Results 261 to 270 of about 5,245 (272)
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1973
It is common, at an elementary level, to define integration as the process that is the inverse of differentiation. Thus, given a function f, we look for a function F such that F′(x) = f(x) for all x. There is, in fact, no reason why such a function should exist. Next, if a and b are real numbers with a < b, it is shown that F(b)−F(a) gives the value of
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It is common, at an elementary level, to define integration as the process that is the inverse of differentiation. Thus, given a function f, we look for a function F such that F′(x) = f(x) for all x. There is, in fact, no reason why such a function should exist. Next, if a and b are real numbers with a < b, it is shown that F(b)−F(a) gives the value of
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2017
This chapter completes the task of establishing the fundamentals of Calculus, this time studying the operation of integration on functions. As we shall see in the next section, in its most simple form this reduces to the computation of areas under the graphs of nonnegative continuous functions \(f: [a,b] \rightarrow \mathbb{R}\), suggesting that ...
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This chapter completes the task of establishing the fundamentals of Calculus, this time studying the operation of integration on functions. As we shall see in the next section, in its most simple form this reduces to the computation of areas under the graphs of nonnegative continuous functions \(f: [a,b] \rightarrow \mathbb{R}\), suggesting that ...
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1995
In this chapter we study elementary integration theory for functions defined on closed intervals. Although we expect that the reader has had experience with integral calculus and that the ideas are familiar, we shall not require any special results to be known. For pedagogical reasons we shall first treat the Cauchy integral.
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In this chapter we study elementary integration theory for functions defined on closed intervals. Although we expect that the reader has had experience with integral calculus and that the ideas are familiar, we shall not require any special results to be known. For pedagogical reasons we shall first treat the Cauchy integral.
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2014
Chapter 9 is about the Riemann integration. It starts with the definition of proper Riemann integral and presents Darboux and Riemann criteria for integrability. It is proved that continuous functions and functions of bounded variation on closed bounded intervals are Riemann integrable.
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Chapter 9 is about the Riemann integration. It starts with the definition of proper Riemann integral and presents Darboux and Riemann criteria for integrability. It is proved that continuous functions and functions of bounded variation on closed bounded intervals are Riemann integrable.
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A Theorem on Riemann Integration
Journal of the London Mathematical Society, 1962E. M. Mandelbaum +2 more
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Riemann integral vs. Lebesgue integral
Real analysis exchange, 1991The author gives a method of construction of Riemann integral. Let \(\Omega\) be a nonempty set and \(\mathcal A\) an algebra on \(\Omega\). Let \(B(\Omega)\) be the set of all bounded real-valued functions on \(\Omega\). \(S=S(\Omega,{\mathcal A})\) denotes the set of all \({\mathcal A}\)-simple functions on \(\Omega\). Let \(I=I(\Omega,{\mathcal A})\)
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Riemann and Lebesgue Integrability
Journal of the London Mathematical Society, 1972openaire +3 more sources