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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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riemann function from $[0,1]$

Riemann function $f:[0,1] \to \mathbb R$ \begin{eqnarray} f(x)= \begin{cases} 0&\quad\text {if } x \in \mathbb R - \mathbb Q \cr 1\over q&\quad\text{if } x = \frac{p}{q} \in \mathbb Q \cr \end{cases} \end{eqnarray} To $f(x)$: in the upper case,…
repulsive23
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Prove a function is odd using Riemann Sums

I've been asked to prove the following: Assume $f(x)$ is continuous on $[-L,\ L]$ and $\int_{-a}^af = 0$ for all $a\in[0,\ L]$. Show that $f(x)$ is odd. The catch? I'm only allowed to use Riemann sums. Note: what makes this question different…
user3002473
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Is there always a partition that gives the Riemann Integral?

The Riemann Integral of a bounded function $f$ on a compact set $A$ is defined if $$U(f) = \inf\{U(f,P) \mid P \in \mathcal{P}\} = \sup \{ L(f, P) \mid P \in \mathcal{P} \} = L(f)$$ where $\mathcal{P}$ is the set of all partitions of $A$ and $L(f,…
Tyler Gaona
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Countable additivity of Rieman integrals with respect to domains and integrands

Lebesgue integrals hav the countable additivity with respect to the domains of integration. This property is also true with respect to the integrands assuming some additional conditions. I don't know if the Rieman integrals also have countable…
Thang
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Example of Riemann Integrable with three requirement

Need example of Riemann Integrable function f: [-1,1] -> R which: 1) Integral from -1 to 1 = 1 2) Integral from -1 to 0 = -1 3) Integral from -1 to 1 of absolute value of f(x) = 2 Any help would be much appreciated as I couldn't come up with any. Is…
ANoobWithAQuestion
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Show f is Riemann integrable if L(-f,[a,b])=-L(f,[a,b])

This question is from Sheldon Axler's real analysis book exercise 1b. Suppose f is a bounded function on [a, b]. Show that f is Riemann integrable on [a,b] iff L(-f,[a,b])=-L(f,[a,b]). This is how I Done. Suppose f is Riemann integrable, then…
Henry Chan
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Riemann integrable function on every subinterval

Suppose $f:[a,b] \rightarrow \mathbb{R}$ is a bounded and Riemann integrable function on [c,b] for every c in (a,b). Prove that f is Riemann integrable on all [a,b]. I am trying to use the Darboux criterion to solve this problem. For a given…
Tim
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What's wrong of my proof of that $f^2$ is Riemann integrable implies $f$ is Riemann integrable?

Suppose $f$ is a bounded real function on $[a,b]$ and $f^2$ is Riemann integrable on $[a,b]$. $f^2$ is Riemann integrable does not imply $f$ is Riemann integrable. As an example, $f = -1$ when $x$ is irrational; $f = 1$ when $x$ is…
Kenton Pendragon Su
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Is it possible to meaningfully define the Riemann Integral for unbounded functions?

$\def\R{\mathbf{R}}$ The Riemann-integral is defined only for bounded functions. So functions such as $f:\R-\{0\}$ defined by $f(x):=1/x$ are not Riemann-integrable over any interval that contains $0$ as a limit point. However, at the core of the…
user1020500
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$f,g$ are Riemann integrals, on $[a,b]$ which is increasingon $R$. prove if $\int_{[a,b]} |f|^pd\alpha=0$ then $\int_{[a,b]} fgd\alpha=0$

$f,g \in {R}(\alpha)$ riemann integrals, on $[a,b]$ which is increasing in $\mathbb{R}$. prove if $\int_{[a,b]} |f|^pd\alpha$=0 then $\int_{[a,b]} fgd\alpha=0$. my attempt: since I can't assume f is continuous, and f and g are Riemann integrable I…
KASUN PRABHATH
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$f,g \in \mathbb{R}(\alpha)$, on $[a,b]$ which is increasing in $R$. prove if $\int_{[a,b]} |f|^pd\alpha=0$ then $\int_{[a,b]} fgd\alpha=0$

$f,g \in {R}(\alpha)$ riemann integrals, on $[a,b]$ which is increasing in $\mathbb{R}$. prove if $\int_{[a,b]} |f|^pd\alpha$=0 then $\int_{[a,b]} fgd\alpha=0$. my attempt: since I can't assume f is continuous, and f and g are Riemann integrable I…
KASUN PRABHATH
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Is it necessary that if a function is Riemann integrable, then it is also Darboux integrable?

I cant find a counter example for a function that is Riemann integrable, but not Darboux integrable. The only source i found on this forum is trying to prove the direction "If Riemann integrable, then Dargboux integrable". But i really doubt its…
Charming Imp
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Integration similar to consequences of Riemann lebesgue lemma

Let $f\in C^1[-\pi,\pi]$. Define , for $n\in N$ b_n=$\int_{-\pi}^\pi f(x)sinntdx$. Which of the following statement are true? a) $b_n\to 0$, as $n\to \infty$ b)$nb_n\to 0$, as $n\to \infty$ Please give me hint for option b) I tried substitution…
Tony
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Example for $\int_a^b f(x)g(x)dx\neq g(a)\int_a^c f(x)dx + g(b) \int_c^b f(x)dx$ for $f\geq 0$ and g a non-monotonic function.

Well we all know about this theorem. Let $f\geq 0$ be a function that is integrable over [a,b] and $g:[a,b]\to \mathbb{R}$ be a monotonic function. Then there is a $c \in [a,b]$ with $$\int_a^b f(x)g(x)dx= g(a)\int_a^c f(x)dx + g(b) \int_c^b…
Analysis_Mark
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Checking for the integrability of $\int_U \ln{\sqrt{x^2+y^2}}$ using exhaustions

The open set to check for integrability is $U = \{(x,y) \in \mathbb{R}^2 | 0
Eduardo V. Kuri
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