Questions tagged [partitions-for-integration]

For questions on finding or constructing a partition of an interval to compute the Riemann integral or Riemann-Stieltjes integral.

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If $f$ is Riemann integrable on a closed interval, does its integral exists for any sequence of partitions with norm converges to $0$?

It can ben shown that if $f$ is continuous on $[a, b]$ then $\int_a^bf\ dx=\lim_{n\to\infty}\ U(f, P_n)=\lim_{n\to\infty}\ L(f, P_n)$ for any sequence of partitions of $[a, b]$ with norm (or mesh) ...
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When defining arclength of a curve, does it matter whether the partition norm approaches $0$ in the domain (interval) or the codomain (space)?

Suppose $f:[0,1]\to\mathbb R^d$ is continuous, and $P=[t_0,t_1,t_2,\cdots,t_{n-1},t_n]$ is a partition, so $0=t_0<t_1<\cdots<t_n=1$. Define $$\sum_P\lVert df\rVert=\sum_{1\leq i\leq n}\lVert ...
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Prove equivalence of two definitions of arclength (for non-differentiable curves) [duplicate]

Suppose $f:[0,1]\to\mathbb R^n$ is continuous. For a partition $P=\{t_0=0,t_1,t_2,\cdots,t_{m-1},t_m=1\}$, with norm $|P|=\max_i(t_i-t_{i-1})$, define $$\sum_P\lVert df\rVert=\sum_{i=1}^m\lVert f(t_i)-...
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For an arbitrary continuous function $f$, is the Stieltjes integral $\int_0^1(df(x))^3=0$?

Suppose $f:[0,1]\to\mathbb R$ is continuous, possibly with unbounded variation. We consider sums of the form $$\sum_{i=1}^n\Big(f(x_i)-f(x_{i-1})\Big)^3$$ where $0=x_0<x_1<x_2<\cdots<x_{n-...
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A question on exercise 10.6(b) in Munkres - Analysis on Manifolds

Let $$I=I_1\times\cdots\times I_n=[a_1,b_1]\times\cdots\times[a_n,b_n]$$ be a $n-$rectangle. Let $f:\ I\longrightarrow\mathbb R$ be bounded and $|f|\leq M$ on $I$. Let $P=P_1\times\cdots\times P_n$ be ...
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Estimate of sum of elements of a partition of an interval

I'm reading the proof of the sewing lemma and i'm not able to proof a quite straightforward result that is implicetly used in the paper. Given an interval $\left[s,t\right]$ and a partition $\mathcal{...
Marco's user avatar
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Is there a driving noise such that it behaves ''Hölderly'' over a uniform partition?

It is well-known that in case of a linear parititon of $[0,1]$, $\{t_n\}_{n=1}^N = \{\frac{n}{N}\}_{n=1}^N$, we have $$ \int_{t_n}^{t_{n+1}} dt = t_{n+1} - t_n = \frac{1}{N} \quad \forall n$$ But ...
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question of partition in double integral

in the definition of double integral,what the first to do is to divide region into small subrectangles . my question is :do these partitions have to be rectangles? for example ,when dealing with polar ...
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Riemann Sums Approximation

Let P = {$-10,-2,0,1,5$} and $f:=[-10,5]-> \mathbb{R}$ given by: $ f(x)= \begin{cases} 4&\text{if}\, x= 0\\ \frac{x+2|x|}{|x|}&\text{if}\, x\not=0 \end{cases} $ I need to find a partition ...
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Why can a partition be a refinement of itself?

Generally speaking, to call something a ‘refinement’ has certain implications. For instance, intuitively if one were to refine a partition of some interval it would make sense that the Darboux sums of ...
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A proof about a condition of Riemann integrable functions on my textbook

I was stuck when proving a theorem in Introduction to real analysis (4th edition). I don't know why the author assumed $c=x_{i}=x_{i-1}$. Is it because $x_{i}$ and $x_{i+1}$ are close enough or ...
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Defining the length of an interval in the Riemann sums

In the Riemann integration theory, the partition of the studied interval $[a,b]$ is $P(x,t)$. But how do we define the length of the subdivided intervals $[x_i , x_{i+1}]$ ? Do we say it is the ...
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Partitions in Definite Integral

Help me to solve it. I don't understand this knowledge!!! Thansk so much ^^ Let f(x)=|x| , [-1,1] . Have P={ x0=-1 < x1< ... <xn = 1}, xi - xi-1 = 2/n. Find: L(f,P) and U(f,P) image
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Upper sum infimum of function with property: ∀ε>0 $\exists N \in \mathbb N \cup \{0\}:\lvert\{x\in[a,b]: |d(x)| \gt \varepsilon\} \rvert = N$

Let $d$ be a real-valued function defined on $[a,b]$, for $a \lt b$, with the following property: $\forall \varepsilon \gt 0: \exists N \in \mathbb N \cup \{0\}: \displaystyle\Big\lvert\{x\in[a,b]: |...
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Show $\{n(\int_0^1 f(t)dt- R_n(f)):n\in\mathbb{N}\}$ is bounded

Let $X=C[0,1]$. For $n\in\mathbb{N}$ and $f\in X$, define the Riemann sum with uniform partition $$ R_n(f)=\frac1n\sum_{k=1}^n f\left(\frac kn\right).$$ Show that if $f\in X$ is Lipschitz, then the ...
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If $S$ is set of lower sums for partitions having $n$-equal subintervals, does $\sup S$ equal the $\sup$ of the set of lower sums for all partitions.

Firstly, here are two relevant definitions: Let $a \lt b$. A partition of the interval $[a,b]$ is a finite collection of points in $[a,b]$, one of which is $a$ and one of which is $b$. Suppose $f$ ...
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Category of partitions of an interval

Revisiting the definition of Riemann integral, Carla noticed that we can define partitions of an interval categorically: given a (nondegenerate) interval $[a,b]$ seen as an ordered set with top and ...
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How to prove that $g(x)=x^2$ is integrable on $[2,5]$ using regular partitions?

So I've been trying to prove that $g(x)=x^2$ is integrable on the interval $[2,5]$ using regular partitions and the theorem that a function is integrable if $$\lim_{n\to\infty}(U(f,P_n)-L(f,P_n)) = 0.$...
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Useful properties for integration

Proposition (1). If $f:[a,b]\to\mathbb{R},g:[a,b]\to\mathbb{R}$ are bounded functions and $P:=\{x_0,x_1,x_2,...,x_n\}$ is a partion of $[a,b]$ then $$m_i(f)+m_i(g)\leq m_i(f+g),$$ where for every ...
Rata mágica's user avatar
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The infimum over all partitions is the same as the infimum over all partitions including a fixed partition $Q$

Take an interval $[a,b]$ on the real line and a bounded function $f:[a,b] \to \mathbb R$. For a partition $P =\{ a = t_0, t_1, \dots, t_n = b\}$ of said interval we define the Upper Darboux Sum of $f$ ...
Francisco José Letterio's user avatar
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Prove a function with infinite discontinuous points is Riemann integrable

\begin{equation} f(x)=\begin{cases} x& x=\frac{1}{n}&n=\mathbb Z/\{0\}\\ \\ 1 & \text{others} \end{cases} \end{equation} The Riemann integration is $ \displaystyle \int^1_{-1} f$ I ...
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When would we want to use uneven subintervals in a Riemann integral?

The formal definition of a Riemann Integral is written such that you can have uneven subintervals and it still works. Why do we need to generalize to the case of uneven subintervals? Why not insist ...
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Show that the Rieman integral of $f$ is ...

Suppose $f:[a,b]\to \mathbb{R}$ is integrable. Show that the Rieman integral of $f$ is the unique real number $r$ satisfying the following condition: For every $\epsilon >0$,$\enspace \exists \...
sameed hussain's user avatar
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3 answers
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Different ways of constructing a partition of an interval

Let $a<b\in\mathbb{R}$. A sequence $P:=(p_0,\ldots,p_n)$ is a called a partition of $[a,b]$ if $$a=p_0<\ldots<p_n=b.$$ The size of $P$ is taken to be $\max_i(p_{i+1}-p_i)$. Now, suppose we ...
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If $f:[a,b]\to\mathbb{R}$ is bounded then $L(f,[a, b])=\lim _{n\to\infty} L(f, P_n,[a, b])$ and $U(f,[a, b])=\lim _{n\to\infty} U(f, P_n,[a, b])$

I am trying to prove the following statement from Axler's MIRA book: "Suppose $f:[a, b]\to\mathbb{R}$ is a bounded function. For $n \in \mathbf{Z}^{+}$, let $P_n$ denote the partition that ...
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Example for partition $ \prod = \{ x_0 < x_1 < ... < x_n \} $ of $ [a,b] $ where $ \max\{ | x_i - x_{i-1} | : i = 1,...,n \} < \delta $

Sometimes when I'm doing proofs regrading Riemann Integrals I would like to create an explicit parition whose norm is less than $ \delta $ . I'm familiar with creating the following partition of $ [a,...
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Function with some partition such that $\underline{S}(f,P) = \overline{S}(f,P)$ implies that $f$ is constant

This posts begins with a question for a homework I had, and that question is rather simple: If $f:[0,1] \rightarrow \mathbb{R}$ is a function and there exists some partition of $[0,1]$ such that $\...
Mr. Bluesky's user avatar
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Finding Limit of Fraction Using Integration

Using ideas around Riemann Integration and Improper Integrals, I am looking to find $$\large\lim_{n\to\infty}\frac{\root^n \of {n!}}{n} $$ I think it is clear that the $\frac{1}{n}$ term can represent ...
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Integrability and Continuity on Interval

If $g$ is continuous on $[a, b]$ and if $f$ is integrable on $g([a, b])$, then $f \circ g$ is integrable on $[a, b]$. One of the famous theorems reveals that if $f$ is continuous on some compact ...
Murad Aghazada's user avatar
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1 answer
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Doubt of a problem of a tagged partition

The problem is: Let $\dot{P}$ be a tagged partition on $[0,3]$. Show that the union $U_{1}$ of all subintervals in $\dot{P}$ with tags in $[0,1]$ satisfies $[0,1-\Vert{\dot{P}}\Vert]\subseteq U_{1} \...
Luis Lapo's user avatar
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Find a partition of a closed interval

I have $P = \{ a = a_0 < a_1 < \dots < a_n = b \}$ to be a partition of $[a, b]$. Define $\lVert P \rVert = \max \{ (a_i - a_{i - 1}): i = 1, \dots, n \} $. I need to find a sequence of ...
little_sky's user avatar
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Is the following cases function Riemann integrable?

I have the following question: Let $f$ be a function defined in the following interval: [0,1], such that:$$f(x)=\begin{cases} 2 & 0\le x<\frac{1}{3} \\\\ 0 & \frac{1}{3}\le x<\frac{2}{3}...
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What's the opposite of the "mesh" of a partition?

The mesh of a partition of an interval is defined as the maximum distance between two consecutive points of the partition, $$ \operatorname{mesh}(P) := \max_{i\ge 1}(x_i-x_{i-1}) \quad\text{where \( P ...
derpy's user avatar
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f Riemann integrable on [0,1]

I was trying to show that if $f$ is Riemann integrable on $[0,1]$ then $$\int_{0}^{1} f dx = \lim \left( \frac{1}{n} \sum_{k=1}^{n} f(\frac{k}{n}) \right)$$ I know that since $f$ is Riemann ...
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What is a sequence of partitions in integration?

Right now in my introductory real analysis course, we are studying integration theory, and in particular the Archimedes-Riemann theorem. The Archimedes-Riemann theorem requires the existence of an ...
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Prove Equivalency to Darboux Integrability

I've been working through a book on Introductory Real Analysis and I've been stumped by part of this problem. Suppose we consider a partition that splits $[a,b]$ into $n$ partitions each with length $\...
Alexander's user avatar
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Please help understand proof of small span theorem in multivariable calculus [closed]

The proof of small span theorem (given in Apostol Volume 2) is as follows: Please (if possible) give a detailed proof of this theorem including all information in the above proof $\mathrm{off}$ is to ...
lorilori's user avatar
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General facts about partitions in integrals

(a) Given two partitions P and P', is it always true that if P has more points than P', then $U_P - L_P \leq U_{P'} - L_{P'}$? My line of though was that if the distance between points is massive then ...
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Common refinement with different intervals

Let $[a,b]$ be an interval. A partition of $[a,b]$ is $N+1$ points such that: $$P:= \{x_1= a, x_2, \ldots,x_{N+1}= b \}.$$ Moreover, for an interval $[a,b]$ we can take two partitions $P^1, P^2$, and ...
independentvariable's user avatar
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Prove that if the partition $\textbf{P}'$ is finer than $\textbf{P}$, then $f$ keeps piecewise constant

Let $I$ be a bounded interval, let $\textbf{P}$ be a partition of $I$, and let $f:I\to\textbf{R}$ be a function which is piecewise constant with respect to $\textbf{P}$. Let $\textbf{P}'$ be a ...
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Prove the following result: if $f(x) \geq g(x)$ for all $x\in I$, then $p.c.\int_{I}f \geq p.c.\int_{I}g$.

Let $I$ be a bounded interval, and let $f:I\to\textbf{R}$ and $g:I\to\textbf{R}$ be piecewise constant functions on $I$. Prove the following result: (a) If $f(x) \geq g(x)$ for all $x\in I$, then $p....
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When the piecewise constant integral independs of the partition's choice?

Proposition Let $I$ be a bounded interval, and let $f:I\to\textbf{R}$ be a function. Suppose that $\textbf{P}$ and $\textbf{P}'$ are partitions of $I$ such that $f$ is piecewise constant both with ...
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Ill-defined derivative?

This is a follow-up question to my previous question, Riemann integral interval confusion, from the same book. Below is a snippet of context from the book: Claim (I should probably be using $\...
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Approximating the Riemann integral for a certain product

Suppose that we have two continuous maps $f,g\colon I\to\mathbb{R}$ on the interval $I:=[0,1]$. Now suppose that $0=x_{0}<x_{2}<\ldots<x_{n}=1$ is some partition of $I$ and that for each $i$ ...
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If $f\in R(\alpha)$ and $C\in\mathbb{R}$, then $Cf\in R(\alpha)$ and $\int_a^b Cf\operatorname{d}\alpha=C\int_a^b f\operatorname{d}\alpha.$

I can figure out how to prove this, assuming C is positive, but I'm not sure how to take into consideration if C < 0. The sup of any partition of $Cf$ will simply be Csupf(x). This makes the proof ...
wooooooo's user avatar
3 votes
1 answer
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How should I partition this interval to utilize the left endpoint $\frac{\epsilon}{2}$?

In the problem above, since f is continuous and thus uniformly continuous on the compact interval $[\frac{\epsilon}{2},1]$, it is possible to directly refer to the general property that all uniformly ...
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Limit of the ratio of two non-Riemann sums.

Let $\left[ {a,b} \right] \subset \mathbb{R}$ and $f,g:\left[ {a,b} \right] \to \mathbb{R}$ be two Riemann-integrable functions. Let $a = {x_0} < {x_1} < {x_2}... < {x_n} = b$ be a ...
Fabrice Pautot's user avatar
1 vote
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Darboux Sum in terms of partitions

I've got a problem to solve, however during the lectures that was explained poorly. I was able to teach myself culculating double integrals. But that is more complex to understand. I will appreciate ...
Михаил Андреев's user avatar
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Proving $\frac{1}{f}$ is Riemann Integrable

Suppose $f\in R(x)$ and $\frac{1}{f}$ is bounded on $[a,b]$. Prove that $\frac{1}{f}\in R(x)$ on $[a,b]$. We need to show $U(P,f)-L(P,f)\leq\epsilon$ to prove Riemann Integrability. To prove this ...
help's user avatar
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Norm of a partition defined by a set, Riemann sum

The set $\{1,1.7,2,3.5,5\}$ determines a partition, $P$. Let $z_1=1$, $z_2=2$, $z_3=3$, $z_4=4$, and $f(x)=\sqrt{4-\left(x-3\right)^2}$. There’s three things I’m told to find: $\lVert P\rVert$ The ...
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