What Does/Did the Integral Sign Represent?

Total Page：16

File Type：pdf, Size：1020Kb

What does/did the Integral Sign represent? Some comments on the origin of the usual integral sign notation b f(x) dx Za might be useful in understanding the ways in which integrals arise in questions of inde- pendent interest. We begin with the limit formula as stated on text page 125 of Guichard (this is page 137 in the pdf file): n−1 b b − a f(x) dx = lim f(ti) dt ; where ∆t = Z n!1 a Xi=0 n If we view the Riemann sums on the right as approximations to the area under the curve y = f(x) for a ≤ x ≤ b, then the sum is actually the sum of the areas of n rectangles of width ∆t, and the crucial fact is that these converge to a limiting value (the \actual area") as n ! 1. The integral symbol is a version of the essentially obsolete letter which is now written as s, and it was first employed to convey the idea that the integral isR a continuous sum of the quantities f(x) dx, which can be viewed as the areas of rectangles whose vertical sides have length f(x) and whose horizontal sides have a so-called infinitesimally thin width which is denoted by dx. Note on infinitesimals. The idea of working with infinitesimally small quantities turns out to be very useful in working with problems in the sciences and engineering, both as a tool for finding solutions and also for explaining them. However, ever since the invention of calculus mathematicians and others have recognized that justifying the use of such objects in a simple, logically rigorous fashion is at best challenging and at worst not feasible. These difficulties are one reason that mathematicians ended up formulating concepts like limits with δ and " terminology. About 50 years ago some mathematicians (most notably Abraham Robinson) finally managed to construct a logically sound theory of infinitesimals, and the calculus text by J. Keisler (cited in Guichard) actually develops calculus in this fashion. However, for a variety of reasons it seems unlikely that infinitesimals will replace deltas and epsilons in the mainstream approach to calculus any time soon. 1.

Copy Link

Recommended publications

ESA Missions AO Analysis
ESA Announcements of Opportunity Outcome Analysis Arvind Parmar Head, Science Support Office ESA Directorate of Science With thanks to Kate Isaak, Erik Kuulkers, Göran Pilbratt and Norbert Schartel (Project Scientists) ESA UNCLASSIFIED - For Official Use The ESA Fleet for Astrophysics ESA UNCLASSIFIED - For Official Use Dual-Anonymous Proposal Reviews | STScI | 25/09/2019 | Slide 2 ESA Announcement of Observing Opportunities Ø Observing time AOs are normally only used for ESA’s observatory missions – the targets/observing strategies for the other missions are generally the responsibility of the Science Teams. Ø ESA does not provide funding to successful proposers. Ø Results for ESA-led missions with recent AOs presented: • XMM-Newton • INTEGRAL • Herschel Ø Gender information was not requested in the AOs. It has been ”manually” derived by the project scientists and SOC staff. ESA UNCLASSIFIED - For Official Use Dual-Anonymous Proposal Reviews | STScI | 25/09/2019 | Slide 3 XMM-Newton – ESA’s Large X-ray Observatory ESA UNCLASSIFIED - For Official Use Dual-Anonymous Proposal Reviews | STScI | 25/09/2019 | Slide 4 XMM-Newton Ø ESA’s second X-ray observatory. Launched in 1999 with annual calls for observing proposals. Operational. Ø Typically 500 proposals per XMM-Newton Call with an over-subscription in observing time of 5-7. Total of 9233 proposals. Ø The TAC typically consists of 70 scientists divided into 13 panels with an overall TAC chair. Ø Output is >6000 refereed papers in total, >300 per year ESA UNCLASSIFIED - For Official Use
[Show full text]
Differential Calculus and by Era Integral Calculus, Which Are Related by in Early Cultures in Classical Antiquity the Fundamental Theorem of Calculus
History of calculus - Wikipedia, the free encyclopedia 1/1/10 5:02 PM History of calculus From Wikipedia, the free encyclopedia History of science This is a sub-article to Calculus and History of mathematics. History of Calculus is part of the history of mathematics focused on limits, functions, derivatives, integrals, and infinite series. The subject, known Background historically as infinitesimal calculus, Theories/sociology constitutes a major part of modern Historiography mathematics education. It has two major Pseudoscience branches, differential calculus and By era integral calculus, which are related by In early cultures in Classical Antiquity the fundamental theorem of calculus. In the Middle Ages Calculus is the study of change, in the In the Renaissance same way that geometry is the study of Scientific Revolution shape and algebra is the study of By topic operations and their application to Natural sciences solving equations. A course in calculus Astronomy is a gateway to other, more advanced Biology courses in mathematics devoted to the Botany study of functions and limits, broadly Chemistry Ecology called mathematical analysis. Calculus Geography has widespread applications in science, Geology economics, and engineering and can Paleontology solve many problems for which algebra Physics alone is insufficient. Mathematics Algebra Calculus Combinatorics Contents Geometry Logic Statistics 1 Development of calculus Trigonometry 1.1 Integral calculus Social sciences 1.2 Differential calculus Anthropology 1.3 Mathematical analysis
[Show full text]
Notes on Calculus II Integral Calculus Miguel A. Lerma
Notes on Calculus II Integral Calculus Miguel A. Lerma November 22, 2002 Contents Introduction 5 Chapter 1. Integrals 6 1.1. Areas and Distances. The Definite Integral 6 1.2. The Evaluation Theorem 11 1.3. The Fundamental Theorem of Calculus 14 1.4. The Substitution Rule 16 1.5. Integration by Parts 21 1.6. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. Partial Fractions 32 1.8. Integration using Tables and CAS 39 1.9. Numerical Integration 41 1.10. Improper Integrals 46 Chapter 2. Applications of Integration 50 2.1. More about Areas 50 2.2. Volumes 52 2.3. Arc Length, Parametric Curves 57 2.4. Average Value of a Function (Mean Value Theorem) 61 2.5. Applications to Physics and Engineering 63 2.6. Probability 69 Chapter 3. Differential Equations 74 3.1. Differential Equations and Separable Equations 74 3.2. Directional Fields and Euler’s Method 78 3.3. Exponential Growth and Decay 80 Chapter 4. Infinite Sequences and Series 83 4.1. Sequences 83 4.2. Series 88 4.3. The Integral and Comparison Tests 92 4.4. Other Convergence Tests 96 4.5. Power Series 98 4.6. Representation of Functions as Power Series 100 4.7. Taylor and MacLaurin Series 103 3 CONTENTS 4 4.8. Applications of Taylor Polynomials 109 Appendix A. Hyperbolic Functions 113 A.1. Hyperbolic Functions 113 Appendix B. Various Formulas 118 B.1. Summation Formulas 118 Appendix C. Table of Integrals 119 Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus.
[Show full text]
Analyzing Applied Calculus Student Understanding of Definite Integrals in Real-Life Applications
Graduate Theses, Dissertations, and Problem Reports 2021 ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS CODY HOOD [email protected] Follow this and additional works at: https://researchrepository.wvu.edu/etd Part of the Mathematics Commons, and the Science and Mathematics Education Commons Recommended Citation HOOD, CODY, "ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS" (2021). Graduate Theses, Dissertations, and Problem Reports. 8260. https://researchrepository.wvu.edu/etd/8260 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Graduate Theses, Dissertations, and Problem Reports 2021 ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS CODY HOOD Follow this and additional works at: https://researchrepository.wvu.edu/etd Part of the Mathematics Commons, and the Science and Mathematics Education Commons ANALYZING APPLIED CALCULUS STUDENT UNDERSTANDING OF DEFINITE INTEGRALS IN REAL-LIFE APPLICATIONS Cody Hood Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Vicki Sealey, Ph.D., Chair Harvey Diamond, Ph.D.
[Show full text]
Two Fundamental Theorems About the Definite Integral
Two Fundamental Theorems about the Definite Integral These lecture notes develop the theorem Stewart calls The Fundamental Theorem of Calculus in section 5.3. The approach I use is slightly different than that used by Stewart, but is based on the same fundamental ideas. 1 The definite integral Recall that the expression b f(x) dx ∫a is called the definite integral of f(x) over the interval [a,b] and stands for the area underneath the curve y = f(x) over the interval [a,b] (with the understanding that areas above the x-axis are considered positive and the areas beneath the axis are considered negative). In today's lecture I am going to prove an important connection between the definite integral and the derivative and use that connection to compute the definite integral. The result that I am eventually going to prove sits at the end of a chain of earlier definitions and intermediate results. 2 Some important facts about continuous functions The first intermediate result we are going to have to prove along the way depends on some definitions and theorems concerning continuous functions. Here are those definitions and theorems. The definition of continuity A function f(x) is continuous at a point x = a if the following hold 1. f(a) exists 2. lim f(x) exists xœa 3. lim f(x) = f(a) xœa 1 A function f(x) is continuous in an interval [a,b] if it is continuous at every point in that interval. The extreme value theorem Let f(x) be a continuous function in an interval [a,b].
[Show full text]
Herschel Space Observatory and the NASA Herschel Science Center at IPAC
Herschel Space Observatory and The NASA Herschel Science Center at IPAC u George Helou u Implementing “Portals to the Universe” Report April, 2012 Implementing “Portals to the Universe”, April 2012 Herschel/NHSC 1 [OIII] 88 µm Herschel: Cornerstone FIR/Submm Observatory z = 3.04 u Three instruments: imaging at {70, 100, 160}, {250, 350 and 500} µm; spectroscopy: grating [55-210]µm, FTS [194-672]µm, Heterodyne [157-625]µm; bolometers, Ge photoconductors, SIS mixers; 3.5m primary at ambient T u ESA mission with significant NASA contributions, May 2009 – February 2013 [+/-months] cold Implementing “Portals to the Universe”, April 2012 Herschel/NHSC 2 Herschel Mission Parameters u Userbase v International; most investigator teams are international u Program Model v Observatory with Guaranteed Time and competed Open Time v ESA “Corner Stone Mission” (>$1B) with significant NASA contributions v NASA Herschel Science Center (NHSC) supports US community u Proposals/cycle (2 Regular Open Time cycles) v Submissions run 500 to 600 total, with >200 with US-based PI (~x3.5 over- subscription) u Users/cycle v US-based co-Investigators >500/cycle on >100 proposals u Funding Model v NASA funds US data analysis based on ESA time allocation u Default Proprietary Data Period v 6 months now, 1 year at start of mission Implementing “Portals to the Universe”, April 2012 Herschel/NHSC 3 Best Practices: International Collaboration u Approach to projects led elsewhere needs to be designed carefully v NHSC Charter remains firstly to support US community v But ultimately success of THE mission helps everyone v Need to express “dual allegiance” well and early to lead/other centers u NHSC became integral part of the larger team, worked for Herschel success, though focused on US community participation v Working closely with US community reveals needs and gaps for all users v Anything developed by NHSC is available to all users of Herschel v Trust follows from good teaming: E.g.
[Show full text]
MATH 1512: Calculus – Spring 2020 (Dual Credit) Instructor: Ariel
MATH 1512: Calculus – Spring 2020 (Dual Credit) Instructor: Ariel Ramirez, Ph.D. Email: [email protected] Office: LRC 172 Phone: 505-925-8912 OFFICE HOURS: In the Math Center/LRC or by Appointment COURSE DESCRIPTION: Limits. Continuity. Derivative: definition, rules, geometric and rate-of- change interpretations, applications to graphing, linearization and optimization. Integral: definition, fundamental theorem of calculus, substitution, applications to areas, volumes, work, average. Meets New Mexico Lower Division General Education Common Core Curriculum Area II: Mathematics (NMCCN 1614). (4 Credit Hours). Prerequisites: ((123 or ACCUPLACER College-Level Math =100-120) and (150 or ACT Math =28- 31 or SAT Math Section =660-729)) or (153 or ACT Math =>32 or SAT Math Section =>730). COURSE OBJECTIVES: 1. State, motivate and interpret the definitions of continuity, the derivative, and the definite integral of a function, including an illustrative figure, and apply the definition to test for continuity and differentiability. In all cases, limits are computed using correct and clear notation. Student is able to interpret the derivative as an instantaneous rate of change, and the definite integral as an averaging process. 2. Use the derivative to graph functions, approximate functions, and solve optimization problems. In all cases, the work, including all necessary algebra, is shown clearly, concisely, in a well-organized fashion. Graphs are neat and well-annotated, clearly indicating limiting behavior. English sentences summarize the main results and appropriate units are used for all dimensional applications. 3. Graph, differentiate, optimize, approximate and integrate functions containing parameters, and functions defined piecewise. Differentiate and approximate functions defined implicitly. 4. Apply tools from pre-calculus and trigonometry correctly in multi-step problems, such as basic geometric formulas, graphs of basic functions, and algebra to solve equations and inequalities.
[Show full text]
Connes on the Role of Hyperreals in Mathematics
Found Sci DOI 10.1007/s10699-012-9316-5 Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics Vladimir Kanovei · Mikhail G. Katz · Thomas Mormann © Springer Science+Business Media Dordrecht 2012 Abstract We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in func- tional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being “vir- tual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnera- ble to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace − (featured on the front cover of Connes’ magnum opus) V.
[Show full text]
Differentiating an Integral
Differentiating an Integral P. Haile, February 2020 b(t) Leibniz Rule. If (t) = a(t) f(z, t)dz, then R d b(t) @f(z, t) db da = dz + f (b(t), t) f (a(t), t) . dt @t dt dt Za(t) That’s the rule; now let’s see why this is so. We start by reviewing some things you already know. Let f be a smooth univariate function (i.e., a “nice” function of one variable). We know from the fundamental theorem of calculus that if the function is defined as x (x) = f(z) dz Zc (where c is a constant) then d (x) = f(x). (1) dx You can gain some really useful intuition for this by remembering that the x integral c f(z) dz is the area under the curve y = f(z), between z = a and z = x. Draw this picture to see that as x increases infinitessimally, the added contributionR to the area is f (x). Likewise, if c (x) = f(z) dz Zx then d (x) = f(x). (2) dx Here, an infinitessimal increase in x reduces the area under the curve by an amount equal to the height of the curve. Now suppose f is a function of two variables and define b (t) = f(z, t)dz (3) Za where a and b are constants. Then d (t) b @f(z, t) = dz (4) dt @t Za i.e., we can swap the order of the operations of integration and differentiation and “differentiate under the integral.” You’ve probably done this “swapping” 1 before.
[Show full text]
International Space Station Basics Components of The
National Aeronautics and Space Administration International Space Station Basics The International Space Station (ISS) is the largest orbiting can see 16 sunrises and 16 sunsets each day! During the laboratory ever built. It is an international, technological, daylight periods, temperatures reach 200 ºC, while and political achievement. The five international partners temperatures during the night periods drop to -200 ºC. include the space agencies of the United States, Canada, The view of Earth from the ISS reveals part of the planet, Russia, Europe, and Japan. not the whole planet. In fact, astronauts can see much of the North American continent when they pass over the The first parts of the ISS were sent and assembled in orbit United States. To see pictures of Earth from the ISS, visit in 1998. Since the year 2000, the ISS has had crews living http://eol.jsc.nasa.gov/sseop/clickmap/. continuously on board. Building the ISS is like living in a house while constructing it at the same time. Building and sustaining the ISS requires 80 launches on several kinds of rockets over a 12-year period. The assembly of the ISS Components of the ISS will continue through 2010, when the Space Shuttle is retired from service. The components of the ISS include shapes like canisters, spheres, triangles, beams, and wide, flat panels. The When fully complete, the ISS will weigh about 420,000 modules are shaped like canisters and spheres. These are kilograms (925,000 pounds). This is equivalent to more areas where the astronauts live and work. On Earth, car- than 330 automobiles.
[Show full text]
The Definite Integral
Mathematics Learning Centre Introduction to Integration Part 2: The Definite Integral Mary Barnes c 1999 University of Sydney Contents 1Introduction 1 1.1 Objectives . ................................. 1 2 Finding Areas 2 3 Areas Under Curves 4 3.1 What is the point of all this? ......................... 6 3.2 Note about summation notation ........................ 6 4 The Definition of the Definite Integral 7 4.1 Notes . ..................................... 7 5 The Fundamental Theorem of the Calculus 8 6 Properties of the Definite Integral 12 7 Some Common Misunderstandings 14 7.1 Arbitrary constants . ............................. 14 7.2 Dummy variables . ............................. 14 8 Another Look at Areas 15 9 The Area Between Two Curves 19 10 Other Applications of the Definite Integral 21 11 Solutions to Exercises 23 Mathematics Learning Centre, University of Sydney 1 1Introduction This unit deals with the definite integral.Itexplains how it is defined, how it is calculated and some of the ways in which it is used. We shall assume that you are already familiar with the process of finding indefinite inte- grals or primitive functions (sometimes called anti-differentiation) and are able to ‘anti- differentiate’ a range of elementary functions. If you are not, you should work through Introduction to Integration Part I: Anti-Differentiation, and make sure you have mastered the ideas in it before you begin work on this unit. 1.1 Objectives By the time you have worked through this unit you should: • Be familiar with the definition of the definite integral as the limit of a sum; • Understand the rule for calculating definite integrals; • Know the statement of the Fundamental Theorem of the Calculus and understand what it means; • Be able to use definite integrals to find areas such as the area between a curve and the x-axis and the area between two curves; • Understand that definite integrals can also be used in other situations where the quantity required can be expressed as the limit of a sum.
[Show full text]
Herschel Space Observatory-An ESA Facility for Far-Infrared And
Astronomy & Astrophysics manuscript no. 14759glp c ESO 2018 October 22, 2018 Letter to the Editor Herschel Space Observatory? An ESA facility for far-infrared and submillimetre astronomy G.L. Pilbratt1;??, J.R. Riedinger2;???, T. Passvogel3, G. Crone3, D. Doyle3, U. Gageur3, A.M. Heras1, C. Jewell3, L. Metcalfe4, S. Ott2, and M. Schmidt5 1 ESA Research and Scientific Support Department, ESTEC/SRE-SA, Keplerlaan 1, NL-2201 AZ Noordwijk, The Netherlands e-mail: [email protected] 2 ESA Science Operations Department, ESTEC/SRE-OA, Keplerlaan 1, NL-2201 AZ Noordwijk, The Netherlands 3 ESA Scientific Projects Department, ESTEC/SRE-P, Keplerlaan 1, NL-2201 AZ Noordwijk, The Netherlands 4 ESA Science Operations Department, ESAC/SRE-OA, P.O. Box 78, E-28691 Villanueva de la Canada,˜ Madrid, Spain 5 ESA Mission Operations Department, ESOC/OPS-OAH, Robert-Bosch-Strasse 5, D-64293 Darmstadt, Germany Received 9 April 2010; accepted 10 May 2010 ABSTRACT Herschel was launched on 14 May 2009, and is now an operational ESA space observatory offering unprecedented observational capabilities in the far-infrared and submillimetre spectral range 55−671 µm. Herschel carries a 3.5 metre diameter passively cooled Cassegrain telescope, which is the largest of its kind and utilises a novel silicon carbide technology. The science payload comprises three instruments: two direct detection cameras/medium resolution spectrometers, PACS and SPIRE, and a very high-resolution heterodyne spectrometer, HIFI, whose focal plane units are housed inside a superfluid helium cryostat. Herschel is an observatory facility operated in partnership among ESA, the instrument consortia, and NASA. The mission lifetime is determined by the cryostat hold time.
[Show full text]