# 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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### 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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### 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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### Prove a function with infinite discontinuous points is Riemann integrable

$$f(x)=\begin{cases} x& x=\frac{1}{n}&n=\mathbb Z/\{0\}\\ \\ 1 & \text{others} \end{cases}$$ 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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### 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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### 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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### 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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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 $\... • 333 3 votes 0 answers 42 views ### Approximating the Riemann integral for a certain product Suppose that we have two continuous maps$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$i$... • 2,958 0 votes 0 answers 47 views ### 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 ... 3 votes 1 answer 80 views ### 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 ... • 829 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 ... 1 vote 0 answers 47 views ### 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 ... 0 votes 1 answer 613 views ### 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 ... • 569 1 vote 0 answers 344 views ### 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 ... • 11 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: $$\left|\Sigma^... • 998 1 vote 1 answer 127 views ### Prove that for each n \in \mathbb{N}, the set P = \{j/n\} is a partition of [0,1]$$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 ... • 2,303 4 votes 2 answers 1k views ### 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 ... • 732 2 votes 0 answers 40 views ### 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 ... 1 vote 1 answer 993 views ### 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 \... • 303 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. ... • 9,229 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] ... • 363 0 votes 0 answers 51 views ### 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 ... 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 ... • 107 1 vote 2 answers 813 views ### 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}... • 299 3 votes 1 answer 873 views ### 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$is continuous except at ... • 151 0 votes 2 answers 83 views ### How to find$\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],...
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 ...