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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 Integration of Parabola

I know from integration that the answer is -4. However, I am messing something up somewhere while working through the Riemann sums. Going cross-eyed trying to find my mistake. I included the pertinent steps and skipped the details. I do have…
user147485
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How to build $f$ differentiable on $[0,1]$ easily, such that $f'$ is not Riemann-integrable?

How to build $f:[0,1]\to \mathbb{R}$ differentiable on $[0,1]$ easily, such that $f'$ is not Riemann-integrable?
Stu
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representation of riemann stieltjes integral as riemann integral

Let $f:[a,b]\to\mathbb{R}$ be a continuous function and $g:[a,b]\to\mathbb{R}$ be an absolutely continuous function. It is known that the absolutely continuous function $g$ is diffentiable a.e. and there holds $$g(y)-g(x)=\int_x^y g'\;\mbox{d}t$$…
elliptic
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If function $f$ and $g$ aren't Riemann integrable, can the function $f+g$ be?

As the question states, if function $f$ and $g$ aren't Riemann integrable, can the sum of the functions be Riemann integrable? Most importantly, in what way should this be proven?
Dole
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Riemann integrability of $f$ on a [0,1]

Suppose $f:[0,1] \rightarrow \mathbb{R}$ is a bounded function such that $f$ is Riemann integrable on $[a,1]$ for every $a \in (0,1)$. Is $f$ Riemann integrable on [0,1]?
Naweed G. Seldon
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Problem Related to Integrability of a Function

Consider $E$ to be the set of all $(x,y) \in [0,1] \times [0,1]$ such that there exists a $p \geq 2$ prime and $m,n \in \mathbb{N}$ such that $(x,y) = (\frac{m}{p},\frac{n}{p}).$ Take $f$ to be the indicator function for the set, i.e. $f(x,y) =…
Richard
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Continuity and non-negative on a closed interval implies integrability?

Suppose $f$ is continuous and non-negative on $[0,1]$ and that $f(0.5)=1$. Then, f is integrable and $\int_0^1 f>0$ This seems very, very intuitive considering the $U(F,P)$ and $L(F,P)$ upper and lower sums respectively are defined $>0$ in this…
user6276
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how to check and solve the following riemann integral if function is integrable?

let $$f(x)= \begin{cases} x^2 \text{ if $x$ is rational}\\ 0 \text{ if $x$ is irrational} \end{cases} $$ let $$g(x)=\begin{cases} \frac{1}{q} \text{ if $x=\frac{p}{q}$ where $(p,q)=1$}\\ 0 \text{ if $x$ is…
PAMG
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Suppose $f:[a,b]\to\mathbb{R}$ is Riemann integrable. Prove that $\int_{a}^{b}f=\lim_{n\to\infty} \frac{b-a}{n}\sum_{j=1}^{n} f(a+\frac{j(b-a)}{n}).$

I am reading "Measure, Integration & Real Analysis" by Sheldon Axler. The following exercises are Exercise 7 and Exercise 8 in Exercises 1A on p.8. Exercise 7 on p.8: Suppose $f:[a,b]\to\mathbb{R}$ is a bounded function. For $n\in\mathbb{Z}^+$, let…
tchappy ha
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The author divided $[a,b]$ into $2^n$ intervals of equal size in the above exercise. But why $2^n$? Measure, Integration & Real Analysis Sheldon Axler

I am reading "Measure, Integration & Real Analysis" by Sheldon Axler. The following exercise is Exercise 7 in Exercises 1A on p.8. Suppose $f:[a,b]\to\mathbb{R}$ is a bounded function. For $n\in\mathbb{Z}^+$, let $P_n$ denote the partition that…
tchappy ha
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Let $f:[a,b]\to\mathbb{R}$ be integrable and $\alpha \in \mathbb{R}$. Then $\int_a^b \alpha f=\alpha \int_a^b f$

I need help with the following task (dealing with the Riemann-Integration), I am doing as an exercise for myself right now. Let $f:[a,b]\to\mathbb{R}$ be integrable and $\alpha \in \mathbb{R}$. Then $\alpha \cdot f$ is integrable with $\int_a^b…
Analysis_Mark
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$\lim_{n\to\infty}\sum_{k=1}^{n}(\frac{k}{n})^{4}\frac{2k-1}{n^{2}}$ as a limit of a Riemann Sum

I'm having problems trying to compute the limit of the sum $\lim_{n\to\infty}\sum_{k=1}^{n}(\frac{k}{n})^{4}(\frac{2k-1}{n^{2}})$ As you can see you can't take $\xi_{k}$ as the extreme of some partition, for example $k/n$. because we obtain…
Iesus Dave Sanz
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Chcking if a function defined for a convergent sequence is Riemann Integrable

Let $\{a_n\}$ be a sequence of real numbers that converges to $1$. Define $f:[0,2]\to \mathbb{R} $ by $$ f(x)= \begin{cases} 1&\text{if }\, x\in \{a_n\}\\ 0& \text{otherwise}\\ \end{cases} $$ Prove $f$ is Riemann integrable on $[0,2]$ I was trying…
robert.d
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Riemann integrability when two functions are not equal at one point.

I am studying from Bartle-Shebert and the book says this- Let $c$ be the point where $f$ and $g$ are not equal.Then, the Riemann sums of $f$ and $g$ are identical with the exception of atmost two points (in the case $c=x_i=x_{i-1}$ is an endpoint).…
Soham
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Help with epsilon and Riemann integral

I posted this question yesterday, but it could have been written better and it was a bit "disorderly". So I try to post it again: Given $$f(x) = \begin{cases} 5 & \quad \text{if } x \text{ <3}\\ 7 & \quad \text{if } x \geq3 …
Mathomat55
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