Riemann integral

Riemann integral

Existence of the Riemann Integral and Properties

Published in James K. Peterson, Basic Analysis I, 2020

Since f is a constant, for any partition π of [a, b], all the individual mj’s and Mj’s associated with π take on the value c. Hence, U(f,π)−L(f,π)=0always. It follows that f satisfies Riemann’s Criterion and hence is Riemann Integrable. Finally, since f is integrable, by Theorem 21.2.2, we havec(b−a)≤RI(f;a,b)≤c(b−a).

Numerical Integration

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Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018

A. C. Faul

Definition 5.2 (Riemann integral). For each n ∈ ℕ let there be a set of numbers a = ξ0 < ξ1 < … < ξn = b. A Riemann integral is defined by∫abf(x)dx=limn→∞,Δξ→0∑i=1n(ξi−ξi−1)f(xi),

Measure and Integration Theory

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Published in Athanasios Christou Micheas, Theory of Stochastic Objects, 2018

Athanasios Christou Micheas

Remark 3.19 (Lebesgue and Riemann integrals) Clearly, the Lebesgue integral (when defined) can be thought of as a generalization of the Riemann integral. For what follows in the rest of the book, Lebesgue integrals will be computed using their Riemann versions, whenever possible, since Riemann integrals are easier to compute. Let f:Ω→[a,b],Ω⊂Rp, be a bounded, measurable function in the space (Ω,ℬp,μp). The following results clarify when the two integrals coincide.

⋄_α-Measurability and combined measure theory on time scales

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Journal Information

Published in Applicable Analysis, 2022

Chao Wang, Guangzhou Qin, Ravi P. Agarwal, Donal O'Regan

A bounded function f on is Riemann -integrable if and only if it is (Darboux)-integrable, in which case the values of the integrals are equal.

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Existence of the Riemann Integral and Properties

Numerical Integration

Measure and Integration Theory

⋄α-Measurability and combined measure theory on time scales

⋄_α-Measurability and combined measure theory on time scales