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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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Proof of Generalized Riemann Integrability Criterion

Suppose f:[a,b] $\rightarrow$ $\mathbb{R}$ is a bounded function and there is a set Z $\subset$ [a,b] sucht that: f is continuous at every point x $\notin$Z. For every $\epsilon$ > 0, the set Z can ...
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Riemann-Stieltjes Integral for Specific Integrators

I'm asked to show that $\lim_{n\to\infty}\frac{b-a}{n}\sum_{k=1}^n f(a+k\frac{b-a}{n}) = \int_{a}^{b}f(x)dx$. I tried to come up with a step function that is constant on each of the open intervals $(...
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Riemann Integrable definition in partition distingued $\mathbb{Q}$

A function $f:[a,b]\rightarrow \mathbb{R}$ is Riemann Integrable on $[a,b]$ if $\exists \thinspace L \in \mathbb{R} : \forall \epsilon >0\thinspace \exists \thinspace \delta >0 :$ if $P$ is any ...
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Function is Riemann integrable in $[a,c]$ and $[c,b]$ then is RI in $[a,b]$

i need to prove that if Riemann integrable in $[a,c]$ and $[c,b]$ then is RI in $[a,b]$, maybe is easy but i can't see it. Definition of Riemann Integral A function $f:[a,b]\rightarrow \mathbb{R}...
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Is this proof that $f$ is Riemann integrable correct?

Let $f:[a,b]\rightarrow \mathbb{R}$ be a bounded, non constant function. We know that f is integrable in $[c,b]$ $\forall c \in [a,b]$. Prove that $f$ is Riemann integrable in $[a,b]$. I have tried ...
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Replacing differentiation with anti-integration?

When thinking about the proof of the differentiability of Taylor series, I noticed that the theorem was proved by using properties of integrals. This got me thinking: To what extent can the role of ...
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Examples of Lebesgue-integrable, but not Riemann-integrable functions

The standard example of this is the characteristic function of the rationals. However this is somewhat pathological as this function is zero almost everywhere. What are other examples that differ from ...
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How to find $f(x)$ if $\int_0^{x^2} (1+t)f'(t)dt=x^4$ and $\int_0^1 f(t)dt=3 $

Knowing that $f:[0,+\infty)\rightarrow \mathbb{R}$ is continuous and derivable, and that: $\int_0^{x^2} (1+t)f'(t)dt=x^4$; $\int_0^1 f(t)dt=3 $ Determine $f(x)$. (Note: This is supposed ...
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Integral inequality $\int_{0}^{1}\left ( \frac{f(x)}{x} \right )^2dx \leq 4\int_{0}^{1}(f'(x))^2dx$

Let $f \in C^1([0;1],\mathbb{R})$ such that $f(0)=0$. $$\text{Prove that} \qquad \int_{0}^{1}\left ( \frac{f(x)}{x} \right )^2dx \leq 4\int_{0}^{1}(f'(x))^2dx$$ My attempt: Let $$g(x)=\begin{cases} ...
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Problem with Definition of Riemann Integrable Function

I have difficulty understanding the definition of a Riemann integrable function $f$ on $[a,b]$. I understand the definition of upper sum $U(f,P) = \sum\limits_{i=1}^{n}M_i \Delta x_i$ and lower sum $...
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If $f(x)=\int_a^x f(t)dt$, is $f(x)=0$?

How can I prove this? Knowing $f:[a,b] \rightarrow \mathbb{R}$ is continuos in $[a,b]$ and $f(x)=\displaystyle\int _a^x f(t)dt$, how can I conclude that $f(x)=0$ $\forall x \in [a,b]$? I can't even ...
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Baby Rudin 6.11 and 6.8

Theorem 6.8 If $f$ is continuous on $[a, b]$ then $f \ ; \epsilon \; R(\alpha)$$f$ is Riemann integrable on$ [a, b]$. $f \epsilon R(\alpha)$ means $f$ is Riemann integrable w.r.t. $\alpha$ ...
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Integral inequality $\int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}x|f'(x)|dx$

Let $f \in C^1([0;1],\mathbb{R})$ such that $f(1)=0$. $$\text{Prove that} \qquad \int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}x|f'(x)|dx$$ My attempt: \begin{align} \int_{0}^{1}x|f'(x)|dx = \int_{0}^{1}...
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How to prove that a function is of bounded outside variation?

Let $E$ be the Vitali set (or any other nonmeasurable subset of $[0,1]$ with respect ot Lebesque measure), so $E\subset [0,1]$ is nonmeasurable. Consider the following finction $f\colon [0,1]\to l_{\...
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Two definitions for Riemann integrability of $f:[a,b]\rightarrow X$, where $X$ is a Banach space.

Let $X$ be a Banach space and $f:[a,b]\rightarrow X$ be a function. Consider the following two definitions of Riemann integrability: Definition 1: there exists $x\in X$ and a sequence of partitions $\...
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Calculating upper Riemann sum. Is it sufficient to only consider “simple” partitions

Let $b > 0$. I'd like to calculate $ \int_{0}^b x^2 dx$ using upper and lower Riemann sums. Since the function is continuous on $[0,b]$ I know that it is integrable so I only need to calculate $$ ...
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Volume of a tilted cylinder

Suppose I have a tilted cylinder of length l inclined to the horizontal by an angle of $\theta$ then it's volume comes out to be same as that of a straight cylinder of height $l\sin\theta$. I tried to ...
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Show that if $f \in R_\alpha$,and g increasing and continuous then $ f(g(x)) \in R_{\alpha(g(x))}$

Let $f \in R_\alpha[a,b]$ and $g:[c,d] \rightarrow \mathbb{R}$ continuous and strictly increasing, such that $g(c) = a$ and $g(d) = b$. Prove that $f(g(x)) \in R_{\alpha(g(x))}$. In calculus I ...
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Question about Riemann integrability: do we need to specify that all Riemann sums converge to the same number in the definition?

Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. Suppose that there is a sequence of partitions $\{P_n\}_{n=1}^\infty$ with mesh tending to $0$, $P_n=\{a=t_0^n<t_1^n<\ldots<t_{r_n}^n=b\}$, ...
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Lower and upper sums of a bounded function

Assume $f:[a,b]\to\mathbb{R}$ is bounded. Let $P=\{x_0,...,x_N\}$ be a partition of $[a,b]$. Why, for $1\leq n \leq N$ do we have that $$\inf\{-f(x),\,\,\,x\in[x_{n-1,x_n}]\}=-\sup\{f(x),\,\,\,x\...
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Derivative under the integral sign

I have a question about derivative under the integral sign, perhaps someone could help me with this issue. The problem is the following: Let us consider two intervals $I_{x} \in (0, +\infty)$ and $...
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1answer
36 views

Showing that $f \in R_\alpha[a,b]$ and $\lim_{n\rightarrow \infty}\int_a^b f d\alpha_n = \int_a^b f d\alpha $

Let $\left(\alpha_n \right)_{n\in \mathbb{N}}$ a succesion in $BV[a,b]$ and $f:[a,b] \rightarrow \mathbb{R}$ such that $f \in R_{\alpha_n} [a,b]$. If $\alpha \in BV[a,b]$ and $V_a^b(\alpha_n - \alpha) ...
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If $f \in BV[0,2\pi], f(0)= f(2\pi)$ Show that $\int_0^{2\pi}f(x)\sin(nx)dx$ exist for each $n$ natural.

This is from Carothers 14.38 If $f \in BV[0,2\pi], f(0)= f(2\pi)$ show that $\int_0^{2\pi}f(x)\sin(nx)dx$ exist for each $n$ natural and $$\left|\int_0^{2\pi}f(x)\sin(nx)dx\right| \leq \frac{V_0^{2\...
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Show that $\sum_{{i = 1}}^{n} f(i) = \lfloor n \rfloor f(n)- \int_{1}^{n}f'(x)\lfloor x\rfloor\, dx$

Where $f$ is a function defined in $\mathbb{R}$ with countinuos derivative in all $\mathbb{R}$, for each $n\in \mathbb{N}$ and the function $\lfloor x \rfloor$ is the floor function. I tried using ...
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Definition of integrability

By the definition of Riemann integrals I have to show that: $\int_{-a}^{a} f(x^2) dx=2\int_{0}^{a} f(x^2) dx$ (given that both integrals exist) I thought to approach it as: $\int_{-a}^{a} f(x^2) dx=\...
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1answer
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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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Sequence of integrals converges to an improper integral proof?

Let $f: (0,1) \rightarrow \mathbb{R}$ be increasing on $(0,1)$. If the improper integral (of second kind) $\int_0^1 f(t)dt$ converges ($f$ is unbounded at $0$ and at $1$), then show that the sequence $...
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trying to proove this limit equality to a specific integral [closed]

can you help me prove this equality? I tried to use Riemann sums but I haven't succeeded to find something useful. $$\lim_{n\to\infty} \sum_{k=1}^n f\left(\frac{k}{n}\right)\frac{1}{n} = \int_0^1f(...
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If $\int_a^b f\, d\alpha $ exists for all continuous $f$ then $\alpha$ is of bounded variation

In this answer the following theorem is proved: Theorem: Let $\alpha:[a, b] \to\mathbb {R} $ be a function such that the Riemann-Stieltjes integral $$\int_{a} ^{b} f(x) \, d\alpha(x) $$ exists for ...
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62 views

Prove that $g(x)=\sqrt{x}$ is not an integral function on $[0,1]$.

Prove that $g(x)=\sqrt{x}$ is not an integral function on $[0,1]$, i.e. $\nexists f \in R[0,1]$ so that $$\sqrt{x}=\int_0^x f(t) \mathrm{d}t$$ I see that $f$ cannot be continuous, because if it ...
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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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Riemann Integrability of Composite functions.

Function $f$ is continuous and strictly increasing on $[a,b]$ and $g$ is Riemann integrable such that $g \circ f$ is defined. Is $g \circ f$ Riemann integrable? I was able to show that if $f$ ...
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A new equivalent characterization of Riemann-Integrability

Question: Given two bounded functions $\,f:[a,b]\to \mathbb R\;$ and $\;\theta:(0,b-a]\to [0,1]$. Suppose $\,P:a=x_0<x_1<⋯<x_n=b\;$ is a partition of $[a,b]$. Let $\,Δx_k=x_k−x_{k−1}$ and ...
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Counterexample for this statement about the Riemann integral?

I'm considering a Riemann integral, and trying to work out if the following statement is true: Let $P$ and $P'$ be two partitions such that $\mu (P') \lt \mu (P)$. $\mu(P)$ denotes the mesh of P, the ...
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Laplace Transform: Piecewise Function Integrability and Existence of Laplace Transform

I am trying to decide whether the function $$f(t) = \begin{cases} 1, & \text{$t$ is even} \\ 0, & \text{$t$ is odd} \end{cases}$$ has a Laplace transform, or is even integrable in the ...
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Between proper integrals and improper integrals

I just started learning about improper integrals. Many of them are improper because the function evaluates to infinity at some point in their domains, e.g. $f(x)=1/x$ on the domain of $(0,1)$. My ...
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Lebesgue measurable but not Riemann integrable

Every bounded function $f:[a,b]\to\mathbb R$, which is Riemann integrable, it is also Lebesgue integrable. On the other hand $$ g(x)=\left\{ \begin{array}{lll} 1 & \text{if} & x\in\mathbb Q\...
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Is this kind of limit defined?

Is there any limit like this $$\lim_{f(x)\to0} g(x)=0?$$ Defined as follows$$\forall \epsilon>0,\exists \delta>0 | |g(x)|<\epsilon ,\forall x |0<|f(x)|<\delta?$$ Where f ...
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If $f$ is integrable, then $\| f\|$ is also integrable.

As usual, a partition of a compact interval $[a, b]$ is, by definition, an strictly increasing family $\Pi = (t_k)_{k = 0}^m$ ($m \geq 0$) of points in the interval such that $t_0 = a$ and $t_m = b;$ $...
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Riemman-Stieltjes Integration-IVT

I need to show this, Suppose that $f,g: [a,b] \rightarrow \mathbb{R}$ are continous. Show that exist $\eta \in (a,b)$ such that $$g(\eta)\int_a^\eta f(x)dx=f(\eta)\int_\eta^b g(x)dx $$ I defined $...
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Existence of antiderivative of a function [duplicate]

Let $f: \mathbb{R}\rightarrow{\mathbb{R}}$ be given by $f(x)=sen(\frac{1}{x})$ si $x\neq 0$ and $f(x)=c$ si $x=0$, where $c\in [-1,1]$ For which values of c there is an antiderivative? Proof (...
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limit of Integral of product of convergent functions

Let $f,g$ be real functions that are Riemann-integrable over $[0,t] \, \forall t \in {\mathbb R^+}$. With: $$\lim_{x \to \infty}(f(x))= p$$ $$\lim_{x \to \infty}(g(x))= q$$ Show that: $$\lim_{t \to \...
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1answer
72 views

Integration and a function increasing

Let $g_{1},g_{2}\in\mathcal{R}([a,b])$ (Riemann-integrable) such that $$\int_{a}^{x}{g_{2}(t)dt}\leq\int_{a}^{x}{g_{1}(t)dt}\phantom{a}\text{for each}\phantom{a}x\in [a,b]$$ and $$\int_{a}^{b}{g_{1}(t)...
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1answer
71 views

Integration and existence of an antiderivative

Let $ f: \mathbb {R} \rightarrow {\mathbb {R}} $ be given by $$ f (x) = \left \{\begin {matrix} \sin ({\frac {1} {x}}), & \text {if} \phantom {a} x \neq 0; \\  c, & \text {if} \phantom {a} x ...
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Example of sequence of continuous function which is not Riemann integrable with following property

$\{f_n\}$ is a sequence of continuous functions with following properties: $0 \leq f_n(x) \leq 1, \forall x \in \Bbb R$ $f_n(x)$ is monotonically decreasing sequnce as $n\to \infty$ The Limiting ...
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1answer
86 views

Find the limit of $\lim\limits_{n\to\infty}\frac{1}{n^2}\sum\limits_{k=1}^{n}k\arctan{\big(\frac{pk-p+1}{pn}\big)}$

I am required to find the limits of two "siblings" using the same idea, they are: $$\lim_{n\to\infty}\sin{\left(\frac{1}{pn+r}\right)}\sum_{k=1}^{n}\sin{\left(\frac{2pk-2p+r}{2pn}\right)}$$ with $0&...
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2answers
75 views

Prove that a function is Riemann Integrable

Let $ f: [0,1] \rightarrow \Bbb R$ defined by: $f(x)= \sin(\frac{1}{x}) $ if $ x \notin \Bbb Q$ , $ 0 $ if $x \in \Bbb Q$ I have to prove that this function is Riemann integrable in that interval. ...
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2answers
74 views

Evaluate using Riemann sums $\int_{\pi/2}^{3\pi/2}(4\sin 3x - 3 \cos 4x)dx$

I have to evaluate this integral using Riemann sums. I've already tried: $$\int_{\pi/2}^{3\pi/2}(4\sin 3x - 3\cos 4x)dx = [\delta x = \frac{\pi}{n}, x_i = \frac{\pi}{2} + \frac{i\pi}{n}] =$$$$= \...
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45 views

$\int_a^b f(x)g(x)\ dx=g(a)\int_a^{x_0}f(x)\ dx+g(b)\int_{x_0}^bf(x)dx$

Let $f,g:[a,b]\to\mathbb{R}$ be Riemann integrable functions such that $g$ is monotone. Show that there exists $x_0\in [a,b]$ such that $\int_a^b f(x)g(x)\ dx=g(a)\int_a^{x_0}f(x)\ dx+g(b)\int_{x_0}^...
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2answers
94 views

Integration and null function [closed]

Let $ f: [a, b] \rightarrow {\mathbb {R}} $ be continuous and such that $$ \int_ {a} ^ {b} {f (x) g (x) dx} = 0 $$ for all $ g: [a, b] \rightarrow {\mathbb {R}} $ continuous with $ g (a) = g (b) = 0 $....