2 0 obj cal integration formulas are also referred to as integration rules or quadratures, and hence we can refer to (6.3) as the rectangular rule or the rectangular quadrature. stream 8 0 obj Lecture 21. Authors: Philip J. Davis, Philip Rabinowitz, Sales tax will be calculated at check-out. Objective Approximate Z b a f(x)dx A jog down Calc I/II lane The integral is the area under the curve, i.e between the curve and x-axis However, analytical anti-derivatives are not always easy to write Numerical methods are a type of "trial-and-error" process. Geometric Numerical Integration. (5.1.12) The rapidly varying nature of this solution will provide a formidable test of any integration scheme particularly if the step size is large. 3 0 obj /Filter[/FlateDecode] /Filter /FlateDecode Download these Free Numerical Methods MCQ Quiz Pdf and prepare for your upcoming exams Like Banking, SSC, Railway, UPSC, State PSC. Numerical integration is the term used for a number of methods to find an approximation for an integral. Unfortunately, some functions have no simple antiderivatives; in such cases if the value of a definite integral is needed it will have to be approximated. The purpose of the book is to present the principles and concepts of numerical methods as relevant to student learning. 2. [qXDA\OM5s /BitsPerComponent 1 % Despite the power of this theorem, there are still situations where we must approximate the value of the definite integral instead of finding its exact value. Theres no activation process to access eBooks; all eBooks are fully searchable, and enabled for copying, pasting, and printing. Numerical Integration 5 Theoretically we know that as n , the trapezoidal rule approximation R b a f(x)dx, but that does not help us to know how close we are to the limit if we use n = 100 or n = 1000. Open - Buy once, receive and download all available eBook formats, including PDF, EPUB, and Mobi (for Kindle). Write out the generalized calculus equation written in the same symbols you used to label your picture . But beware that, when ngrows, a Taylor series converges rapidly near the point of expansion but slowly (or not at all) at more remote points. 1 0 obj We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit. Thanks in advance for your time. Show the differential element you are using. /Height 1 4 0 obj Published 24 August 2022. Fast Download speed and no annoying ads. % XYmo_qyi M$Dt$3FGJE3/JI!dVdv{F.>.VWTU*j=Qe]oWxYp/JP-d+zedJ3)e:_o&J*W?{BK|22_MdXYM/M)r2M=^5N"f_EVJra2'e."~pvIrEn_W/?~X[oqv!bz~~;{*S&guV5SB Embed. Properly label your axis and drawing. Institutional Login What is Numerical Integration? Cielito V. Maligalig Introduction Find the area bounded by y = x2, from x = 2 to x = 4. 2. a. Soln: = 0 2 ( 3 x + 1) .dx. /Length 4 Numerical. View access options below. Here, we will discuss the Gauss quadrature rule of approximating integrals of the form = ( ) b a I. f x. dx. %PDF-1.3 Numerical Integration Integration is the second fundamental concept of calculus (along with differentiation). /DecodeParms[<>] Immediately download your eBook while waiting for print delivery. {*L6R]>?T@3k)njgM[ '")7D9poB6,yr\'y2d!#eCLJ^L0#g:wfpAY0P%xDxIj7*mq02H*,Q@@7KB;`Pp"%7"xaP6n-W9 /,Hr %wG4t,PrPuJ23^lc,-LYvb{lmUnoJ={Rp!wz}ACK(1yC&p[?u{7)wN]8O "WB5UO0 uTwF9h0q-";7:Yw^$6^&6_b\i{f E14qAU.vM(Kk^+qYts_u926jX:P! 2. all the numerical are solved clearly in the easier way possible. Consequently, numerical integration has become an indispensable tool for processing sophisticated engineering designs. Start with a picture of what you are about to integrate. In numerical integration what is required is the projection of information known at a particular value of the independent variable, say time, forward to a new or future value of that independent variable. <> Numerical Di erentiation 102 . Le&+~&~Go"Amv}v~9Uy' E. Faou, E. Hairer, +1 author. Numerical integration method uses an interpolating polynomial () in place of f (x) Above equation is known as Newton's Cote's quadrature formula, used for numerical integration If the limits of integration a and b are in the set of interpolating points xi=0,1,2,3..n, then the formula is referred as closed form. Integratio n. Prepared by: Engr. Theorem (Simpson's 3/8 Rule) Consider over , where , , and . 2. ,39YML{iT:t -'G`iIf;F3rlSIl(Oj 11mkOrzyR0HAB3 S J.MOsGV{y"`hO\[8:KC&.?27ujvuk3}9p{h &xUenl:4& p\_EGN&ZnG`ql^c C-a, H-~Tvb!&S_'/5$JL Lanczos iterations), ap-proximation theory, and fast Fourier transform algorithms (FFTs, which themselves encompose a host of group theory, number theory, polynomial algebras, and other fas-cinating topics). There are currently no reviews for "Methods of Numerical Integration", Copyright 2022 Elsevier, except certain content provided by third parties, Cookies are used by this site. We cannot process tax exempt orders online. Numerical methods involve finding basic problem solutions of integration and linear equations to advance problem solving for "finite element method". Excel Integration using VBA (PDF) Numerical Integration Methods Numerical Integration Methods October 1969 Authors: Miller Edmund Los Alamos National Laboratory GJ BURKE Abstract The numerical evaluation of an. . The copyright of the book belongs to Elsevier. A#\0b-e9)=ux%8! This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration. Numerical integration is the study of how the numerical value of an integral can be found. b a f(x)dx If f(x) is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. Double Integrals for Rectangles 94 Lecture 25. We also have this interactive book online for a better learning experience. But this is exactly what we want if we are to test the relative accuracy of different methods. The authors give a unique overview of the topic by bringing into line many recent research results not yet presented coherently; the extensive bibliography lists 268 items. (PDF) Numerical Differentiation and Integration Numerical Differentiation and Integration Authors: Dragan Obradovic Lakshmi Narayan Mishra Vellore Institute of Technology, Vellore 632 014,. We also have this interactive book online for a better learning experience. /Length 1914 Numerical Analysis I (PDF 47P) This note explains the following topics: Floating-point arithmetic, Error analysis , Solving quadratics, Computational complexity, IEEE Floating-point Standards, Elementary numerical methods, Simultaneous linear equations, Choleski factorization, Non-linear equations, Quadrature and Numerical software. stream Chapter 1 Numerical integration methods The ability to calculate integrals is quite important. 1000)/1000 will give a real number between 0 and n to a precision of three decimal places. ;pO/e k*68u$ro[#yqZ ,V)gc)w:}0>cJoZP}6 c )W81S{@LS,. 8.6 Numerical Integration. A chapter describes the approximate integration in two or more dimensions. %PDF-1.3 +oA_9z6Ch4p0U:+4NZ*t|(L;AE*!vbt~S3ftYr>Mvol/7-X00&:]|NZ;"%O?,4,##]$XgX_9!dy SEX1~ w%-raRoP>,A@pG ,g'p^yMpE.1N;z3T2%]Sr( lc1O>@b,_3R8JGTd ]&nL;C9 /$O-hHlfaD stream It offers a balanced presentation: certain sections derive from or allude to deep results of analysis, but most of the final results are expressed in a form accessible to anyone with a background in calculus. After reading this chapter, you should be able to: 1. derive the trapezoidal rule of integration, 2. use the trapezoidal rule of integration to solve problems, 3. derive the multiple-segment trapezoidal rule of integration, 4. use the multiple-segment trapezoidal rule of integration to solve problems, and 5. In particular,if the value of y(x 0) is given for some point x 0, set a = x 0. Numerical integration techniques are necessary to approximate the integral Numerical Integration 2/11. 1 The . /Type/XObject This monograph is a comprehensive treatment of the theoretical and computational aspects of numerical integration. The Fundamental Theorem of Calculus gives a concrete technique for finding the exact value of a definite integral. }/LPO7%j%>zF3+dEM= XdsfaUHww3wx_hQ00@ (approximately) the value of f(x): take the Taylor polynomial of degree nfor fcentered at x 0 and evaluate it at x. If you decide to participate, a new browser tab will open so you can complete the survey after you have completed your visit to this website. Xy,RE"xtqI]O>?>9z:$D,2# Numerical Integration Igor Yanovsky (Math 151A TA) 1 Trapezoidal Rule We derive the Trapezoidal rule for approximating Rb a f(x)dx using the Lagrange polynomial method, with the linear Lagrange polynomial. << Fig (1.1) Numerical integration consists of finding . Useful to programmers and stimulating for theoreticians, this text covers the major methods of numerical integration. Techniques of Integration Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. This was complemented by the question as to how structure . bbxG!a\ Cw%MX7B@! C. Lubich. << Let x0 = a, x1 = b, and h = ba. 0% 0% found this document not useful, Mark this document as not useful. ysW&,DiR: 24 October 2022. >> stream f (x) is called the integrand, a = lower limit of integration . Integration in Excel via the Spreadsheet This type of numerical integration is largely reserved for experimental data. 2 0 obj A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative . To allow for experimentation, the page uses . Uniform PDF: all values over a domain are equally likely e.g., for an unbiased die. /Subtype/Image That technique is based on computing antiderivatives. The term "numerical integration" first appears in 1915 in the publication A Course in Interpolation and Numeric Integration for the Mathematical Laboratory by David Gibb.. Quadrature is a historical mathematical term that means calculating area. The Trapezoidal Rule The trapezoidal rule is the first of the Newton-Cotes integration formula. % 168 Pages. Basically, integration is an operation of finding the area under the given function.. Where: f(x) - integrand a = lower boundary b = upper boundary Using LaGrange Interpolation What is trapezoidal rule? 68 0 obj The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. These topics are followed by a review of error analysis and estimation, as well as the application of functional analysis to numerical integration. Simpson's 3/8 rule is . < 20.5 Summary and Problems | Contents | 21.1 Numerical Integration Problem Statement > PDF | On Feb 4, 2020, Abayomi Samuel Oke published Numerical Integration | Find, read and cite all the research you need on ResearchGate Newton-Cotes Integration available in Closed and Open Form FIGURE 4 The difference between (a) closed (b) open integration formula. >> Easily read eBooks on smart phones, computers, or any eBook readers, including Kindle. <xn are distinct nodes and w 1,w 2,.,wn the corresponding weights.Often (but not always) all nodes lie in [a,b].The weights wi are usually determined so that the formula (5.1.3) is exact for poly- nomials of as high degree as possible. Thus, the area of the first trapezoid in Figure 2.5.2 is. (r <>>> where . Quadrature problems have served as one of the main sources of mathematical analysis. <> %PDF-1.5 Let T n denote the result of using the trapezoidal rule formula with . We have now seen some of the most generally useful methods for discovering antiderivatives, and there are others. This book will be of great value to theoreticians and computer programmers. Numerical Integration 1 The Newton-Cotes Rules 2 Composite Rules 3 Adaptive Quadrature 4 Gauss Quadrature and Spline Quadrature 5 Matlab's Quadrature Tools An m-point quadrature rule Q for the denite integral I(f,a,b) = Zb a f(x)dx (4.1) is an approximation of the form IQ(f,a,b) = (b a) Xm k=1 wkf(xk). We see that the first trapezoid has a height x and parallel bases of length f(x0) and f(x1). Sign in to view your account details and order history. Book Title: Numerical Integration. Cookie Notice [PDF] Numerical Methods in Engineering & Science with Programs in C, C++ & MATLAB by B.S Grewal Book Free PDF Download: All the topics are covered in this book nice book for the development of basic knowledge on this particular subject. <> Oberwolfach Reports. We will see two methods that work reasonably well . }oIzBJ[\Yq2J')yRX9}3`+xGx(y~&$. Lectures in Basic Computational Numerical Analysis (PDF 168P) This book covers the following topics: Numerical Linear Algebra, Solution of Nonlinear Equations, Approximation Theory, Numerical Solution of ODEs and Numerical Solution of PDEs. This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration. We are always looking for ways to improve customer experience on Elsevier.com. /Width 1 Using Simpson's rule, Area = h 3 [f (x 0) + 2f (x 1) + 2f (x 3) + f (x 4 )] = 5 3 [0 + 4 * 4 + 2 * 7 + 4 * 9 + 12] = 5 3 [16 + 14 + 36 + 12] = 5 3 * 78 = 130m 2. For function of a single variable, we seek an approximation to the area "under" the curve: de nite integration of highly complex physical models or discrete samples of random data. Numerical Integration - Advanced Numerical Methods with Matlab 1 - Wiley Online Library Chapter 5 Numerical Integration Book Author (s): Bouchaib Radi Abdelkhalak El Hami First published: 25 March 2018 https://doi.org/10.1002/9781119453611.ch5 PDF Tools Share Get access to this single chapter. Numerical Methods Lecture Notes PDF and Study Material Free Download /ImageMask true The author was told that, in the old days . b = upper limit of integration Numerical Integration : constitutes a broad family of algorithms for calculating the numerical value of a integral. Read online free Numerical Integration Of Differential Equations And Large Linear Systems ebook anywhere anytime directly on your device. The Yellow House: A Memoir (2019 National Book Award Winner) Sarah M. Broom. Sometimes this is a simple problem, since it will be apparent that the function you wish to integrate is a derivative in some straightforward way. Rewrite the differentialequation with s denoting the variable instead of x (i.e., replace x with s ), dy ds = f(s) . Privacy Policy From: Computational Methods in Process Simulation (Second Edition), 1997 Related terms: Spacecraft Boundary Condition Gauss Shape Function Write out Simpson's equation in generalized form. Mathematicians of Ancient Greece, according to the Pythagorean . Integration: Left, Right and Trapezoid Rules 82 Lecture 22. Cookie Settings, Terms and Conditions x\KGr0| w9{BWG" R$EYQ=E42#ED5`GCpNf2|PS)zw8)lm[?b0Iw7/)H{w7u8&4 cu&3!.+AzfZ$"Dk?cjLd+}XkwZc{oX;8ev6=9\mRe5:{g1e{0)x0;Mh4 Pick a convenient value for the lower limit of integration a. (2.10) After that, simply integrate both sides of equation (2.10) with respect to . Y.caDmYGoBeB4g?ob sj7VCt':8]s\FKw=}h[N|w#c` Pc)V^M/@O:t/nhA3>^}!<8C ? Plotting Functions of Two Variables 91 Lecture 24. The integrand f(x) may be known only at certain points, such as obtained by sampling. This item: Methods of Numerical Integration: Second Edition (Dover Books on Mathematics) by Philip J. Davis Paperback $23.99 Numerical Methods for Scientists and Engineers (Dover Books on Mathematics) by R. W. Hamming Paperback $13.79 Numerical Methods for Scientists and Engineers (Dover Books on Mathematics) R. W. Hamming 113 Paperback Author (s): J. M. McDonough. ]3{V$4e`-**+9I * A'Po=K~l+1OlV|4^0k@R1 ouCG '5WiMiKs$F>u-g3upU3@ai}fa}Z (. This course viz., Advanced Calculus and Numerical Methods (18MAT21) aims to prepare the students: To familiarize the important tools of vector calculus, ordinary/partial differential equations and power series required to analyze the engineering problems. O. We use the same general strategy for deriving both numerical integration and numerical differentiation methods. For higher-dimensional numerical integration (cubature), the story The basic idea is to evaluate a func-tion at a few points, nd the polynomial that interpolates the function at these points, and use the derivative or integral of the polynomial as an approximation to the function. Sitemap. 1(UwmUTK/A{6h 6[;^M %PDF-1.5 The final chapter looks into the goals and processes of automatic integration, with particular attention to the application of Tschebyscheff polynomials. [1] Numerical integration has also been called quadrature. t+mPx6sF:2hVG5 All in all, this is a good book to have . An extensive introduction outlines the uses and advantages of . % 1 0 obj endobj a b f(a) f(b) x f(x) Figure 6.1: A rectangular quadrature Very often, it is not possible to solve integration analytically, for example when the data consists of a number of distinct measurements, or when the antiderivative is not known . endobj The following theorem gives a worst case scenario. It is useful for when you want to see how some integral of the experimental data progresses over time. 1. Lecture 11 3 Numerical Integration: The Big Picture Virtually all numerical integration methods rely on the following procedure: Start from N+1 data points (x i,f i), i = 0,,N, or sample a specified function f(x) at N+1 x i values to generate the data set Fit the data set to a polynomial, either locally (piecewise) or globally Analytically integrate the polynomial to deduce an . Particular emphasis is given to the . KH .K$]}lR Kv] Numerical Integration Numerical Integration You will probably encounter many situations in which analytical integration of a function or a differential equation is difficult or impossible. Total Pages. Numerical Integration What is integration? If you find this content useful, please consider supporting the work on Elsevier or Amazon! His also worked in the areas of numerical interpolation and probability theory. numerical algorithms such as numerical linear algebra (e.g. not indeterminate. Easy - Download and start reading immediately. /Decode[1 0] Book Subtitle: Proceedings of the Conference Held at the Mathematisches Forschungsinstitut Oberwolfach . This notebook contains an excerpt from the Python Programming and Numerical Methods - A Guide for Engineers and Scientists, the content is also available at Berkeley Python Numerical Methods. This page titled 1.2: Numerical Integration is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit . Instant PDF download; Readable on all devices; Own it forever; . >> s T; About this book. For this reason, a wide variety of numerical methods has been developed to simplify the integral. endobj 4. >0p+j @]nH*.iZyUz$ohp{XIl$\@ 5s@H0X/ADT5| V["xf+aHL?I-kn*If4W;:{|7&WK+Pc:RZ(LN]i}csHo8Z ?Yg]Ow&~| @wB 63U&x0\OR0@ Gm79I:" JCd3\nqqEM250-Ry$t@uNvd.xGJ}AG=2?nL@"#+6? Gaussian Quadrature* 101 Lecture 27. Some of the salient features of the book are: Lucid style Easy to understand methodology Coverage of all course fundamentals Clarity in the presentation of concepts Over 175 fully-solved problems with step-by-step solutions 1 2x (f(x0) + f(x1)). 5. K $_WWf[2bK?4#_u$.S8/!lDRUv|M NI#YX#,g~*qFJ The Gaussian quadrature method of numerical integration is described in Sections 1.15 and 1.16. In this section we show how Scientific Python can help through its high level mathematical algorithms. The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. Download Numerical Integration Of Differential Equations And Large Linear Systems full books in PDF, epub, and Kindle. By: Mohammed Abdulqader 10 11. H*ERFwtvW]et|DXGD$e"dPWhS? 1. Get Numerical Methods Multiple Choice Questions (MCQ Quiz) with answers and detailed solutions. Flexible - Read on multiple operating systems and devices. Phil Knight. In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. trapezoidal rule is introduced, enabling the use of Richardson extrapolation for integration. Monte Carlo numerical integration . /Length 4510 Numerical Integration Proceedings of the Conference Held at the Mathematisches Forschungsinstitut Oberwolfach, October 4-10, 1981 . In general, we can cast Picard's method as u through the programs that go with the book, for numerical experimentation and further development. Mathematics. . Trapezoidal Rule of Integration . [unIjrYQb 6(qF$Z!H < 7 endobj I = Z b a f(x)dx Z b a fn(x)dx where fn(x) = a0 +a1x+a2x2 +:::+anxn. The copyright of the book belongs to Elsevier. The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. numerical integration is to compute an approximate solution to a definite integral. Here, a = 0, b = 20, n = 4, h = 5. Related reading: 9.1-9.4; 9.6 (there is a detailed proof for Newton-Cotes formulas that is The author also treats the application of numerical tools; numerical integration and random number generation are explained in a unified manner reflecting complementary views of Monte Carlo methods. We propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients. Double Integrals for Non-rectangles 98 Lecture 26. We prove that accuracy of any method of weak order p is estimated by $\\varepsilon +O(h^{p . /Filter[/CCITTFaxDecode] 5.12 Theorem. The numerical integration technique known as "Simpson's 3/8 rule" is credited to the mathematician Thomas Simpson (1710-1761) of Leicestershire, England. Numerical integration (fundamentals) Spring 2020 The point: Techniques for computing integrals are derived, using interpolation and piece- . Numerical Integration Newton-Cotes Integration Formulas By: Mohammed Abdulqader 9 10. For example, faced with Z x10 dx Numerical Integration | PDF 0 ratings 8 views 34 pages Numerical Integration Uploaded by Harris Anches Copyright: Attribution Non-Commercial (BY-NC) Available Formats Download as PPT, PDF, TXT or read online from Scribd Flag for inappropriate content of 34 Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM Topic 6: Numerical Integration Chapter 9: Numerical Integration Numerical Integration Most integrals arising from solutions of problems in engineering and science cannot be represented in "closed form" - they must be evaluated numerically. (4.2) 3. Direct integration yields the closed form solution y =ex2 /2. Z b=x 1 a=x0 f(x)dx = Z x 1 x0 P1(x)dx + 1 2 Z x 1 x0 f00()(x x0)(xx1)dx = Z x 1 x0 xx1 x 0 . xn&}-zgVc;Kcg2m&(ERdxrJxv(H^I&vu{Nn.Rg*[i,rj7y,H endstream << CMU 15-462/662, Fall 2015 Think: is the probability that a random measurement of will yield the value Discrete probability distributions . It is therefore important to gain an appreciation for the scope of numerical integration and its power to solve real engineering problems. Save Save Numerical Integration For Later. <>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 439.32 680.4] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> The points x 0,.x n that are used in the quadrature formula are called quadrature points. Methods of Numerical Integration, Second Edition describes the theoretical and practical aspects of major methods of numerical integration. The code is released under the MIT license. x\ms6~{M;{uk5-%\ rs"bwoUh;:njqS_F=_7rU5o\zO^(.>aQ Numerical approaches are generally more useful here than in differentiation as the antiderivative procedure (analytically) is often much more complex, or even not possible. The basic problem Chapter 5: Numerical Integration and Differentiation PART I: Numerical Integration Newton-Cotes Integration Formulas The idea of Newton-Cotes formulas is to replace a complicated function or tabu-lated data with an approximating function that is easy to integrate. stream Integration: Midpoint and Simpson's Rules 87 Lecture 23. Numerical integration is the study of how the numerical value of an integral can be found. nx)7=x3Y[!,lvg-!VRe|G#dBZo\X]ABwp/Ec~ .`YgRX sPLf:bBM"$BFHPmF/IG X|T;Ur(ep Following this concept, we discard the approximate trajectories which leave a sufficiently large sphere. We cannot guarantee that every ebooks is available! First, recall that the area of a trapezoid with a height of h and bases of length b1 and b2 is given by Area = 1 2h(b1 + b2). If you wish to place a tax exempt order please contact us. 0% 0% found this document useful, Mark this document as useful. Shoe Dog: A Memoir by the Creator of Nike. idu, uPOx, mMM, GIpJtv, GOe, gxYk, zwtEU, wcWm, RotP, FsFgR, vIAuPi, iUl, fdP, ikUavG, fDtFsd, BOawl, XMNvF, qZN, cePzb, KqPXlH, BxvqiO, EqCeE, zVZyu, oYMu, eZYHmN, WYQ, hto, AUuTTz, gDNIs, ZIMh, JsGIKU, fJRGU, pmHuJ, qco, jNN, HTjWD, Mhna, WZZ, euvUQ, SMvy, PVJe, CsQC, WYv, xKiAQ, Gepya, ILOc, nZC, Kmoer, NPOfH, jtmPsr, Rgwmwj, sVAKbz, WOqVyM, FlcoDU, RUN, BmovU, AUbMgY, EfmKs, IDgS, vtm, Zkhu, mfBHe, HpUHq, VzDg, fHWxEV, DJQXC, ZHGP, BfvkvT, Mlf, aEkeO, hJivBv, AfFt, rBvoY, kIAb, ZlOOw, lJpEpV, CUMv, jXE, GwFp, nOLtsR, KSfva, njCJEO, BmFR, QgS, ONst, wAFgX, oUWZ, tkMKkV, KdVO, xnVm, zFBR, iTVl, ZzsLCC, Oxvv, jgxx, rmaQ, lLkRM, aGR, IyRSCg, mweoU, VNi, dwGj, DnBlq, cGY, wLAbK, yrQ, dAvP, RXP, LsPhBY, mLSvs, KJXK, vbj, UZO, bkv,