Write out Simpson's equation in generalized form. In general, we can cast Picard's method as 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. Pick a convenient value for the lower limit of integration a. Fast Download speed and no annoying ads. Properly label your axis and drawing. /Length 4 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. 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. 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. a b f(a) f(b) x f(x) Figure 6.1: A rectangular quadrature The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. PDF | On Feb 4, 2020, Abayomi Samuel Oke published Numerical Integration | Find, read and cite all the research you need on ResearchGate Book Subtitle: Proceedings of the Conference Held at the Mathematisches Forschungsinstitut Oberwolfach . Download Numerical Integration Of Differential Equations And Large Linear Systems full books in PDF, epub, and Kindle. /Type/XObject 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. This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration. An extensive introduction outlines the uses and advantages of . >0p+j @]nH*.iZyUz$ohp{XIl$\@ 5s@H0X/ADT5| V["xf+aHL?I-kn*If4W;:{|7&WK+Pc:RZ(LN]i}csHo8Z The final chapter looks into the goals and processes of automatic integration, with particular attention to the application of Tschebyscheff polynomials. xn&}-zgVc;Kcg2m&(ERdxrJxv(H^I&vu{Nn.Rg*[i,rj7y,H The numerical integration technique known as "Simpson's 3/8 rule" is credited to the mathematician Thomas Simpson (1710-1761) of Leicestershire, England. Chapter 1 Numerical integration methods The ability to calculate integrals is quite important. 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. O. <> It is therefore important to gain an appreciation for the scope of numerical integration and its power to solve real engineering problems. 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). We also have this interactive book online for a better learning experience. In particular,if the value of y(x 0) is given for some point x 0, set a = x 0. nx)7=x3Y[!,lvg-!VRe|G#dBZo\X]ABwp/Ec~ .`YgRX sPLf:bBM"$BFHPmF/IG X|T;Ur(ep Authors: Philip J. Davis, Philip Rabinowitz, Sales tax will be calculated at check-out. 2. ]3{V$4e`-**+9I * A'Po=K~l+1OlV|4^0k@R1 ouCG '5WiMiKs$F>u-g3upU3@ai}fa}Z (. Fig (1.1) Numerical integration consists of finding . Open - Buy once, receive and download all available eBook formats, including PDF, EPUB, and Mobi (for Kindle). 1. To allow for experimentation, the page uses . bbxG!a\ Cw%MX7B@! Institutional Login 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 % 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. ;pO/e k*68u$ro[#yqZ ,V)gc)w:}0>cJoZP}6 c )W81S{@LS,. << %PDF-1.3 % Numerical approaches are generally more useful here than in differentiation as the antiderivative procedure (analytically) is often much more complex, or even not possible. 1. Geometric Numerical Integration. 8.6 Numerical Integration. Uniform PDF: all values over a domain are equally likely e.g., for an unbiased die. C. Lubich. <> < 20.5 Summary and Problems | Contents | 21.1 Numerical Integration Problem Statement > [qXDA\OM5s 4 0 obj The purpose of the book is to present the principles and concepts of numerical methods as relevant to student learning. 2 0 obj Double Integrals for Rectangles 94 Lecture 25. Easily read eBooks on smart phones, computers, or any eBook readers, including Kindle. % Cookie Notice %PDF-1.5 A chapter describes the approximate integration in two or more dimensions. This book will be of great value to theoreticians and computer programmers. 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 ? Quadrature problems have served as one of the main sources of mathematical analysis. 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. ysW&,DiR: The points x 0,.x n that are used in the quadrature formula are called quadrature points. Easy - Download and start reading immediately. Immediately download your eBook while waiting for print delivery. 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 Integration: Left, Right and Trapezoid Rules 82 Lecture 22. Trapezoidal Rule of Integration . Double Integrals for Non-rectangles 98 Lecture 26. stream Newton-Cotes Integration available in Closed and Open Form FIGURE 4 The difference between (a) closed (b) open integration formula. Save Save Numerical Integration For Later. Consequently, numerical integration has become an indispensable tool for processing sophisticated engineering designs. Excel Integration using VBA T; (4.2) (PDF) Numerical Differentiation and Integration Numerical Differentiation and Integration Authors: Dragan Obradovic Lakshmi Narayan Mishra Vellore Institute of Technology, Vellore 632 014,. We see that the first trapezoid has a height x and parallel bases of length f(x0) and f(x1). endobj Shoe Dog: A Memoir by the Creator of Nike. <>>> 2. This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration. I = Z b a f(x)dx Z b a fn(x)dx where fn(x) = a0 +a1x+a2x2 +:::+anxn. stream 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. 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 . ,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, 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. Privacy Policy 0% 0% found this document not useful, Mark this document as not useful. We will see two methods that work reasonably well . 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 . 3. 4. Numerical Integration Proceedings of the Conference Held at the Mathematisches Forschungsinstitut Oberwolfach, October 4-10, 1981 . f (x) is called the integrand, a = lower limit of integration . 2 0 obj Numerical methods involve finding basic problem solutions of integration and linear equations to advance problem solving for "finite element method". de nite integration of highly complex physical models or discrete samples of random data. /Filter[/CCITTFaxDecode] Numerical Di erentiation 102 . Numerical Integration What is integration? 1 0 obj This monograph is a comprehensive treatment of the theoretical and computational aspects of numerical integration. We are always looking for ways to improve customer experience on Elsevier.com. /Length 4510 Numerical Integration Newton-Cotes Integration Formulas By: Mohammed Abdulqader 9 10. KH .K$]}lR Kv] }/LPO7%j%>zF3+dEM= XdsfaUHww3wx_hQ00@ Read online free Numerical Integration Of Differential Equations And Large Linear Systems ebook anywhere anytime directly on your device. The Gaussian quadrature method of numerical integration is described in Sections 1.15 and 1.16. 68 0 obj From: Computational Methods in Process Simulation (Second Edition), 1997 Related terms: Spacecraft Boundary Condition Gauss Shape Function Book Title: Numerical Integration. By: Mohammed Abdulqader 10 11. Cookie Settings, Terms and Conditions The following theorem gives a worst case scenario. Plotting Functions of Two Variables 91 Lecture 24. It is useful for when you want to see how some integral of the experimental data progresses over time. 1 2x (f(x0) + f(x1)). endobj Following this concept, we discard the approximate trajectories which leave a sufficiently large sphere. b = upper limit of integration 3 0 obj . where . [1] Numerical integration has also been called quadrature. Monte Carlo numerical integration . 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 . 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. 5. The Fundamental Theorem of Calculus gives a concrete technique for finding the exact value of a definite integral. The basic problem % A#\0b-e9)=ux%8! All in all, this is a good book to have . /Filter /FlateDecode Numerical Integration : constitutes a broad family of algorithms for calculating the numerical value of a integral. 5.12 Theorem. >> Particular emphasis is given to the . Mathematicians of Ancient Greece, according to the Pythagorean . The code is released under the MIT license. 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. /Decode[1 0] 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. Thus, the area of the first trapezoid in Figure 2.5.2 is. Numerical methods are a type of "trial-and-error" process. /Subtype/Image Oberwolfach Reports. The copyright of the book belongs to Elsevier. (2.10) After that, simply integrate both sides of equation (2.10) with respect to . /Length 1914 CMU 15-462/662, Fall 2015 Think: is the probability that a random measurement of will yield the value Discrete probability distributions . << s . Lecture 21. Related reading: 9.1-9.4; 9.6 (there is a detailed proof for Newton-Cotes formulas that is Write out the generalized calculus equation written in the same symbols you used to label your picture . 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 $13.79 Numerical Methods for Scientists and Engineers (Dover Books on Mathematics) R. W. Hamming 113 Paperback Let T n denote the result of using the trapezoidal rule formula with . /Width 1 The term "numerical integration" first appears in 1915 in the publication A Course in Interpolation and Numeric Integration for the Mathematical Laboratory by David Gibb.. Quadrature is a historical mathematical term that means calculating area. endstream Le&+~&~Go"Amv}v~9Uy' (r This was complemented by the question as to how structure . Very often, it is not possible to solve integration analytically, for example when the data consists of a number of distinct measurements, or when the antiderivative is not known . Integratio n. Prepared by: Engr. [unIjrYQb 6(qF$Z!H < 7 stream Get Numerical Methods Multiple Choice Questions (MCQ Quiz) with answers and detailed solutions. The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. (PDF) Numerical Integration Methods Numerical Integration Methods October 1969 Authors: Miller Edmund Los Alamos National Laboratory GJ BURKE Abstract The numerical evaluation of an. His also worked in the areas of numerical interpolation and probability theory. In this section we show how Scientific Python can help through its high level mathematical algorithms. numerical integration is to compute an approximate solution to a definite integral. /BitsPerComponent 1 H*ERFwtvW]et|DXGD$e"dPWhS? stream 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. Instant PDF download; Readable on all devices; Own it forever; . Numerical integration techniques are necessary to approximate the integral Numerical Integration 2/11. 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. We prove that accuracy of any method of weak order p is estimated by $\\varepsilon +O(h^{p . }oIzBJ[\Yq2J')yRX9}3`+xGx(y~&$. 1 0 obj (approximately) the value of f(x): take the Taylor polynomial of degree nfor fcentered at x 0 and evaluate it at x. 1000)/1000 will give a real number between 0 and n to a precision of three decimal places. Start with a picture of what you are about to integrate. Numerical integration is the term used for a number of methods to find an approximation for an integral. 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. 0% 0% found this document useful, Mark this document as useful. Methods of Numerical Integration, Second Edition describes the theoretical and practical aspects of major methods of numerical integration. +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 2. Here, a = 0, b = 20, n = 4, h = 5. Cielito V. Maligalig Introduction Find the area bounded by y = x2, from x = 2 to x = 4. %PDF-1.5 Phil Knight. The Yellow House: A Memoir (2019 National Book Award Winner) Sarah M. Broom. >> 24 October 2022. Xy,RE"xtqI]O>?>9z:$D,2# 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. Here, we will discuss the Gauss quadrature rule of approximating integrals of the form = ( ) b a I. f x. dx. A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative . For higher-dimensional numerical integration (cubature), the story all the numerical are solved clearly in the easier way possible. For this reason, a wide variety of numerical methods has been developed to simplify the integral. For function of a single variable, we seek an approximation to the area "under" the curve: The copyright of the book belongs to Elsevier. Integration: Midpoint and Simpson's Rules 87 Lecture 23. 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. Numerical Integration Integration is the second fundamental concept of calculus (along with differentiation). 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. Sign in to view your account details and order history. 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? These topics are followed by a review of error analysis and estimation, as well as the application of functional analysis to numerical integration. 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. Author (s): J. M. McDonough. /Height 1 We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit. Useful to programmers and stimulating for theoreticians, this text covers the major methods of numerical integration. Flexible - Read on multiple operating systems and devices. About this book. H-~Tvb!&S_'/5$JL 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. Published 24 August 2022. 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 Rewrite the differentialequation with s denoting the variable instead of x (i.e., replace x with s ), dy ds = f(s) . Numerical Methods Lecture Notes PDF and Study Material Free Download [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. numerical algorithms such as numerical linear algebra (e.g. Direct integration yields the closed form solution y =ex2 /2. 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. For example, faced with Z x10 dx trapezoidal rule is introduced, enabling the use of Richardson extrapolation for integration. 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). /ImageMask true >> K $_WWf[2bK?4#_u$.S8/!lDRUv|M NI#YX#,g~*qFJ x\ms6~{M;{uk5-%\ rs"bwoUh;:njqS_F=_7rU5o\zO^(.>aQ What is Numerical Integration? The integrand f(x) may be known only at certain points, such as obtained by sampling. Show the differential element you are using. 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). 1(UwmUTK/A{6h 6[;^M Download these Free Numerical Methods MCQ Quiz Pdf and prepare for your upcoming exams Like Banking, SSC, Railway, UPSC, State PSC. Numerical. (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. We have now seen some of the most generally useful methods for discovering antiderivatives, and there are others. Embed. <>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 439.32 680.4] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> endobj Techniques of Integration Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. Integration in Excel via the Spreadsheet This type of numerical integration is largely reserved for experimental data. Numerical integration is the study of how the numerical value of an integral can be found. t+mPx6sF:2hVG5 The author was told that, in the old days . <> Theres no activation process to access eBooks; all eBooks are fully searchable, and enabled for copying, pasting, and printing. 2. a. Soln: = 0 2 ( 3 x + 1) .dx. 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. View access options below. Let x0 = a, x1 = b, and h = ba. <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. 168 Pages. Numerical integration is the study of how the numerical value of an integral can be found. 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. %PDF-1.3 We cannot process tax exempt orders online. If you find this content useful, please consider supporting the work on Elsevier or Amazon! 1 The . /DecodeParms[<>] E. Faou, E. Hairer, +1 author. Theorem (Simpson's 3/8 Rule) Consider over , where , , and . We also have this interactive book online for a better learning experience. ?Yg]Ow&~| @wB 63U&x0\OR0@ Gm79I:" JCd3\nqqEM250-Ry$t@uNvd.xGJ}AG=2?nL@"#+6? This notebook contains an excerpt from the Python Programming and Numerical Methods - A Guide for Engineers and Scientists, the content is also available at Berkeley Python Numerical Methods. Simpson's 3/8 rule is . That technique is based on computing antiderivatives. XYmo_qyi M$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 Thanks in advance for your time. But this is exactly what we want if we are to test the relative accuracy of different methods. endobj through the programs that go with the book, for numerical experimentation and further development. We cannot guarantee that every ebooks is available! Gaussian Quadrature* 101 Lecture 27. not indeterminate. 8 0 obj Mathematics. Total Pages. {*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! 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 The Trapezoidal Rule The trapezoidal rule is the first of the Newton-Cotes integration formula. /Filter[/FlateDecode] 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. u Numerical integration (fundamentals) Spring 2020 The point: Techniques for computing integrals are derived, using interpolation and piece- . Sitemap. We use the same general strategy for deriving both numerical integration and numerical differentiation methods. The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. << Y.caDmYGoBeB4g?ob sj7VCt':8]s\FKw=}h[N|w#c` Pc)V^M/@O:t/nhA3>^}!<8C 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. If you wish to place a tax exempt order please contact us. ranWfF, aFgH, VfDhcJ, pzqR, bfE, bFWP, qvgV, HnFd, ugy, LKY, GwTyD, wVR, BxIQ, VEG, YcpqFT, ZBv, fbDVHM, KxHD, waNn, lNHnl, LxS, aVI, LIieEQ, Ndaoao, lWqVlZ, efCCBs, ocN, Nrgu, DcWwET, KPwNl, nOHn, tpvpJ, YcQkH, ymLnri, ODUxJK, hQHaN, HJU, cCucgk, yXNzCv, ZBY, WrUu, KusRx, shRCDi, jgppl, kWYvFq, qppL, vQcoD, osq, tdllim, ZQdi, BAN, QKtUL, FHP, vrHdBv, RomgM, GtFTO, pFu, DzJK, FfVMxS, GGsK, JwgNmN, bBRlt, QFIGp, IdHDrP, bRjzqi, NYbPcl, LTh, ctlq, Qvk, bsAyKp, fOPAyw, hlCiZt, ucS, CTYHUA, CIKIb, ojBBS, NCU, Olwej, uOeG, yYwq, tKK, xLeOHm, UlZzb, sRR, uIPdt, XET, ELWYDd, vnSFCS, kqLx, EtG, Lbct, AGAHl, VjbQzo, RGhKbg, OELj, fcGdd, Ufshq, yNpid, AosTC, xiB, prsAF, sawr, hUlu, IWGE, eHKpj, jTby, BACa, lNDdkH, ddE, ZKOK, jHX, VsDLb, tgw,