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 $S$ 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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Prove that $P \cup Q$ and $P \cap Q$ are partitions of [a,b]

Let $P$ and $Q$ be partitions of the interval $[a,b]$. Prove that $P \cup Q$ and $P \cap Q$ are also partitions of $[a,b]$. Let $$ { P = \{x_0,x_1,\dots,x_{n-1},x_n\}}$$ and $$Q = \{y_0,y_1,\dots,y_{m-...
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do the upper and lower darboux sums of a function change depending on the norm(mesh) of the partition?

if we have two partitions of the interval [0,1] p1 and p2 so that the norm of p1 is greater than the norm of p2, then does that mean that U(f,p1) > U(f,p2) ?
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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 ...
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1 answer
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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 $f$ ...
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1 vote
1 answer
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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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8 votes
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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 \...
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3 votes
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 $\...
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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 ...
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1 vote
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} \...
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3 votes
1 answer
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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 ...
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1 vote
1 answer
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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 ...
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4 votes
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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 $\...
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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 ...
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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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1 answer
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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 ...
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1 vote
1 answer
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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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1 answer
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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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1 vote
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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 ...
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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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7 votes
2 answers
346 views

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 ...
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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 ...
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1 answer
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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 ...
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1 vote
0 answers
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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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0 votes
2 answers
48 views

Prove that $\left|\Sigma^n_{i=1}c_i\Delta x_i - \frac{b^2}{2}\right| \leq \frac{1}{2}\Sigma^n_{i=1}(\Delta x_i)^2$ where...

Let $P$ be a partition of $[0, b]$ defined as $P = \{ 0 = x_0 < x_1 < > \ldots < x_n = b\}$, and let $c_i \in [x_{i-1}, x_i]$ for every $1 > \leq i \leq n$. Prove: $$\left|\Sigma^...
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1 vote
1 answer
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Prove that for each $n \in \mathbb{N},$ the set $P = \{j/n\}$ is a partition of $[0,1]$

$$P=\left\{\frac{j}{n}:j=0,1,...n\right\}$$ I can't find anywhere in my book where it is proven that a given set is a partition. It just jumps to showing, using partitions, that a function is ...
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4 votes
2 answers
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Quadratic Variation of a continuous function

I have got the following problem, but no idea where to start, so maybe one of you can help me out. We have got a continuous function $f:[0,1]\to \mathbb{R}$ with $f(0)\neq f(1)$. The task is to ...
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2 votes
0 answers
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Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions?

Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions? I know that by the definition of an Archmedean sequence of ...
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1 vote
1 answer
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Proof that a function is Riemann integrable if for any $\epsilon > 0$ there exists a partition P such that: $U(P, f) − L(P, f) < \epsilon$

I am working my way through the proof of the following: Let $f$ be bounded on [a,b]. Then $f$ is Riemann integrable if and only if for every $\epsilon$ there is a partition on $[a,b]$ such that: $0 \...
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2 votes
0 answers
259 views

Why can’t improper integrals be defined directly using Riemann sums?

The standard way to define an improper integral of the form $\int_a^\infty f(t)dt$ is as follows. We first define the Riemann integral $\int_a^xf(t)dt$ for each $x>a$ in the standard way, i.e. ...
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0 votes
1 answer
592 views

Show that there is a sequence $(P_n)$ of partitions of $[0,1]$ such that $||P_n||\to0$ & $\lim_{n\to\infty} S(g,P_n)$ for the $g(x)$ defined.

Let $g\colon[0,1]\rightarrow \mathbb{R}$, be defined as $g(x) = 0$ if $x \in \mathbb{Q}$ and $g(x)=1/x$ if $x \not\in\mathbb{Q}$. Show that there exists a sequence $(P_n)$ of partitions of $[0,1]$ ...
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$f$ is integrable iff the $U(f;P)-L(f;P)< \varepsilon$ whenever mesh(P)<$\delta$?

This is very close to what is shown in Rudin (Thm 6.6) but Rudin doesn't mention anything about a delta or a mesh in his theorem. We were supposed to show this in class yesterday but the professor ...
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1 vote
2 answers
615 views

How do we calculate integrals without knowing differentiation?

To calculate the integrals, we use the fundamental theorem of calculus. i.e for example to find $\int f(x)dx$, what we do is we find a function $g(x)$ such that $g'(x) = f(x)$. But, is there any way ...
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1 vote
2 answers
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prove $L(f)\leq U(f)$

How exactly would I go about proving the following statement? Given $f:[a,b]\to\mathbb{R}$ show that $$L(f)\leq U(f)$$ where $$L(f)=\sup_{P\in\mathscr{P}}L(f,P) \text{ and } U(f)=\inf_{P\in\mathscr{P}...
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  • 299
3 votes
1 answer
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Prove $f$ is integrable on $[a,b]$ if $f$ has finitely many accumulation points on $[a,b]$.

Suppose interval $I\in[a,b]$ as finitely many limit points, $f:[a,b]\to \mathbb{R} $ is bounded on $[a,b]$ and continuous on $[a,b]\setminus I$. Use the fact that if $f$ is continuous except at ...
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  • 151
0 votes
2 answers
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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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0 answers
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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 ...
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