Teaching ideas and activities for classroom: integrating technology into the pedagogy of integral calculus and the approximation of definite integrals

The purpose of this article is to offer teaching ideas in the treatment of the definite integral concept and the Riemann sums in a technology-supported environment. Specifically, the article offers teaching ideas and activities for classroom for the numerical methods of approximating a definite integral via left- and right-hand Riemann sums, along with midpoint and trapezoidal Riemann sums. The activities demonstrate innovative and creative ways of integrating technology, in particular, GeoGebra dynamic software, into the pedagogy of college-level integral calculus. It also provides, among other things, interesting and original teaching ideas incorporating technology, and an evaluation of these activities by the students themselves who experienced these activities with the GeoGebra dynamic software.     

Filling an area discretely and continuously

The literature dealing with student understanding of integration in general and the Fundamental Theorem of Calculus in particular suggests that although students can integrate properly, they understand little about the process that leads to the definite integral. The definite integral is naturally connected to the antiderivative, the area under the curve and the limit of Riemann sums; these three conceptualizations of the definite integral are useful in different contexts and provide students with what it takes to interpret the definite integrals. Research shows that students rarely invoke the multiplicatively-based summation conception of the definite integral although it is essential for evaluating line integrals, surface integrals and volumes. This paper describes a teaching module that promotes understanding as well as activating all three conceptualizations of the definite integral through motivating the accumulation area function and the results in the Fundamental Theorems of Calculus.     

基于结构元的模糊值函数的一般表示方法

文[1]提出了模糊结构元的概念,并给出了模糊数与模糊值函数的模糊结构元表示,以及一类由模糊结构元线性生成的模糊值函数的微分和积分(黎曼意义下的)的表达形式。本文在文[1]的基础上进一步给出了由模糊结构元生成的模糊值函数的一般表达形式,并得到了一般表达形式下的模糊值函数的连续性和微分、黎曼积分的定义,它们与传统模糊分析中相应定义是等价的。     

重积分和线面积分中的一个典型题目

针对授课班级出错率较高的一道曲面积分题目,给出四种解法.分析出错的原因在于练习不够外,主要是对重积分概念理解不够透彻.     

Harary's conjectures on integral sum graphs

Let N denote the set of positive integers and Z denote all integers. The (integral) sum graph of a finite subset S N(Z) is the graph (S, E) with uv ε E if and only if u + v ε S. A graph G is said to be an (integral) sum graph if it is isomorphic to the (integral) sum graph of some S N(Z). The (integral) sum number of a given graph G is the smallest number of isolated nodes which when added to G result in an (integral) sum graph.

We show that the integral sum number of a complete graph with n 4 nodes equals 2n − 3, which proves a conjecture of Harary. And we disprove another conjecture of Harary by showing that there are infinitely many trees which are not caterpillars but are integral sum graphs.     

模糊黎曼积分的拓广

给出了模糊黎曼积分的拓广定义,并证明了拓广的模糊黎曼积分在下方图度量和d1度量下可以通过有限个层次集逼近.     

Areas,anti-derivatives,and adding up pieces: Definite integrals in pure mathematics and applied science contexts

Research in mathematics and science education reveals a disconnect for students as they attempt to apply their mathematical knowledge to science and engineering. With this conclusion in mind, this paper investigates a particular calculus topic that is used frequently in science and engineering: the definite integral. The results of this study demonstrate that certain conceptualizations of the definite integral, including the area under a curve and the values of an anti-derivative, are limited in their ability to help students make sense of contextualized integrals. In contrast, the Riemann sum-based “adding up pieces” conception of the definite integral (renamed in this paper as the “multiplicatively-based summation” conception) is helpful and useful in making sense of a variety of applied integral expressions and equations. Implications for curriculum and instruction are discussed.     

“数学分析”课的教学与研究

根据教学实践,可采取四项措施以加深学生对极限概念的理解,即借助几何直观,通过对典型例子的分析,对“ε-δ”语言进行多次反复,在讲定积分的应用时讲一讲微积分的发展史．对数列极限理论的几个主要定理,可用分散难点的办法给予处理,并以比较直观的区间套定理作为主要证明工具．要教好“数学分析”课,必需具备三个条件,即教学方法要好,包括口齿清楚,黑板书写整洁．讲话流利、风趣等;要具有后继课程“实变函数”、“泛函分析”、“拓扑学”等的基本知识,这样,才能居高临下,讲深讲透;要不断搞科研,因为科研搞得好的教师,讲课常能抓住重点,讲出思想和深度,把书上没有写出来的某些关键之处讲出来．     

An Itô formula for a family of stochastic integrals and related Wong–Zakai theorems

The aim of this paper is to generalize two important results known for the Stratonovich and Itô integrals to any stochastic integral obtained as limit of Riemann sums with arbitrary evaluating point: the ordinary chain rule for certain nonlinear functions of the Brownian motion and the Wong–Zakai approximation theorem. To this scope we begin by introducing a new family of products for smooth random variables which reduces for specific choices of a parameter to the pointwise and to the Wick products. We show that each product in that family is related in a natural way to a precise choice of the evaluating point in the above mentioned Riemann sums and hence to a certain notion of stochastic integral. Our chain rule relies on a new probabilistic representation for the solution of the heat equation while the Wong–Zakai type theorem follows from a reduction method for quasi-linear SDEs together with a formula of Gjessing’s type.     

Elementary estimates for a certain typeof Soto-Andrade sum

This paper shows that a certain type of Soto-Andrade sum can be estimated in an elementary way which does not use the Riemann hypothesis for curves over finite fields and which slightly sharpens previous estimates for this type of Soto-Andrade sum. As an application, we discuss how this implies that certain graphs arising from finite upper half planes in odd characteristic are Ramanujan without using the Riemann hypothesis.

     

Spectral analysis and the Riemann hypothesis

The explicit formulas of Riemann and Guinand-Weil relate the set of prime numbers with the set of nontrivial zeros of the zeta function of Riemann. We recall Alain Connes’ spectral interpretation of the critical zeros of the Riemann zeta function as eigenvalues of the absorption spectrum of an unbounded operator in a suitable Hilbert space. We then give a spectral interpretation of the zeros of the Dedekind zeta function of an algebraic number field K of degree n in an automorphic setting.

If K is a complex quadratic field, the torical forms are the functions defined on the modular surface X, such that the sum of this function over the “Gauss set” of K is zero, and Eisenstein series provide such torical forms.

In the case of a general number field, one can associate to K a maximal torus T of the general linear group G. The torical forms are the functions defined on the modular variety X associated to G, such that the integral over the subvariety induced by T is zero. Alternately, the torical forms are the functions which are orthogonal to orbital series on X.

We show here that the Riemann hypothesis is equivalent to certain conditions bearing on spaces of torical forms, constructed from Eisenstein series, the torical wave packets. Furthermore, we define a Hilbert space and a self-adjoint operator on this space, whose spectrum equals the set of critical zeros of the Dedekind zeta function of K.     

Learning the integral concept by constructing knowledge about accumulation

We propose an approach to the integral concept for advanced high school students and provide evidence for the potential of this approach to support students in acquiring an in-depth proceptual view of the integral. The approach is based on the mathematical idea of accumulation. A ten-lesson unit has been implemented with four pairs of students. The students’ learning processes were micro-analysed using the methodological–theoretical framework of Abstraction in Context. In this paper, we focus on the lessons in which the notions of approximation and accumulation are introduced. The work of one student pair is analysed in detail, and the work of the other pairs is summarized. Our results show that most of the students reached a proceptual understanding of the integral that prepared them for the next step in the curriculum, namely the Fundamental Theorem of Calculus.     

Expanding Students' Conceptions of the Arithmetic Mean

The arithmetic mean is the most commonly used measure of central tendancy; nevertheless, many students who can add all the elements of a data set and then divide that sum by the number of elements do not truly understand the concept of mean. This article presents four activities designed to help elementary and middle school students develop a concept of mean. To bring about a desirable level of understanding, all computational formulae and algorithms in mathematics should be preceded by experience emphasizing conceptual understanding. Since that is not the normal instructional sequence for the arithmetic mean, the activities presented in the article assume previous exposure to the computational algorithm for the arithmetic mean.     

了解积分——求和

求一个函数的黎曼积分,实际上就是一个分割、近似代替、求和、取极限的过程.求和运算是整个积分计算的轴心.就积分四部曲中的求和问题,做一个一般性的讨论.文中使用的是分析和讨论的语言,不去追求数学语言本身的严格性.目的不仅是探讨求和这个步骤,在黎曼积分意义下具体实现的过程和隐含的内容,而且对一般的积分中的求和实现的可能性、应该满足的条件、实现的过程,以及应该注意那些基本问题,也做一点儿逻辑上的探讨.已达到以知识为媒介,提高认知能力的目的.     

Finding the coefficients in the new representation of the solution of the Riemann–Hilbert problem using the Lauricella function

The solution of the Riemann–Hilbert problem for an analytic function in a canonical domain for the case in which the data of the problem is piecewise constant can be expressed as a Christoffel–Schwartz integral. In this paper, we present an explicit expression for the parameters of this integral obtained by using a Jacobi-type formula for the Lauricella generalized hypergeometric function F _D ^(N). The results can be applied to a number of problems, including those in plasma physics and the mechanics of deformed solids.     

The Riemann Function for the Third-Order One-Dimensional Pseudoparabolic Equation

In this work, some classes of initial boundary-value problems are investigated in the Sobolev space for the third-order linear pseudoparabolic equation having, in general, nonsmooth coefficients. A new type of Riemann function concept is given for these problems, which is more natural than the classical Riemann-type function concept, and an integral form of the solutions of nonhomogeneous problems can be found more naturally using this concept.     

A fundamental solution to the Cauchy problem for a fourth-order pseudoparabolic equation

The Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied. Under certain summability and boundedness conditions imposed on the coefficients, the operator of this problem and its adjoint operator are proved to be homeomorphism between certain pairs of Banach spaces. Introduced under the same conditions, the concept of a θ-fundamental solution is introduced, which naturally generalizes the concept of the Riemann function to the equations with discontinuous coefficients; the new concept makes it possible to find an integral form of the solution to a nonhomogeneous problem.     

黎曼相关积分研究

通过证明和反例讨论黎曼积分、直接黎曼积分、黎曼-斯蒂尔切斯积分三者间的联系与区别.结果显示：若函数直接黎曼可积,则它黎曼可积,并且两者积分值相同,但反之不成立;若函数黎曼可积,则任意连续函数关于该函数不一定黎曼-斯蒂尔切斯可积.从讨论结果中还获得直接黎曼可积和黎曼可积各自的一个充分条件.     

ANTICIPATIVE STOCHASTIC INTEGRALS EQUATIONS DRIVEN BY SEMIMARTINGALES

In this paper, we use the Riemann sum approach to construct the anticipative stochastic integrals and consider the Cauchy problem (non-adapted initial value) for stochastic integral equations driven by discontinuous semimartingales. For general equations with Lipschitz coefficients, we prove the existence of the solutions. Apropos of semilinear equations, we find that under some conditions uniqueness of solutions will also hold.