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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.

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33 views

Double integration: $\sin(2y+3x)$. [on hold]

I have to solve this double integral: $$\int_C \sin(2y+3x)\,dx\,dy$$ where $C$ is the sector of the circle with radius $1$ centered at the origin between the angles $\pi/2$ and $3\pi/4$.
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1answer
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If f is not Riemann integrable on an interval can the absolute of f be Riemann Integrable on that interval? [on hold]

If f(x) is not Riemann integrable on some interval say I, can the absolute value of f(x) be Riemann integrable on that interval I ?
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58 views

Counterexample to Riemann sum limit

For convergent Riemann sums of $f \in C^1([0,1])$ there is the property: $$\tag{A}\lim_{n \to + \infty} \left[\, \sum_{k=0}^{n} f \left( \frac{k}{n+1} \right) - \sum_{k=0}^{n-1} f \left( \frac{k}{n} \...
2
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1answer
43 views

Rudin real anlysis: prove that $f$ is Riemann integrable

Let $f : [0,1] →\Bbb R$ satisfy the property that for every $ε>0$ the set $\{x∈[0,1]: |f(x)|\ge ε\}$ is finite. Prove that for every continuous increasing function $α :[0,1]→\Bbb R$, that $f$ is ...
1
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1answer
39 views

Show that $ \lim_{n\to\infty} \int_{0}^{1} f_n $ exists

Let $(X,d)= (C[0,1],d)$ where $C[0,1]$ is the set of real-valued continuous functions on $[0,1]$ and $d= \int_{0}^{1} |f-g|$ is the Riemann Integral. Suppose $(f_n)$ is a Cauchy sequence in $(X,d) $ ...
1
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1answer
44 views

Tricky analysis proof involving the sum of integrals

The question is as follows: Let $P:a=t_{0}<t_{1}<...<t_{m}=b $ be a partition of $[a,b]$, $f_{j}$ be a Riemann integrable function on $[t_{j-1},t_{j}] \ \forall \ j=1,...,m$, and let $f:[...
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51 views

Intuition behind Riemann integrabillity and sequences of partitions [on hold]

I started to self-study measure theory and as a review the book Im using(Rana) begin explaining some theorems on Riemann Integrals and im trying to understand to the fullest the intuition and proofs ...
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1answer
17 views

Integrability of composite functions [duplicate]

Let $f$ be a Riemann-integrable function on a closed interval $[a,b] \subset \mathbb{R}$. Let g be a function on $\mathbb{R}$. What conditions must g satisfy so that $g \circ f$ is also Riemann-...
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0answers
12 views

Riemann-Stieltjes Integral - Changing Order of Integration

I am new to Riemann-Stieltjes integral. I want to ask a very basic question regarding changing the order of integration. Let $ t > 0 $ and I have an integral that looks like this $$ \int_\mathbb{R}...
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1answer
20 views

Prove that $f \in R(F_s)$ on the interval $[a, b]$ and that $\int f dF_s = f(s)$

Let $a<s<b$ and let $f:[a, b] \to \mathbb{R}$ be a bounded function that is continous at the point s. Define $F_s(x) = \begin{cases} 0, & \text{if $a \le x \lt s$} \\ 1, & \text{if $s \...
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1answer
31 views

How to evaluate the sum for definite integrals using limit definition?

If $f$ is integrable on $[a,b]$, then $$ \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x_i $$ where $\Delta x = (b-a)/n$ and $x_i = a + i\Delta x$. Use this definition of the ...
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2answers
44 views

Integrability of the derivative of $f(x) = x^2 \sin (1/x^2)$ if $x \ne 0$ and 0 otherwise.

why is the derivative of the function $f(x) = x^2 \sin (1/x^2)$ if $x \ne 0$ and 0 otherwise, not integrable?
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0answers
47 views

A bounded function $f : [a, b] → \Bbb R$ is Riemann integrable [closed]

A bounded function $f : [a, b] →\Bbb R$ is Riemann integrable, show that there is a sequence $(P_n)$ of partitions such that $$\lim_{n→∞} [U(f, P_n) − L(f, P_n)] = 0$$ Since $f$ is bounded, then we ...
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2answers
36 views

How to find $\int_0^1x^3$ using sums and partitions?

The problem statement is to. Calculate $\int_0^1x^3dx$ by partitioning $[0,1]$ into subintervals of equal length. This is my attempt: Let $p=3.$ Let $\delta x = 0.5$ so that the partition is $[0,0.5],...
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0answers
18 views

Compute the R-S Integral when there are infinite number of points of discontinuities.

The question is as follows, Let $f(x)$ be a function such that: $$f(x) = \begin{cases} 0 & \text{if $x \in \left\{\frac{n}{n+1},\frac{n+1}{n}\right\}$}\\ 1 & \text{else} \end{...
1
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1answer
24 views

How to determine if an improper integral converges or diverges?

Let's say we have $\int_{0}^{\infty}\frac{e^{-x}}{\sqrt{x}}dx$. I'd like to find the antiderivative and then analyze the integral going from 0 to N as N goes to infinity. However, I don't know how ...
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2answers
26 views

If f is continuous on [a,b] and $f(x) \geq 0$, for $x \in [a,b]$, but $f$ is not the zero function, prove that $\int_{a}^{b} f(x) dx > 0$ [duplicate]

If f is continuous on [a,b] and $f(x) \geq 0$, for $x \in [a,b]$, but $f$ is not the zero function, prove that $\int_{a}^{b} f(x) dx > 0$ Could anyone give me a hint for this proof please?
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2answers
55 views

Limit, Riemann Sum, Integration, Natural logarithm

For any natural number $m$, $\lim_{n\rightarrow \infty }\left ( \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\cdots +\frac{1}{mn} \right )=\ln (m)$. I tried to prove the statement in the following way. ...
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2answers
53 views

Prove that $\lim_{n \rightarrow \infty} \int_{0}^{2 \pi} \frac {\sin (nx)}{x^2 + n^2} dx = 0$.

Prove that $\lim_{n \rightarrow \infty} \int_{0}^{2 \pi} \frac {\sin (nx)}{x^2 + n^2} dx=0.$ I think I will use Riemann integrability, but how I do not know, could anyone help me in this?
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1answer
14 views

Find a specific partition to make $U(f,P) < 2/100$

" Suppose $f : [0,1] \to \mathbb{R}$ is given by $f(1/n) = 1/n$ when $n \in \mathbb{N}$ and $f(x) = 0$ for all other $x \in [0,1]$. Show that for some partition $P$ of $[0,1]$, we have $U(f,P) <...
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2answers
27 views

Integrability of of $f$ over an Infinite Partition

Let $a_n$ be a strictly monotonic sequence such that $a_n \rightarrow b$. Moreover, $a_0 = a$ and $a<b$. If $f[a,b] \rightarrow \mathbb{R}$ is a bounded function, and $f$ is integrable on $[a_n, ...
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1answer
39 views

How to show $\lim_{n\to\infty}\sum_{k=1}^n(1+(k)/n)^3(1/n)=15/4$?

How would I show $$\lim_{n\to\infty}\sum_{k=1}^n(1+(k)/n)^3(1/n)=15/4?$$ My attempt is using the Riemann Sum technique. We know $(1+(k)/n)^2=f(\zeta_k)$ and $(1/n)=\Delta x$. So the definite integral ...
2
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1answer
53 views

How do i show following equality [closed]

Let $f(x)$ be monotone on $[a,b]$.Show that there exists $c∈[a,b]$ such that $$\int_a^b f(x)dx = f(a)(c-a) + f(b)(b-c) $$ I want to show it from basics and by using first intermediate value theorem ...
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2answers
47 views

How to show $f(x)$ is $0$ in following problem? [duplicate]

Let $f$ be a continuous function defined on $[a, b]$. Assume that there exist constants $α$ and $β$ with $(α ≠ β)$ such that $$\alpha\int_a^x f(u)du + β\int_x^bf(u)du = 0 $$ for all $x$ belonging to ...
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3answers
215 views

How to find lower Riemann integral in given function?

$f(x)$ defined on $[0,1]$ as following - $$ \begin{align} f(x) = \begin{cases} 0 & \text{if $x=0$}\\ \frac{1}{n} & \text{if $1/(n+1)<x\le 1/n$} \end{cases} \end{align} $$ How to find lower ...
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2answers
62 views

How to find the upper Riemann integral of following function?

$f(x)$ is defined on $[a,b]$ as - $= 0$ if $x ∈ [a, b] ∩ Q$ $= x$ otherwise Lower integral is $0$ and i think that upper integral should be $(b^2 - a^2)/2$(i may be wrong also) but i am not able to ...
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2answers
52 views

Showing $\lim_{n \to \infty} L(f,P_n,[a,b]) = L(f,[a,b])$, where $P_n$ is partition of $[a,b]$ into $2^n$ subintervals

Suppose $f : [a,b] \to \mathbb{R}$ is bounded. With a partition $P$ of the form $a = x_0, \dots ,x_n = b$ of $[a,b]$, the lower Riemann sum is $L(f,P,[a,b]) := \sum_{i=1}^{n} (x_i - x_{i-1}) \inf_{[x_{...
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1answer
135 views

Calculating $S=\sum\limits_{n=1}^\infty\left(\frac{1}{\Gamma^2(n+1)}\right)^{{1}/{n}}$

I tried to find the answer for the question: Numerical evaluation of $\sum_{N=1}^\infty\left(\frac{1}{\Gamma(N+1)^2}\right)^{\frac{1}{N}}$. I think my result is $4$ times than the expected value. Is ...
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4answers
55 views

Is $1 / \sqrt{x}$ Riemann integrable on $[0,1]$?

If $\int_0^1 1 / \sqrt{x}$ Riemann integrable then using second fundamental theorem of calculus i can easily say that $\sqrt{x}$ is uniformly continuous. Basically it has one point i.e $0$ where it ...
3
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1answer
85 views

Calculate $\int_{0}^{1} x^2 dx$ using the definition of the integral using Riemann Sums

Okay, so my Real Analysis textbook defines a definite integral as follows: Let $[a,b]$ be an interval and $f$ a function with domain $[a,b]$. We say that the Riemann sums of $f$ tend to a limit $l$ ...
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1answer
44 views

Riemann integration for $f:\mathbb{R} \rightarrow \mathbb{R^2}$

I wish to show that in case $f:\mathbb{R^2}\rightarrow\mathbb{R}^2$ is continuous then for $g(t) :[a,b]\rightarrow \mathbb{R^2}$ , defined as: $$g(t) = f(z+ht),\quad z,h\in \mathbb{R}^2,$$ then $$\...
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3answers
73 views

Proving that $f(x)=0$ in all points of continuity if $f$ is orthogonal to all polynomials

Suppose that the function $f$ is: 1) Riemann integrable (not necessarily continuous) function on $\big[a,b \big]$; 2) $\forall n \geq 0$ $\int_{a}^{b}{f(x) x^n} = 0$ (in particular, it means that ...
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1answer
18 views

If $f_m$ is a sequence of continuous functions which converges uniformly to a function. then that function is continuous as well. [duplicate]

I was looking for the proof is this theorem, but I couldn't find it anywhere. the theorem is stated formally: If $f_m$ is a sequence of continuous functions defined on $D$ (subset of $R$) such that $...
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1answer
24 views

Changing value of a Riemann integrable function on a Lebesgue measure 0 set implies the new function has the same Riemann integral?

Suppose $[a,b]$ is a compact interval of $\mathbb{R}$ and $f:[a,b]\to\mathbb{R}$ be integrable in the Riemann sense. Then, by Lebesgue's criterion, $f$ is bounded on $[a,b]$ and it's set of ...
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1answer
26 views

Improper integral of bounded functions over bounded open interval

Suppose $(a,b)$ is an open and bounded interval of $\mathbb{R}$. Also, let $$f:(a,b)\to \mathbb{R}$$ be a bounded function. Set $\tilde{f}:[a,b]\to\mathbb{R}$ as the function $$\tilde{f}(x)=\begin{...
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0answers
34 views

Ahlfors page 171 Poisson Integral

Tl;dr : compute the last integral with $z$ fixed. If $C_1$ and $C_2$ are complementary arcs on the unit circlw, set $U = 1$ on $C_1$ and $U=0$ on $C_2$. Let $0 \leq \theta_0 < \theta_1 \leq 2 \...
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0answers
26 views

If $0 \le f(x) \le M$, prove $\lim\limits_{n \to \infty} \left[ \int_0^1 f(t)^n \, dt \right]^{1/n} = M$ [duplicate]

Q: Suppose that $f$ is a continuous, nonnegative function on the interval $[0,1]$. Let $M$ be the maximum of $f$ on the interval. Prove that: \begin{align*} \lim\limits_{n \to \infty} \left[ \int_0^...
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1answer
50 views

A Basic Question about Integrals

I have a basic question about integrals. Suppose that $f: [0,1] \mapsto \mathbb{R}$ is a bounded function. Do not assume that $f$ is continuous. For each $n \geq 1$, define $f_n: [0,1]\mapsto \mathbb{...
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0answers
30 views

Trying to integrate Thomae's Function with Riemann sums definition

Let $f:[1,2] \to \mathbb{R}, f(x) = 0$ at irrationals and $f(x)=\frac1q$ at rationals $\frac{p}{q},q>0$ in lowest terms. (Thomae function in $[1,2])$ I need to prove that $\int_{1}^{2}dx = 0$. I ...
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0answers
30 views

Show the relationship between the supremum and infimum of f^2 and |f|

Suppose f: [a,b] $\to$ $\mathbb{R}$ and B satisfy |f(x)| $\le$ B for every x $\epsilon$ [a,b]. Show that if P = {x$_{0}$,...,x$_{n}$} is a partition of [a,b], then M(f$^{2}$,[x$_{i-1}$,x$_{i}$]) - ...
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1answer
16 views

Relaxing hypotheses for the mean value theorem for integrals

If a function $f$ is continuous in $[0,\Delta]$ it is pretty easy to prove that $$ \exists c\in(0,\Delta):\frac{1}{\Delta}\,\int_{0}^{\Delta}f(t)dt=f(c) $$ It is enough to apply Lagrange's to the ...
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vote
1answer
25 views

Integrability of bounded functions

This question arises from my studies concerning first order partial differential equations, in particular from the definition of weak solution for a conservation law. Let $q \in C^1(\mathbb{R})$ and $...
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votes
1answer
31 views

Is $f$ and $g$ is Riemann-integrable?

let $f : [0,1] \rightarrow \mathbb{R}$ and $g:[0,1] \rightarrow \mathbb{R}$ be two function define by $$f(x) = \begin{cases} \frac {1}{n},\text{ if }x = \frac{1}{n},n \in \mathbb{N}\\ 0, \text{...
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0answers
30 views

Help for this problem involving rieman integral and partitions

If $f: I--->\mathbb R$ is bounded, let $||f||:= Sup {|f(x)|: x \in I}$, and if $P =(x_0,...,x_n)$ is a partition of $I:=[a,b]$, let $||P||:=Sup [x_1-x_0,...,x_n-x_{n-1}]$ (a) If $P'$ is the ...
2
votes
1answer
52 views

Riemann integrability on open subset of $\mathbb{R}$

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $f$ is Riemann integrable over $\mathbb{R}$. Is it true that $f$ is Riemann integrable over any open subset $A\subset \mathbb{R}$??
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1answer
20 views

Continuous and bounded functions and Riemann integrability

Suppose that $v=v(t,x)\in C^1([0,+\infty]\times\mathbb{R})$ is a compactly supported function. Is it true that $$v(0,x):\mathbb{R}\to \mathbb{R}$$ is Riemann integrable over $\mathbb{R}$? Certainly $...
1
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2answers
65 views

When does the Riemann Integral-like characterization of Lebesgue Integrals fail?

Wheeden and Zygmund's book "Measure and Integral" gives an interesting characterization of the Lebesgue Integral that is reminiscent of the Riemann Integral. If $E$ be a Lebesgue measurable set, then ...
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0answers
23 views

If f is a bounded function and Riemann integrable on a subset [a',b] of [a,b], prove that f is integrable on [a,b].

Let f : [a,b] $\to$ $\mathbb{R}$ be a bounded function such that f $\epsilon$ $\mathcal{R}$[a',b] (ie: f is Riemann integrable) for every a' $\epsilon$ (a,b). Prove that f $\epsilon$ $\mathcal{R}$ [a,...
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0answers
26 views

Function with countably infinite number of discontinuities is integrable [duplicate]

Suppose that $g:[a,b]\to\mathbb{R}$ is bounded and continuous on $[a,b]$ except at $\{x_n\}_{n=1}^\infty\subset[a,b]$ and $\{x_n\}$ converges to $x_0$. Prove that $g$ is Riemann integrable on $[a,b]$. ...
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votes
2answers
39 views

Thomae function is Riemann integrable [duplicate]

Let $f:[0,1]\to \mathbb{R}$, $f(x)=0$ if $x\not\in \mathbb{Q}$ and $f(x)=\frac{1}{q}$ if $x=\frac{p}{q}$, $p,q$ coprime. $p$ integer, $q$ natural. I want prove that $f$ is Riemann integrable. I ...