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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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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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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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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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Maximizing the Riemann sum for partitions of fixed size

As I am doing again some elementary maths (for teaching), I have this following problem regarding Riemann sums. Let's say we consider a function $\,f$ on $[0,1]$ and we only consider partitions of ...
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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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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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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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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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93 views

Prove the product of two distinct integrable functions is integrable.

Let $R_1$ ⊂ $\mathbb{R}^n$ and $R_2$ ⊂ $\mathbb{R}^m$ be closed generalized rectangles, and let R = $R_1 ×R_2$ ⊂ $\mathbb{R}^{n+m}$. Let $g:R_1 → \mathbb{R}$ and $h:R_2 → \mathbb{R}$ be integrable ...
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How to compete the proof of that for a map defined on an arbitrary subset $S$ of $\mathbb{R}^n$ , we can extend $f$ to $C^r$ map on $\mathbb{R}^n$

In the book of Analysis on Manifolds, by Munkres, at page 144, question 3, it is asked that Questions: First of all, I was writing this question to ask whether my proof was correct or not, but ...
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What partial orders on tagged partitions generate the Riemann Integral?

Let $[a,b]$ be a closed interval in $\mathbb{R}$, and let $X$ be the set of tagged partitions of $[a,b]$. Now let’s define two partial orders over $X$. Let $P_1\geq_1 P_2$ if $P_1$ is a refinement ...
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Lower and Sup Sums [Integration, partitions]

So i have this problem. Let $f(x)$= {$x^2$ if -3$\le$ $x$ $\le$ 1, $-2x$ if $1$$\lt$$x$$\le$$2$ And $P$={$x_0$,...,$x_n$} a partition of [-3,2]. If $1$$\in$[$x_i-1$,$x_i$], find $m_i$, $M_i$, i.e, $...
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Can you explain how to write the proof of Section 1.15 Exercise 4(a) of Apostol's Calculus

I am attempting to self-study Calculus from Apostol's book, but I have gotten stuck on the proof for Exercise 4(a). There are indeed other solutions, such as the below two links, but they don't ...
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58 views

Prove that, piecewise constant integral is independent of partitions.

This is a proposition from the book of Terence Tao, Analysis I. My question may seem duplicate, but I don't want the proof in detail. I have solved it and it looks like- Suppose, $f:I\to\Bbb{R}$ be ...
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Riemann Sums and Upper Sums for a discontinuous function.

This question is from Abbott's Understanding Analysis: "If $f$ is not continuous, it may not be possible to find tags for which $R(f,P)=U(f,P)$. Show, however, that given an arbitrary $\epsilon>0 ...
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Question about Tagged Partitions and $\delta$-fine partitions

Let $f:[a,b]\rightarrow \mathbb{R}$ where $|f|$ is bounded by $M>0$. Assume $f$ is Riemann Integrable. Let $\epsilon>0$ be arbitrary but fixed. Let $P_{\epsilon}$ be a partition of $[a,b]$ such ...
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Showing that g is integrable and $\int^b_a{f}$ = $\int^b_a{g}$

Let $f$ be integrable on $[a,b]$, and suppose g is a function on $[a,b]$ such that $g$($x$) = $f$($x$) except for finitely many $x$ in $[a,b]$. Show $g$ is integrable and $\int^b_a{f}$ = $\int^b_a{g}$....
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Embedding of weighted Holder space into $L^p$

Suppose we have a bounded metric space $(X,d)$ and a countable measurable partition $Q$ of $X$ (with respect to some probability measure $\mu$) such that $\mu(q)>0$ for all $q\in Q$. We know that ...
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45 views

For two partitions of an interval any subinterval of the one with the strictly smaller mesh is contained in one or two adjacent ones of the other.

If for an interval $[a,b]$ you have two partitions $Z_1: a = x_0 < \cdots < x_m = b$ and $Z_2: a = y_0 < \cdots < y_n = b$ where the mesh of $Z_1$ is strictly greater than the one of $Z_2$...
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Help with partition/width

Let $f:[0,2] \to \mathbb{R}$ be defined by $$f(x)= \begin{cases} 1,& \text{if } x=\frac{2}{n} \text{ for some $n \in \mathbb{N}$};\\ 0, & \text{otherwise}. \end{cases}$$...
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1answer
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Bounded variation of $\frac1f$ when $\inf(|f|)>0$ & $f$ bounded variation

I want to show if $\frac{1}{f}\in BV[a,b]$ when $\inf(|f|)>0 \land f\in BV[a,b]$. I tried to find a partition that $V(\frac{1}{f},P)$ is upper-bounded using the partition that makes $V(f,P)$ ...
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109 views

Riemann Stieltjes integral. Prove that $\lim_{|P| \to 0} S(f,P,T) = \int_a^bfd\alpha$

Problem: Let $\alpha \in BV[a,b]$ and let $f$ be continuous on $[a,b]$. Prove that $\lim_{|P| \to 0} S(f,P,T) = \int_a^bfd\alpha$ Background: BV stands for bounded variation. I am using $|P|$ to ...
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Checking Riemann integrability for a function defined separately for rationals and irrationals in any given interval.

I was working on the following function, trying to find its Riemann integrability: $$f(x) = \begin{cases} x & x \in \mathbb{Q}\cap [-1,0] \\ -x & x \in \mathbb{R}\backslash \mathbb{Q}\cap [-1,...
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Non-standard partition for Riemann Sums?

I know I definitely saw an example of this in this site in the past, but I can no longer find it. In many (dare I say most?) Calc. I classes, if I, say, wanted to evaluate $$\int_{a}^{b}f(x)\text{ ...
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1answer
123 views

Difference between gauge and partition in Integration Theory

I'm studying generalized Riemann integral using Bartle's Introduction to Real Analysis textbook. What's the difference between gauge and partition? Seems like the same concept to me.
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How Many Points are in the Partition?

This is a theorem proven in my book. As usual, I covered up the proof of the theorem so that I might prove it myself. I came up with essentially the same proof, the only point of contrast being that I ...
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1answer
122 views

If the Partition $P\subset P'$, then $L(f, P)\leq L(f, P')$ and $U(f, P))\geq U(f, P')$

I have this proof but I don't really understand it, mathematically and visually (if that makes sense). So the definition is: A partition $P'$ is a refinement of the partition $P$ is $P\subset P'$. ...
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1answer
247 views

Piecewise constant integral is independent of partitions

I'm trying to prove the following proposition (from T.Tao's Analysis 1 book): Let $I$ be a bounded interval, $f\colon I\to\mathbb{R}$ function piecewise constant with respect to both P and P' ...
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118 views

Change of integration limits for uniformly continuous function

I have arrived at the inequality $$ V(\gamma) \le \sup\limits_{P\in\mathcal{P}[a,b]}\sum\limits_{i=1}^n\int\limits_{t_{i-1}}^{t_i} \left| \gamma'(s)\right|ds $$ where $\gamma(t):[a,b]\to\mathbb{C}$ ...
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480 views

Show $f$ is integrable in each subinterval, $[x_{i−1}, x_i]$, and further, $\int_a^bfdx=\sum\limits_{x_{k-1}}^{x_k}\int\limits_{x_{k-1}}^{x_k}fdx$.

Let $f$ be integrable on $[a,b]$, and let Let $P = \{x_0,x_1,x_2,...,x_{n−1},x_n\}$ be any partition of $[a,b]$. Show that $f$ is integrable in each subinterval, $[x_{i−1}, x_i]$, and further, $\int_a^...
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$f(x) = x^2$. For each positive integer $n$, let $P_n$ be the partition $P_n = \{0, \frac{1}{n}, \frac{2}{n},…, \frac{n-1}{n},1 \}$ of $[0,1]$.

Let $f(x) = x^2$. For each positive integer $n$, let $P_n$ be the partition $P_n = \{0, \frac{1}{n}, \frac{2}{n},..., \frac{n-1}{n},1 \}$ of $[0,1]$. Show that $S(P)=\frac{1}{3}+\frac{1}{2n}+\frac{1}...
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Is this right? Showing that if $S\subset \mathbb{R}^n$ arbitrary and $f$ a $C^r$ function then there is $A$ open such that $f$ is $C^r(A)$.

Let $f : S \to \mathbb{R}$ a $C^r$ function, where $S$ is any subset of $\mathbb{R}^n$. We say that $f$ is differentiable on $x_o \in S$ if there is $U_{x_0}$ and $g : U_{x_0} \to \mathbb{R}$ such ...
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Would an integral defined using partitions of an interval into infinitely many intervals make sense?

In the definition of Riemann integral or Darboux-integral we first study partitions (or tagged partition) of the given interval determined by finitely many points. To each partition and a function $f$ ...
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evaluate the $\int_{0}^1 x\,dx$ using the definition

From the definition -"$f$" is integrable on [a,b] if there exists a number $A$ so that for any $\epsilon > 0$ there exists a $\delta > 0$ such that if the sequence ${X_i}$ from $i=0,n$ is a ...
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Riemann Sum to show convergence help?

I have the function $f(x)=\sin\left(\dfrac{\pi x}{2}\right)$ on the partition of $[0,1]$ given by $$P_{n}: 0 < \frac{1}{n} < \frac{2}{n} < ... < \frac{n-1}{n} < 1$$ I have shown that $$...
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Describing (tagged) partition.

Can someone please explain what a partition is? And a tagged partition? Preferably with pictures and very few assumptions of previous knowledge.
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$C_1$ curve length proof deficiency

On page 5 of these notes is this theorem: "If $\alpha: I \rightarrow \Bbb{R}^n$ is a $C^1$ curve, then $\operatorname{length}[\alpha] = \int_I ||\alpha'(t)||dt$ " The proof begins "It suffices to ...
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Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$

Let $f$ be a generalized function on $\Omega$. Let $G$ be an open subset of $\Omega$. Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$ Let $\phi ...
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Proof of Riemann integral as limit of Riemann integral sum

I want to Prove the following statement, I will be appreciate if some one help me to do that. Let $f:[a,b]\to R$ and $f$ is bounded, show that if $f \in R$ ( Riemann integrable) and $\int_a^b ...
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1answer
88 views

Proving Riemann Integrability when $x$ is a function of $n$?

(Context: I have been using this resource from UC Davis to help me out with Riemann integrability.) So, I more or less get how Riemann integrability works when we specify $x$ over an interval, say $x&...
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343 views

Prove that $\inf{U(f,P)}\geq 1/2$

Define $f(x)=\begin{cases} x\;\;\;\text{if the point $x\in[0,1]$ is rational}\\ 0\;\;\;\text{if the point $x\in[0,1]$ is irrational} \end{cases}$ Prove that $\inf{U(f,P)}\geq 1/2$. Let $P=\{x_1,...
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469 views

Prove: Partitions and refinements

Problem: Let $ R $ be the set of partitions of a real interval. Then for all elements in $ R $, every pair of elements has an upper bound. I am having trouble structuring the proof; and intuitively ...
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820 views

If $f$ is continuous on $[a,b)$ and $[b,c]$, then $f$ is Riemann integrable on $[a,c]$.

True or False: If $f$ is continuous on $[a, b)$ and on $[b, c]$, then $f$ is Riemann integrable on $[a, c]$. I was unsure if the $)$ in $[a,b)$ completely changed the problem and made it false and I ...
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558 views

If L(P, f) = U(P, f), prove that f is a constant function on [a, b]

That is what I have for the beginning of my proof, but I'm not sure how to conclude that the function is constant.
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Real Analysis: Show that g is integrable on [a,b] and that $\int_a^b$ $g(x)dx=$ $\int_a^b$ $f(x)dx$

Suppose f is integrable and g is bounded on [a,b], and g differs from f only at points in a set H with the following property: For each $\epsilon>0$, H can be covered by a finite number of closed ...
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1answer
260 views

Real Analysis Riemann integrals with piece wise function

This is part of my homework assignment and I have been stuck on it for a few days. Let f be the function on [0,1] given by f(x)= { 1 if x does not = 1/2 and 2 if x=1/2 Prove f is Riemann integrable ...
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886 views

Definition of a Regular Partition of a Closed Generalized Rectangle in $\mathbb R^n$

What the heck does this definition of a regular partition $P$ of $R$ mean? I follow what it is saying until we get to the last part, "the $k_1\cdot k_2\cdot \cdots \cdot k_n$ subrectangles of the form ...
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3answers
814 views

Integral $ \int e^{x} \sqrt{e^{x} - 1} dx $

I want to determinate the following integral $\int e^{x} \sqrt{e^{x} - 1} dx$ My try and steps were as follow $$ \int e^{x} \sqrt{e^{x} - 1} dx $$ let $ u = \sqrt{e^{x} - 1} $ and $ v' = e^{x} $ ...
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370 views

Prove $\int \limits_0^b x^3 = \frac{b^4}{4} $ by considering partitions $[0, b]$ in $n$ equal subinvtervals.

I was given this question as an exercise in real analysis class. Here is what I came up with. Any help is appreciated! Prove $\int \limits_0^b x^3$ = $\frac{b^4}{4} $ by considering partitions [0, ...