List questions from partitions-for-integration

Published on 09 Dec 2019 - 16:05

407

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Limit of the ratio of two non-Riemann sums.

Published on 19 Dec 2019 - 8:41

137

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How should I partition this interval to utilize the left endpoint $\frac{\epsilon}{2}$?

Published on 21 Jan 2020 - 5:16

105

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If $f\in R(\alpha)$ and $C\in\mathbb{R}$, then $Cf\in R(\alpha)$ and $\int_a^b Cf\operatorname{d}\alpha=C\int_a^b f\operatorname{d}\alpha.$

Published on 04 Mar 2020 - 12:09

43

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Approximating the Riemann integral for a certain product

Published on 28 Mar 2020 - 15:06

91

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Ill-defined derivative?

Published on 14 Apr 2020 - 0:47

72

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When the piecewise constant integral independs of the partition's choice?

Published on 24 May 2020 - 1:25

62

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Prove the following result: if $f(x) \geq g(x)$ for all $x\in I$, then $p.c.\int_{I}f \geq p.c.\int_{I}g$.

Published on 04 Jun 2020 - 19:39

226

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Prove that if the partition $\textbf{P}'$ is finer than $\textbf{P}$, then $f$ keeps piecewise constant

Published on 05 Jun 2020 - 0:41

494

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Common refinement with different intervals

Published on 04 Jul 2020 - 22:23

47

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General facts about partitions in integrals

Published on 17 Jul 2020 - 1:36

126

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$f(x) = x^2$. For each positive integer $n$, let $P_n$ be the partition $P_n = \{0, \frac{1}{n}, \frac{2}{n},..., \frac{n-1}{n},1 \}$ of $[0,1]$.

Published on 05 Nov 2016 - 20:33

2.1k

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Show $f$ is integrable in each subinterval, $[x_{i−1}, x_i]$, and further, $\int_a^bfdx=\sum^n_{k=1}\int\limits_{x_{k-1}}^{x_k}fdx$.

Published on 06 Nov 2016 - 14:00

401

Views

Change of integration limits for uniformly continuous function

Published on 24 Jan 2017 - 20:46

55

Views

question of partition in double integral

Published on 27 Mar 2023 - 17:53

List Question

Darboux Sum in terms of partitions

Limit of the ratio of two non-Riemann sums.

How should I partition this interval to utilize the left endpoint $\frac{\epsilon}{2}$?

If $f\in R(\alpha)$ and $C\in\mathbb{R}$, then $Cf\in R(\alpha)$ and $\int_a^b Cf\operatorname{d}\alpha=C\int_a^b f\operatorname{d}\alpha.$

Approximating the Riemann integral for a certain product

Ill-defined derivative?

When the piecewise constant integral independs of the partition's choice?

Prove the following result: if $f(x) \geq g(x)$ for all $x\in I$, then $p.c.\int_{I}f \geq p.c.\int_{I}g$.

Prove that if the partition $\textbf{P}'$ is finer than $\textbf{P}$, then $f$ keeps piecewise constant

Common refinement with different intervals

General facts about partitions in integrals

$f(x) = x^2$. For each positive integer $n$, let $P_n$ be the partition $P_n = \{0, \frac{1}{n}, \frac{2}{n},..., \frac{n-1}{n},1 \}$ of $[0,1]$.

Show $f$ is integrable in each subinterval, $[x_{i−1}, x_i]$, and further, $\int_a^bfdx=\sum^n_{k=1}\int\limits_{x_{k-1}}^{x_k}fdx$.

Change of integration limits for uniformly continuous function

question of partition in double integral

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