttion of the mteril given in Jmes Stewrt s textook Clculus, Concepts nd Contexts (2nd edition) These notes re intended primrily for inclss presenttion nd should not
More information5.5 The Substitution Rule
5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n ntiderivtive is not esily recognizble, then we re in
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationTHE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS
THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS CARLOS SUERO, MAURICIO ALMANZAR CONTENTS 1 Introduction 1 2 Proof of Gussin Qudrture 6 3 Iterted 2Dimensionl Gussin Qudrture 20 4
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 informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationRiemann Integrals and the Fundamental Theorem of Calculus
Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums
More informationLecture 23: Interpolatory Quadrature
Lecture 3: Interpoltory Qudrture. Qudrture. The computtion of continuous lest squres pproximtions to f C[, b] required evlutions of the inner product f, φ j = fxφ jx dx, where φ j is polynomil bsis function
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More informationCalculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties
Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwthchen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion:
More information