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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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Integrability of $\text{ sgn}(\sin(\frac{\pi}{x}))$ on $[0,1]$

Is $\text{ sgn}(\sin(\frac{\pi}{x}))$ fit the basic setting of Riemann Integrability on $[0,1]$(bounded function, closed and bounded interval)? Since $sgn(x)$ is bounded for all $x$, so $\text{ sgn}(\sin(\frac{\pi}{x}))$ is bounded on $[0,1]$, but…
Belive
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What is the definition of Riemann-Stieltjes integrability?

For some documents, $$ if \displaystyle\inf_{\text{partition}}U_{\text{partition}}(f)=\sup_{\text{partition}}L_{\text{partition}}(f),\\then~f~is~Riemann\text{-}Stieltjes~integrable. $$ like the following: However, my textbook said the definition…
Danny_Kim
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Darboux integral definition in wikipedia

At some point, wikipedia says that An equivalent and sometimes useful criterion for the integrability of $f$ is to show that for every $ε > 0$ there exists a partition $P_ε$ on $[a,b]$ such that $$U_{f,P_{\epsilon}}-L_{f,P_{\epsilon}} <…
Miguel
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Constructing a Riemann integral...

I want to evaluate the Riemann integral $\int_0^1 {{x^2}dx} $ I want to find upper and lower estimates of the form: $$U \ge {1 \over {6{N^3}}}(N(N + 1)(2N + 1))$$ $$L \ge {1 \over {6{N^3}}}(N(N - 1)(2N - 1))$$ Then show they're equal and then…
CoffeeToTheorems
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Riemann-integrability of $f(x) \geq (\frac{1}{\lfloor x\rfloor})^\alpha$

Let $f$ be Riemann-integrable in the interval $[1,\infty)$ and let for all $x \geq 1$ $f(x) \geq \left(\frac{1}{\lfloor x\rfloor}\right)^\alpha$. Then $\alpha > 1$. How to prove this statement? Ok for example for $\alpha = 1$ I know that $1/x$ is…
Mekanik
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Sum of functions is Riemann integrable implies that the functions are Riemann integrable?

If $f,g: [a,b] \to \mathbb{R}$ are bounded, if $f+g$ is Riemann integrable does this imply that $f$ and $g$ are Riemann integrable?
yerman
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Help with understanding this Riemann integral

Okay, so I have a hard time understanding this math problem: Given $$[f(x)] = \begin{cases} 5 & \quad \text{if } x \text{ <3}\\ 7 & \quad \text{if } x \geq3 \end{cases} $$ were the partitioning is…
Mathomat55
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prove or find counter example to the statement : Riemann integralable

If the function $f:[a, b]\rightarrow R $ is Riemann integralable on $[a, b]$, then $F:[a, b]\rightarrow R$ is differentiable on [a, b] where $F(x)=\int_a^xf(t)dt$
Jun Hyeok Choi
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Let $ [a,b] \subseteq \mathbb{R} $ be a non-degenarate closed interval

Let $ [a,b] \subseteq \mathbb{R} $ be a non-degenarate closed interval . For $ \epsilon >0 $ prove that there is a partition $ P $ of $ [a,b] $ such that $ || P|| < \epsilon $. Answer: Let $P: \ a=x_{0}
MAS
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$\lim\limits_{n \to \infty }\sum_{i=1}^{n}{1 \over n+i-1} = \int_{1}^{2}{1 \over x}dx$

$\lim\limits_{n \to \infty }\sum_{i=1}^{n}{1 \over n+i-1} = \int_{1}^{2}{1 \over x}dx$ For the above equations, could one can prove it without using any derivatives?
snapper
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Integrable or not ???

How to check whether a function is integrable or not: $f(x,y)=\frac{1}{1+x^2+y^2}$ over $\Bbb{R}^2$. I am not able to start the problem. Is there some theorem which deals with such problems?
user315483
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