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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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Existence of $c$ such that $\int_{a}^{b} f(x)g(x)dx = f(c)\int_{a}^{b} g(x)$ with g taking on negative values as well.

I am doing a task right now as an excercice for myself. Let $f,g:[a,b] \to \mathbb{R}$ continuous, $g(x)>0 \forall x \in [a,b]$. Then there is $c \in (a,b)$ such that $\int_{a}^{b} f(x)g(x)dx = f(c)\int_{a}^{b} g(x)$. There are many good solutions…
Analysis_Mark
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$f,g:[a,b]\to\mathbb{R}$ integrable. Then f+g is integrable with $\int_a^b (f+g)=\int_a^b f + \int_a^b g$

First of all I know there are many posts where someone asked for the proof of this sentence. But I just need help to understand one step of the first part of the proof, I hope there is someone who could help me. Here is the first part of the…
Analysis_Mark
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Is this piecewise function Riemann integrable on $[0,1]$?

How to check if function is a Riemann integrable on $[0,1]$? I don't know how to find lower and upper Darboux sum. $$f(x) =\left\{ \begin{array}{ll} 1,\text{when}\ \exists \ m,n \in \mathbb{N}, x = \frac{m}{2^n}\\ …
anna
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Is $f(x) =x^{-2}$ Riemann-Integrable on [0,2] if we set f(0) = 0?

Let's consider the function $$f: [0,2] \to \mathbb{R}, f(x) = \begin{cases}x^{-2} &\text{for}\;x\in(0,2] \\ 0 & \text{for} \;x =0 \end{cases}$$ Would this function be Riemann-integrable? I'm having trouble thinking about this integral, can it be…
maibrl
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Riemann Integrable Function is Zero

I am working through a Real Analysis textbook and found the following exercise. Let $f\,:\,[a,b]\rightarrow\mathbb R$ be Riemann integrable and nonnegative. Suppose $$\int_a^bf(x)\,dx=0.$$ Find $f$. I know that this should come out to be that…
Alexander
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Riemann integrability of a certain function

Let $ f $ defined at $ [0,1] $ by $$f(x)=\frac{1}{p+q}\; \;\text{if }x=\frac pq \; \text{with }\; \gcd(p,q)=1$$ and $$f(x)=0 \; \;\text{if }\; x=0\; \text{or} \;x\notin \Bbb Q$$ I want to prove that $ f$ is Riemann integrable at $ [0,1] \;$ and …
hamam_Abdallah
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Help with understanding solution of Riemann integral problem

The problem is following: $${f}(x) = \begin{cases} 1, & \text{if $x=\frac{1}{n}, \ \ n=1, 2,3 ,\cdots$ } \\ 0, & \text{if else} \end{cases}$$ I'm supposed to prove that this function is Riemann integrable over $[0,1]$. I know that this exact…
Mathomat55
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Can a Riemann integrable function $f$ on $[-1,1]$ such that $F(x)=\int_{-1}^{x}f$ be differentiable at every point except for $x=0$?

Is it possible to have a Riemann integrable function $f$ on $[-1,1]$ such that $F(x)=\int_{-1}^{x}f$ is differentiable at every point except for $x=0$? My initial thoughts are that such function is not possible, but I can't find a way to demonstrate…
KeyboardHunter
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Prove $f$ is not Riemann Integrable?

Let $f:[0,1]\rightarrow\mathbb{R}, f(x)=\begin{cases} x & \text{when $x$ is rational} \\ 1-x & \text{when $x$ is irrational} \end{cases}$ Show that $f$ is not Riemann Integrable. Can you show provide an argument with upper sum…
Chandramauli Chakraborty
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show that f is not riemann integrable over [a,b]

Given $$ f(x) = \begin{cases} x, & \text{when x is rational} \\ -x, & \text{when x is not rational} \end{cases} $$ Show that $f$ is not Riemann-integrable over $[a,b]$, but $|f|$ is. How to go about this problem? I started by taking a partition…
user244722
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Bound arc-length of certain continuously differentiable function $f:[0,1]\to[0,1]$

A continuously differentiable function $f:[0,1]\to[0,1]$ has the properties (a) $f(0)=f(1)=0$ (b) $f'(x)$ is a non-increasing function of $x$. Prove that the arc-length of the graph does not exceed $3$. As I understand from the question that we…
user786
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relation between the integrals $f(x)=(\int_{0}^{x}e^{-t^2} dt)^2,g(x)=\int_{0}^{1}\frac{e^{-x^2(t+1)}}{t^2+1}$

Is there any special relation between the integrals $$f(x)=\left(\int_{0}^{x}e^{-t^2} dt\right)^2,g(x)=\int_{0}^{1}\frac{e^{-x^2(t^2+1)}}{t^2+1} dt$$ as these two integrals can be used to show that, $\int_{0}^{\infty}e^{-t^2}…
user694028
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infimum of the Upper Darboux sums & supremum of the Lower Darboux sums

Hi guys I am doing a question. Well, attempting a few questions about evaluating darboux sums. I fully understand the evaluation of setting up the upper and low sum integrals what I am having some confusion with is when evaluating the…
John
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Riemann Integration Question with unknown range

Hi Guys I am trying to understand what is going on as it relates to the range of the X values since from the question, the range is stated in terms of a and b. I have been exposed to a few question where the closed interval is give example([0,1] or…
John
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Limit of integrals of continuous functions

If $g:[0,1] \rightarrow \mathbb{R}$ be a continuous function such that $\lim\limits_{x \rightarrow 0^+}\dfrac{g(x)}{x}$ exist and is finite, then prove that $$\lim\limits_{n \rightarrow \infty} \int_{0}^{1}g(x^n)dx= \int_{0}^{1}…
Akash Banerjee
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