Results 261 to 270 of about 130,230 (316)
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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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1983
We now consider the legitimacy of passing to the limit under the integral sign. If the sequence 〈f n 〉 of R-integrable functions converges to a limit f on an interval [a, b] does it necessarily follow that $$_{n \to \, + \,\infty }^{\lim }\int_a^b {{f_n}\left( x \right)dx = \int_a^b {f\left( x \right)dx?} }$$ (1.1)
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We now consider the legitimacy of passing to the limit under the integral sign. If the sequence 〈f n 〉 of R-integrable functions converges to a limit f on an interval [a, b] does it necessarily follow that $$_{n \to \, + \,\infty }^{\lim }\int_a^b {{f_n}\left( x \right)dx = \int_a^b {f\left( x \right)dx?} }$$ (1.1)
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2003
As was pointed out in the previous chapter, the second fundamental topic covered in calculus is the Riemann integral, the first being the derivative.
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As was pointed out in the previous chapter, the second fundamental topic covered in calculus is the Riemann integral, the first being the derivative.
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2014
An account is given of the Riemann integral for real-valued functions defined on intervals of the real line, a rapid development of the topic made possible by use of the Darboux approach in place of that originally adopted by Riemann. The sense in which integration is the inverse of differentiation is investigated. To cope with the demands of the later
R. H. Dyer, D. E. Edmunds
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An account is given of the Riemann integral for real-valued functions defined on intervals of the real line, a rapid development of the topic made possible by use of the Darboux approach in place of that originally adopted by Riemann. The sense in which integration is the inverse of differentiation is investigated. To cope with the demands of the later
R. H. Dyer, D. E. Edmunds
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2018
This chapter develops results concerning the Riemann integration of one variable real functions. The standard necessary and sufficient condition of zero Lebesgue measure for the set of discontinuities for the Riemann integrability of a one variable real function is addressed in detail.
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This chapter develops results concerning the Riemann integration of one variable real functions. The standard necessary and sufficient condition of zero Lebesgue measure for the set of discontinuities for the Riemann integrability of a one variable real function is addressed in detail.
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2017
Der Ursprung der Integralrechnung liegt im Problem der Flachenberechnung. Die bekannten Formeln fur den Flacheninhalt eines Rechtecks, eines Dreiecks oder eines Trapezes lassen sich nicht auf krummlinig berandete Flachen verallgemeinern. Im 4. Jahrhundert vor Christus entwickelte Eudoxos von Knidos die Exhaustionsmethode, deren Idee darin bestand ...
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Der Ursprung der Integralrechnung liegt im Problem der Flachenberechnung. Die bekannten Formeln fur den Flacheninhalt eines Rechtecks, eines Dreiecks oder eines Trapezes lassen sich nicht auf krummlinig berandete Flachen verallgemeinern. Im 4. Jahrhundert vor Christus entwickelte Eudoxos von Knidos die Exhaustionsmethode, deren Idee darin bestand ...
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1996
In this chapter we give an exposition of the definite integral of a real-valued function defined on a closed bounded interval. We assume familiarity with this concept from a previous study of calculus, but want to develop the theory in a more precise way than is typical for calculus courses, and also take a closer look at what kind of functions can be ...
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In this chapter we give an exposition of the definite integral of a real-valued function defined on a closed bounded interval. We assume familiarity with this concept from a previous study of calculus, but want to develop the theory in a more precise way than is typical for calculus courses, and also take a closer look at what kind of functions can be ...
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An overview of real‐world data sources for oncology and considerations for research
Ca-A Cancer Journal for Clinicians, 2022Lynne Penberthy +2 more
exaly