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Questions tagged [riemann-integration]

The Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.

Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be Riemann-integrable. The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough.

The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral.

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What is the geometrical meaning of a Riemann-Stieltjes integral?

Geometrical meaning of Reimann integration is to find the area under the curve of function and $x$-axis. Due to this geometrical interpretation, all theorems on Riemann integration are easily understandable. I want to know what is the geometrical…
The 7th sense
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Can Rudin's PMA Theorem 6.17 be proven by using Riemann (as opposed to Darboux) integrals?

Throughout this post I refer to these as "Darboux integrals/integration" and to these as "Riemann integrals/intnegration". It is a standard result of Analysis that these notions of integrability coincide. I've tried (and failed) looking for a book…
Sam
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Can the space of Riemann integrable function can be defined as the closure of step functions?

Q1) We often defined the Riemann integral of a function with Darboux sum, but could we define the space of Riemann integrable function as the closure of step functions ? (but in "$L^1$"-sense) (as Lebesgue Integrable function are defined as the…
user349449
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Pythagorean Integral Inequality

The question is to show that, given $f$ is a non-negative Riemann integrable function over an interval $[a,b]$, such that $\int_a^b f=1$, then we have that $$\left(\int_a^b f(x) \cos(x)\right)^2+\left(\int_a^b f(x) \sin(x)\right)^2\leq 1$$ And I…
Brandon Myers
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A Riemann integrable, non-regulated function

What's an example of a Riemann integrable, non-regulated function? Definitions: Let $X$ be a normed space and $[a,b]$ be a compact of $\mathbb R$. Step Functions: A function $f: [a,b] \longrightarrow X$ is said to be a step function if: there…
user230734
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Prove Riemann integrability if $g(x) = f(x)$

I'm not sure if the following prove is correct. It seems incomplete. Hope you can help me: Theorem: Suppose $f:[a,b] \to \mathbb R$ is Riemann integrable. Suppose $g: [a,b] \to \mathbb R$ is a function such that $g(x) = f(x)$ for all except finite…
songwriter93
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Limit of a nearly Riemann sum

On of my student asked me to find $$\lim_{n\to+\infty}\sum_{k=1}^n\frac{1}{\sqrt{(n+k)(n+k+1)}}$$ I tried to write it as a Riemann sum, but the term $ \frac 1n $ disturbs. I tried to use the fact that $$n+k\le n+k+1 \le 2n+1$$ but no way. Thank you…
hamam_Abdallah
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Show the lower Riemann integral is the limit of lower sums of a sequence of refined partitions

I just finished the first seven chapters of Baby Rudin, and start working on Measure, Integration, and Real Analysis by Sheldon Axler. The following is Exercise 7 of Section 1A. Suppose $f\colon [a,b]\to \mathbb{R}$ is a bounded function. For $n\in…
Min
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Question about Riemann integral and Norm of partition

Why do we assume that when the Riemann integral is defined, the partition norm tends to zero? And if it did not tend to zero, but remained constant, would it happen?
Jack J.
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Rudin 6.10 but with countable discontinuities

This is theorem 6.10 in baby Rudin: Suppose $f$ is bounded on $[a, b]$, $f$ has only finitely many points of discontinuity on $[a, b]$, and $\alpha$ is continuous at every point at which $f$ is discontinuous. Then $f \in \mathscr{R}(\alpha)$. Is…
Jamie Bellend
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If $f$ is a derivative then is $|f|$ also a derivative?

If $f$ is a derivative then, is $|f|$ also a derivative? If $f$ is Riemann integrable then it's true. But, if it's not the case,then is it true?
SOUL
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Right Riemann sum Error bound proof

How do we prove the right Riemann sum error bound? In wikipedia (https://en.wikipedia.org/wiki/Riemann_sum#Right_Riemann_sum) they have mentioned the following bound, but no proof. $$\left | \int_a^bf(x)dx - A_{\mbox{right}}\right | \leq…
jnxd
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Prove that the function $f$ defined by $f(x)=x$ if $x$ is rational, $f(x)=-x$ if $x$ is irrational, is not Riemann integrable on $[0,1]$

Prove that $f(x)=\begin{cases}x & \text{ if $x$ is rational } \\ -x & \text{ if $x$ is irrational} \end{cases}$ is not Riemann integrable on [0,1] I'm trying to workout the upper integral and the lower integral. And to say that they are not…
DD90
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Show $ \ f \in \mathcal{R} [0,1] \ $

Let $ \ f: [0,1] \to \mathbb{R} \ $ be defined by $$ f(t)=\frac{1}{2^n} , \ \ \ \frac{1}{2^{n+1}}
MAS
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