The Generalized Riemann Integral Robert M
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Riemann Sums Fundamental Theorem of Calculus Indefinite
Riemann Sums Partition P = {x0, x1, . , xn} of an interval [a, b]. ck ∈ [xk−1, xk] Pn R(f, P, a, b) = k=1 f(ck)∆xk As the widths ∆xk of the subintervals approach 0, the Riemann Sums hopefully approach a limit, the integral of f from a to b, written R b a f(x) dx. Fundamental Theorem of Calculus Theorem 1 (FTC-Part I). If f is continuous on [a, b], then F (x) = R x 0 a f(t) dt is defined on [a, b] and F (x) = f(x). Theorem 2 (FTC-Part II). If f is continuous on [a, b] and F (x) = R R b b f(x) dx on [a, b], then a f(x) dx = F (x) a = F (b) − F (a). Indefinite Integrals Indefinite Integral: R f(x) dx = F (x) if and only if F 0(x) = f(x). In other words, the terms indefinite integral and antiderivative are synonymous. Every differentiation formula yields an integration formula. Substitution Rule For Indefinite Integrals: If u = g(x), then R f(g(x))g0(x) dx = R f(u) du. For Definite Integrals: R b 0 R g(b) If u = g(x), then a f(g(x))g (x) dx = g(a) f( u) du. Steps in Mechanically Applying the Substitution Rule Note: The variables do not have to be called x and u. (1) Choose a substitution u = g(x). du (2) Calculate = g0(x). dx du (3) Treat as if it were a fraction, a quotient of differentials, and dx du solve for dx, obtaining dx = . -
Definition of Riemann Integrals
The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) Let [a; b] be a given interval. By a partition P of [a; b] we mean a finite set of points x0; x1;:::; xn where a = x0 ≤ x1 ≤ · · · ≤ xn−1 ≤ xn = b We write ∆xi = xi − xi−1 for i = 1;::: n. The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) Now suppose f is a bounded real function defined on [a; b]. Corresponding to each partition P of [a; b] we put Mi = sup f (x)(xi−1 ≤ x ≤ xi ) mi = inf f (x)(xi−1 ≤ x ≤ xi ) k X U(P; f ) = Mi ∆xi i=1 k X L(P; f ) = mi ∆xi i=1 The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) We put Z b fdx = inf U(P; f ) a and we call this the upper Riemann integral of f . We also put Z b fdx = sup L(P; f ) a and we call this the lower Riemann integral of f . The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) If the upper and lower Riemann integrals are equal then we say f is Riemann-integrable on [a; b] and we write f 2 R. We denote the common value, which we call the Riemann integral of f on [a; b] as Z b Z b fdx or f (x)dx a a The Definition and Existence of the Integral Left and Right Riemann Integrals If f is bounded then there exists two numbers m and M such that m ≤ f (x) ≤ M if (a ≤ x ≤ b) Hence for every partition P we have m(b − a) ≤ L(P; f ) ≤ U(P; f ) ≤ M(b − a) and so L(P; f ) and U(P; f ) both form bounded sets (as P ranges over partitions). -
The Definite Integrals of Cauchy and Riemann
Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Analysis Mathematics via Primary Historical Sources (TRIUMPHS) Summer 2017 The Definite Integrals of Cauchy and Riemann Dave Ruch [email protected] Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_analysis Part of the Analysis Commons, Curriculum and Instruction Commons, Educational Methods Commons, Higher Education Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits ou.y Recommended Citation Ruch, Dave, "The Definite Integrals of Cauchy and Riemann" (2017). Analysis. 11. https://digitalcommons.ursinus.edu/triumphs_analysis/11 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Analysis by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. The Definite Integrals of Cauchy and Riemann David Ruch∗ August 8, 2020 1 Introduction Rigorous attempts to define the definite integral began in earnest in the early 1800s. A major motivation at the time was the search for functions that could be expressed as Fourier series as follows: 1 a0 X f (x) = + (a cos (kw) + b sin (kw)) where the coefficients are: (1) 2 k k k=1 1 Z π 1 Z π 1 Z π a0 = f (t) dt; ak = f (t) cos (kt) dt ; bk = f (t) sin (kt) dt. 2π −π π −π π −π P k Expanding a function as an infinite series like this may remind you of Taylor series ckx from introductory calculus, except that the Fourier coefficients ak; bk are defined by definite integrals and sine and cosine functions are used instead of powers of x. -
Generalizations of the Riemann Integral: an Investigation of the Henstock Integral
Generalizations of the Riemann Integral: An Investigation of the Henstock Integral Jonathan Wells May 15, 2011 Abstract The Henstock integral, a generalization of the Riemann integral that makes use of the δ-fine tagged partition, is studied. We first consider Lebesgue’s Criterion for Riemann Integrability, which states that a func- tion is Riemann integrable if and only if it is bounded and continuous almost everywhere, before investigating several theoretical shortcomings of the Riemann integral. Despite the inverse relationship between integra- tion and differentiation given by the Fundamental Theorem of Calculus, we find that not every derivative is Riemann integrable. We also find that the strong condition of uniform convergence must be applied to guarantee that the limit of a sequence of Riemann integrable functions remains in- tegrable. However, by slightly altering the way that tagged partitions are formed, we are able to construct a definition for the integral that allows for the integration of a much wider class of functions. We investigate sev- eral properties of this generalized Riemann integral. We also demonstrate that every derivative is Henstock integrable, and that the much looser requirements of the Monotone Convergence Theorem guarantee that the limit of a sequence of Henstock integrable functions is integrable. This paper is written without the use of Lebesgue measure theory. Acknowledgements I would like to thank Professor Patrick Keef and Professor Russell Gordon for their advice and guidance through this project. I would also like to acknowledge Kathryn Barich and Kailey Bolles for their assistance in the editing process. Introduction As the workhorse of modern analysis, the integral is without question one of the most familiar pieces of the calculus sequence. -
Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration Over a Polygonal Element Sulaiman Abo Diab
Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration over a Polygonal Element Sulaiman Abo Diab To cite this version: Sulaiman Abo Diab. Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration over a Polygonal Element. 2019. hal-02306578v1 HAL Id: hal-02306578 https://hal.archives-ouvertes.fr/hal-02306578v1 Preprint submitted on 6 Oct 2019 (v1), last revised 5 Nov 2019 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration over a Polygonal Element Sulaiman Abo Diab Faculty of Civil Engineering, Tishreen University, Lattakia, Syria [email protected] Abstracts: In this paper, the Riemann integral and the fundamental of calculus will be used to perform double integrals on polygonal domain surrounded by closed curves. In this context, the double integral with two variables over the domain is transformed into sequences of single integrals with one variable of its primitive. The sequence is arranged anti clockwise starting from the minimum value of the variable of integration. Finally, the integration over the closed curve of the domain is performed using only one variable. -
Lecture 15-16 : Riemann Integration Integration Is Concerned with the Problem of finding the Area of a Region Under a Curve
1 Lecture 15-16 : Riemann Integration Integration is concerned with the problem of ¯nding the area of a region under a curve. Let us start with a simple problem : Find the area A of the region enclosed by a circle of radius r. For an arbitrary n, consider the n equal inscribed and superscibed triangles as shown in Figure 1. f(x) f(x) π 2 n O a b O a b Figure 1 Figure 2 Since A is between the total areas of the inscribed and superscribed triangles, we have nr2sin(¼=n)cos(¼=n) · A · nr2tan(¼=n): By sandwich theorem, A = ¼r2: We will use this idea to de¯ne and evaluate the area of the region under a graph of a function. Suppose f is a non-negative function de¯ned on the interval [a; b]: We ¯rst subdivide the interval into a ¯nite number of subintervals. Then we squeeze the area of the region under the graph of f between the areas of the inscribed and superscribed rectangles constructed over the subintervals as shown in Figure 2. If the total areas of the inscribed and superscribed rectangles converge to the same limit as we make the partition of [a; b] ¯ner and ¯ner then the area of the region under the graph of f can be de¯ned as this limit and f is said to be integrable. Let us de¯ne whatever has been explained above formally. The Riemann Integral Let [a; b] be a given interval. A partition P of [a; b] is a ¯nite set of points x0; x1; x2; : : : ; xn such that a = x0 · x1 · ¢ ¢ ¢ · xn¡1 · xn = b. -
On the Approximation of Riemann-Liouville Integral by Fractional Nabla H-Sum and Applications L Khitri-Kazi-Tani, H Dib
On the approximation of Riemann-Liouville integral by fractional nabla h-sum and applications L Khitri-Kazi-Tani, H Dib To cite this version: L Khitri-Kazi-Tani, H Dib. On the approximation of Riemann-Liouville integral by fractional nabla h-sum and applications. 2016. hal-01315314 HAL Id: hal-01315314 https://hal.archives-ouvertes.fr/hal-01315314 Preprint submitted on 12 May 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the approximation of Riemann-Liouville integral by fractional nabla h-sum and applications L. Khitri-Kazi-Tani∗ H. Dib y Abstract First of all, in this paper, we prove the convergence of the nabla h- sum to the Riemann-Liouville integral in the space of continuous functions and in some weighted spaces of continuous functions. The connection with time scales convergence is discussed. Secondly, the efficiency of this approximation is shown through some Cauchy fractional problems with singularity at the initial value. The fractional Brusselator system is solved as a practical case. Keywords. Riemann-Liouville integral, nabla h-sum, fractional derivatives, fractional differential equation, time scales convergence MSC (2010). 26A33, 34A08, 34K28, 34N05 1 Introduction The numerous applications of fractional calculus in the mathematical modeling of physical, engineering, economic and chemical phenomena have contributed to the rapidly growing of this theory [12, 19, 21, 18], despite the discrepancy on definitions of the fractional operators leading to nonequivalent results [6]. -
Intuitive Mathematical Discourse About the Complex Path Integral Erik Hanke
Intuitive mathematical discourse about the complex path integral Erik Hanke To cite this version: Erik Hanke. Intuitive mathematical discourse about the complex path integral. INDRUM 2020, Université de Carthage, Université de Montpellier, Sep 2020, Cyberspace (virtually from Bizerte), Tunisia. hal-03113845 HAL Id: hal-03113845 https://hal.archives-ouvertes.fr/hal-03113845 Submitted on 18 Jan 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Intuitive mathematical discourse about the complex path integral Erik Hanke1 1University of Bremen, Faculty of Mathematics and Computer Science, Germany, [email protected] Interpretations of the complex path integral are presented as a result from a multi-case study on mathematicians’ intuitive understanding of basic notions in complex analysis. The first case shows difficulties of transferring the image of the integral in real analysis as an oriented area to the complex setting, and the second highlights the complex path integral as a tool in complex analysis with formal analogies to path integrals in multivariable calculus. These interpretations are characterised as a type of intuitive mathematical discourse and the examples are analysed from the point of view of substantiation of narratives within the commognitive framework. -
General Riemann Integrals and Their Computation Via Domain
Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1999 General Riemann Integrals and Their omputC ation via Domain. Bin Lu Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Lu, Bin, "General Riemann Integrals and Their omputC ation via Domain." (1999). LSU Historical Dissertations and Theses. 7001. https://digitalcommons.lsu.edu/gradschool_disstheses/7001 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, phnt bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. -
More Information About Line and Contour Integrals
More information about line and contour integrals. Real line integrals. Let P (x, y)andQ(x, y) be continuous real valued functions on the plane and let γ be a smooth directed curve. We wish to define the line integral Z P (x, y) dx + Q(x, y) dy. γ This can be done in several ways. One of which is to partition the curve γ by a very large number of points, define Riemann sums for the partition, and then take a limit as the “mesh” of the partition gets finer and finer. However an argument like the one we did in class for complex line integrals shows that if c(t)=(x(t),y(t)) with a ≤ t ≤ b is a smooth parameterization of γ then Z Z b P (x, y) dx + Q(x, y) dy = P (x(t),y(t))x0(t)+Q(x(t),y(t)) dt. γ a R This reduces computing γ P (x, y) dx + Q(x, y) dy to first finding a parameterization of γ and then computing a standard Riemann integral such as we have all seen in calculus. Here is an example. Let γ be the upper half of the circle |z| =1 transversed counter clockwise. Then compute Z (1) xy2 dx − (x + y + x3y) dy. γ This half circle is parameterized by x(t)=cos(t),y(t)=sin(t) with 0 ≤ t ≤ π. Now the calculation proceeds just as if we were doing a change of variables. That is dx = − sin(t) dt, dy = cos(t) dt 1 2 Using these formulas in (1) gives Z xy2 dx − (x + y + x3y) dy γ Z π = cos(t)sin2(t)(− sin(t)) dt − (cos(t)+sin(t)+cos3(t)sin(t)) cos(t) dt Z0 π = cos(t)sin2(t)(− sin(t)) − (cos(t)+sin(t)+cos3(t)sin(t)) cos(t) dt 0 With a little work1 this can be shown to have the value −(2/5+π/2). -
Fractional Integrals and Derivatives • Yuri Luchko Fractional Integrals and Derivatives: “True” Versus “False”
Fractional Integrals and Derivatives • Yuri Luchko Yuri • Fractional Integrals and Derivatives “True” versus “False” Edited by Yuri Luchko Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Fractional Integrals and Derivatives: “True” versus “False” Fractional Integrals and Derivatives: “True” versus “False” Editor Yuri Luchko MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editor Yuri Luchko Beuth Technical University of Applied Sciences Berlin Germany Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) (available at: https://www.mdpi.com/journal/mathematics/special issues/Fractional Integrals Derivatives). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Volume Number, Page Range. ISBN 978-3-0365-0494-0 (Hbk) ISBN 978-3-0365-0494-0 (PDF) © 2021 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editor .............................................. vii Preface to ”Fractional Integrals and Derivatives: “True” versus “False”” ............. ix Yuri Luchko and Masahiro Yamamoto The General Fractional Derivative and Related Fractional Differential Equations Reprinted from: Mathematics 2020, 8, 2115, doi:10.3390/math8122115 ............... -
Using the TI-89 to Find Riemann Sums
Using the TI-89 to find Riemann sums Z b If function f is continuous on interval [a, b], f(x) dx exists and can be approximated a using either of the Riemann sums n n X X f(xi)∆x or f(xi−1)∆x i=1 i=1 (b−a) where ∆x = n and n is the number of subintervals chosen. The first of these Riemann sums evaluates function f at the right endpoint of each subinterval; the second evaluates at the left endpoint of each subinterval. n X P To evaluate f(xi) using the TI-89, go to F3 Calc and select 4: ( sum i=1 The command line should then be completed in the following form: P (f(xi), i, 1, n) The actual Riemann sum is then determined by multiplying this ans by ∆x (or incorpo- rating this multiplication into the same command line.) [Warning: Be sure to set the MODE as APPROXIMATE in this case. Otherwise, the calculator will spend an inordinate amount of time attempting to express each term of the summation in exact symbolic form.] Z 4 √ Example: To approximate 1 + x3 dx using Riemann sums with n = 100 subinter- 2 b−a 2 i vals, note first that ∆x = n = 100 = .02. Also xi = 2 + i∆x = 2 + 50 . So the right endpoint approximation will be found by entering √ P( (2 + (1 + i/50)∧3), i, 1, 100) × .02 which gives 10.7421. This is an overestimate, since the function is increasing on the given interval. To find the left endpoint approximation, replace i by (i − 1) in the square root part of the command.