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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Replacement of the tagged partition displacement of the Riemann Integral

In specific, my question is if I can make the following by the definition on Riemann integral... $$\int_a^bf(x)dx = \lim_{\lambda \rightarrow 0}\sum_{i=1}^n f(c_i)\Delta x_i = \lim_{\lambda \...
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70 views

Show that $f=\sin(x)$ if $x\in\mathbb{Q}$ and $f(x)=\cos(x)$ if $x\in\mathbb{R}\setminus\mathbb{Q}$ is not Riemann integrable on $[0,1]$

Show that $f=\sin(x)$ if $x\in\mathbb{Q}$ and $f(x)=\cos(x)$ if $x\in\mathbb{R}\setminus\mathbb{Q}$ is not integrable on $[0,1]$. I know that there is already a post on this function, but I didn't ...
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32 views

Let $f(x)=2|x|+1$ if $x\in\mathbf{Q}$ and $f(x)=0$ if not. Show that $f$ is not Riemann integrable

Let $f(x)=2|x|+1$ if $x\in\mathbf{Q}$ and $f(x)=0$ if $x\in \mathbf{R}/\mathbf{Q}$. Show that $f$ is not Riemann integrable on $[-2,3]$. I would like to have a feedback on my proof and to know if it ...
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61 views

Riemann integrable function which is discontinuous at an uncountable number of points on every open set?

Let $n$ be some positive integer. Does there exist a Riemann integrable function $$f: [0,1]^n \to \mathbb{R} $$ with the following property: for every non-empty open subset of $U \subset (0,1)^n$, the ...
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21 views

Is $g$ integrable, given $f$ integrable and $g=f$ except for a set of Lebesgue measure 0 (not necessarily finite)?

If $f$ is Riemann integrable on $[a, b]$, and the value of $g$ agrees with $f$ at almost every point (except on a set of Lebesgue measure 0), is it true that $g$ Riemann integrable and $\displaystyle \...
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47 views

Show that $f(x)=x$ if $x\in\mathbf{Q}$ and $f(x)=1-x$ if $x \in\mathbf{R}/\mathbf{Q}$ is Riemann integrable on $[0,1]$

Let $f(x)=x$ if $x\in\mathbf{Q}$ and $f(x)=1-x$ if $x \in\mathbf{R}/\mathbf{Q}$. Show that $f$ is Riemann integrable on $[0,1]$. I know that there are a few posts on this function, but I didn't see ...
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79 views

show that $f$ is Riemann integrable on $[0,1]$

Let $f(x)=\sin\left(\frac{1}{x}\right)$ if $0<x\le1$ and $f(x)=0$ if $x=0$. Show that $f$ is Riemann integrable on $[0,1]$ and calculate it's integral on $[0,1]$. I would like to know if my proof ...
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Exercise 5.S. the elements of integration and lebesgue measure

I'm having problems with this exercise. I've tried to apply the DOMINATED CONVERGENCE THEOREM but I couldn't. Could someone gives me any hint? Suppose the function $x\rightarrow f(x,t)$ is $X$-...
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48 views

Let $f(x)=1$ if $x=0$ and $f(x)=0$ if $x>0$. Show that $f$ is integrable.

Let $f(x)=1$ if $x=0$ and $f(x)=0$ if $x>0$. Show that $f$ is Riemann integrable on $[0,1]$. I think for everyone this question is really basic, but I'm just training myself on proving the ...
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88 views

Show that $f$ is Riemann Integrable on $\left[0,\pi/2\right]$

Let $f(x)=\cos^2(x)$ if $x\in \mathbb{Q}$ and $f(x)=0$ if not. Show that $f$ is Riemann Integrable on $\left[0,\pi/2\right]$. The problem that I have, is that I don't really see why this function ...
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109 views

Show that $f$ is Riemann-Integrable

Let $f$ be a function defined on $[0,1]$ by: $f(x)=1$ if $x=\frac{1}{n}, n \in \mathbf{N}^*$ and $f(x)=0$ if not. Show that $f$ is Riemann integrable on $[0,1]$. I know that there is already a post on ...
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Riemann integrable implies Lebesgue Integrale on a Plane

Let $f$ be a bounded real-valued function on $[a,b]$. If $f$ is Riemann integrable, then $f$ is Lebesgue measurable and $\int_{a}^b f(x) dx = \int_{[a,b]}f dm$. The proof of this involves showing that ...
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Riemann subtotal of $e$ on partition $Z = (0, \frac{1}{2^n}, \frac{2}{2^n},.., 1)$

Given the intermediate points $\alpha = (\frac{1}{2^n}, \frac{2}{2^n},...,1)$ and $y=\frac{1}{2^n}$. Prove that, the subtotal of $e$ on $(Z, \alpha)$ equals: $$e^{y}\frac{e - 1}{\frac{e^y - 1}{y}} = e^...
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49 views

Writing the Real Part of Complex Integrand

I'm having a hard time to understand how's Eq. $(6.73)$ become Eq. $(6.75)$. It's taken from Numerical Methods for Laplace Transform Inversion by Cohen. Here's the problem: [...]. The basis of their ...
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20 views

Clarity on a property of infimum and supremum

I came across this property while studying material on Reimann Integral, i.e when is a function integrable. (When $inf{U} = sup{L}$, where U and L are the upper and lower bound of sums). For a set S, ...
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Difference between upper Riemann sum and lower Riemann sum [closed]

Let $f(x)$ be differentiable on $[0,1]$ with $|f'(x)|$ less than or equal to $1$.let $P$ be a partition of $[0,1]$ then $U$ $(F,P)-L(f,P)\leq||P||$ Is this true or false?
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Monotonic and non monotonic function's integrability(Riemann) question

I was told that all functions that have a finite number of discontinuities, are Riemann-integrable. Then, I found a proposition that tells the following: All monotonic functions on closed interval $[a,...
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36 views

Convergence or divergence of $\int_0^2 \frac{\sin^2x}{x^p}\,dx$

I am starting to learn a little about analysis and convergence of Riemann integrals. I came across this problem and it stomped me. I want to find the values of $p$ for which the integral converges. $$\...
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21 views

Improper Riemann sums

Let $f,g:\mathbb{R}\to\left[0,\infty\right)$ be two functions such that the improper Riemann integrals $$ \int_\mathbb{R} f \text{ and } \int_\mathbb{R} g $$ both exist. The definition of the improper ...
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25 views

What is the dual space of the set of all riemann integrable functions

I was going through the dual of the basis of a vector space - which is essentially the set of linear functionals such that if $\{\alpha_1,\alpha_2,...,\alpha_n\}$ is the basis vectors then $\{f_1,..,...
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34 views

Convergence of the logarithmic integral $\int_{0.5}^1\frac 1{\ln(x)} \,dx$

I computed the integral using Wolfram Alpha, and it gave me values in terms of the logarithmic integral, but I am unsure on how to prove convergence or divergence using analysis. So I want to prove ...
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1answer
49 views

Riemann Sum with Definite integral [closed]

I have a sum called $$S_n = \bigg(1+\frac{1}{n}\bigg)\sin\bigg(\frac{\pi}{n}\bigg) +\bigg(1+\frac{2}{n}\bigg)\sin\bigg(\frac{2\pi}{n}\bigg) +....+\bigg(1+\frac{n-1}{n}\bigg)\sin\bigg(\frac{(n-1)\pi}{n}...
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21 views

Show that the limit can be moved inside the integral for this function

For $x \in [-1,1]$ and $n \in \mathbb{N}$, consider $f_n(x) = \frac{x^{2n}}{(1 + x^{2n})}$ I have found that the function converges pointwise to $f(x) = \begin{cases} \frac{1}{2} & \mathrm{if} x = ...
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20 views

Tag condition for Riemann integrability

As part of the definition for a definite integral we say that the choice of tag in the partition should not affect the limit of the Riemann sum for that function to be considered Riemann integrable. ...
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65 views

Let $g$ be a continuous function on $[0,1]$ with certain conditions. Show that $g(x) = x$

Let $g$ be a continuous function on $[0,1]$ s.t. $$\int_{0}^{1} g^2(x) dx = \frac{1}{3}$$ $$\int_{0}^{1} xg(x)dx = \frac{1}{3}$$ Show that $g(x) = x \forall x \in [0,1]$. My work so far. I wrote it as ...
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86 views

Completing a proof on integrability: $f$ is Riemann integrable under the mesh definition iff $\sup L_\Gamma = \inf U_\Gamma$

$ \newcommand{\II}{\mathcal{I}} \newcommand{\R}{\mathbb{R}} \newcommand{\ve}{\varepsilon} \newcommand{\d}{\delta} \newcommand{\g}{\Gamma} \newcommand{\norm}[1]{\left \Vert #1 \right \Vert} \...
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11 views

calculating a riemann integral of Thomaes function

problem: calculate $\int_0^1 f(x) dx $ with the given conditions: $ f(0)=38 , f(r)=0 $ for every r in $[0,1]/\Bbb Q $ and $f(p/q) = 1/q$ for every $p,q\,in\, \Bbb N/0$ with $ p\le q \,and\, lcm(p,q)=1$...
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1answer
50 views

Do I need to prove $\lim_{\|P\|\to 0} L(f,P)=\sup_P L(f,P)$, and $\lim_{\|P\|\rightarrow 0} U(f,P)=\inf_P U(f,P)$? Or is it trivial?

I'm trying to show that $f$ is bounded and Riemann integrable on $[a, b]$ implies that there exists a sequence $\{P_1, P_2, P_3, \ldots\}$ of partitions of $[a, b]$ s.t. $$ \lim_{n\to \infty} [U_f(P_n)...
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181 views

Riemann-Stieltjes integral of a continuous function w.r.t. a step function

$ \newcommand{\para}[1]{\left( #1 \right)} \newcommand{\abs}[1]{\left| \, #1 \, \right|} \newcommand{\set}[1]{\left\{ #1 \right\}} \newcommand{\ds}{\displaystyle} \newcommand{\ol}[1]{\overline{#1}} \...
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1answer
49 views

Integration of a radial function over a bounded domain

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$. Let $f(x)=|x|^\alpha$. Then $f\in L^1(\Omega)$ if $\alpha>-N$. The above fact seems to hold for the following reason. Since $\Omega$ is bounded, ...
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1answer
76 views

Convergence of the Laplace Transform

The bilateral Laplace transform is defined as $$X(s) = \int_{-\infty}^{\infty}x(t)e^{-st}dt$$where $s = \sigma +j\omega$. I'm trying to prove rigorously The ROC(region of convergence) of $X(s)$ ...
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30 views

Riemann Integrable Function is Zero

I am working through a Real Analysis textbook and found the following exercise. Let $f\,:\,[a,b]\rightarrow\mathbb R$ be Riemann integrable and nonnegative. Suppose $$\int_a^bf(x)\,dx=0.$$ Find $f$. ...
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1answer
29 views

How to prove that if $S(S,f,\{p_i\})=\sum f(p_j)A(D_j)$ then $|\iint_{R}f-S|(S,f,\{p_i\})|<\varepsilon$

Let $$s(S,f,\{p_i\})=\sum_{j=1}^m f(p_j)A(D_j)$$ where $p_j\in D_j$ and $S$ is a general division of a close rectangle $R$, i.e., a finite number of domains $D_j$ each having area, and together covers ...
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117 views

Integral of $\sqrt{4-x^2}$ by Riemann sum

I was trying to compute $\int_{0}^{2}\sqrt{4-x^2}\,dx$ by Riemann sum like this $$\sum_{i=1}^{n}\sqrt{4-x_i^2}\,\Delta x_i$$ where $x_i = \frac{2i}{n}$ and $\Delta x_i = \frac{2}{n}$, but I couldn't ...
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48 views

Prove that $ f(x) = |x| $ is Riemann integrable on the interval [-1, 2] using lower and upper integrals

Prove that $ f(x) = |x| $ is Riemann integrable on the [-1, 2] using lower and upper integrals I am confused how to partition ($P_N$) in this interval. Should we consider two different intervals [-1, ...
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1answer
41 views

Reverse Characteristic Function of the Cantor Set

Let $f: [0,1] \to \mathbb{R}$ $f(x) = 0$ if $x\in C$ $f(x) = 1$ otherwise Show that $f$ is Riemann Integrable and compute $$\int_{0}^{1} f(x) dx$$ I called this the "reverse" characteristic ...
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4answers
118 views

How do I show that if $f$ is integrable on $[c,b]$ for all $c \in [a,b]$, then $f$ is integrable on $[a,b]$?

Let $a < b$. Suppose that the function $f: [a,b] \to \mathbb{R}$ is bounded and Riemann integrable on $[c,b]$ for every $a < c < b$. Prove that $f$ is Riemann integrable on $[a,b]$ and that $$...
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2answers
31 views

Consider the sequence of functions and compute the limit of the integral

Consider the sequence of functions $f_n:[0,1] \to R$ given by $$f_n(x) = \frac{e^{-(n + x^2)}}{n^2 + x}$$ for $x \in [0,1]$ and $n \geq 1$ Compute $$\lim_{n \to \infty} \int_{0}^{1} f_n(x) dx$$ Maybe ...
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81 views

Prove that the integral of a differential form on an open set is well defined.

What to follow is a summary of part of the sixth chapter of the text Analysis on Manifolds by James Munkres. Definition Given $x\in\Bbb R^n$ we define a tangent vector to $\Bbb R^n$ at $x$ to be a ...
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40 views

How can we show that the equality doesn't occur? [duplicate]

Let $f:[0,1] \to [1,2]$, continuous on $[0,1]$ with $1<f(1)<2$. Show that $$\int_0^1\left (f^2(x)-2f(x)\right )\, dx>-1.$$ We have that $$(f(x)-1)^2\geq 0\implies \int_0^1 (f(x)-1)^2\, dx\geq ...
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1answer
64 views

Show that there exists a partition and sample points such that the Riemann sum equals any intermediate value

Let $f:[0, 1]\rightarrow \mathbb{R}$ be a bounded function. Show that for any c such that $f(a) < c < f(b)$ where $a,b \in [0, 1]$ there exists a partition $P$ and sample points such that the ...
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1answer
52 views

Prove that the map $f:[0,1]\to l_{\infty}([0,1];\Bbb{R})$ defined by $t\mapsto 1_{[t,1]}$ is Riemann integrable.

A map $f:[0,1]\to F$ (where $F$ is a banach space) is said to be Riemann integrable if $\exists I\in F$ such that for every $\epsilon>0$ there is $\delta>0$ satisfying $$\left\lVert I-\sum\...
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1answer
74 views

Prove that $R_n < M_n < L_n$ ,for right, mid-point and left Riemann sum

Let $a, b \in \mathbb{R}, a<b$. Suppose that $f$ is continuous and strictly decreasing over $[a, b]$. Let $L_n$ be the left Riemann sum, $R_n$ the right Riemann sum, and $M_n$ the midpoint (Riemann)...
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23 views

riemann integrable function on closed interval is lebesgue measurable

I have recently learned that if a function $f:\left[a,b\right]\to\mathbb{R}$ is reimann integrable then it is measurable as a function from $\left(\mathbb{R},\mathcal{L}\right)$ to $\left(\mathbb{R},...
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1answer
42 views

Query about a proof of Riemann integrability of step/simple functions

Let $f:[a,b]\to\Bbb{R}$ be a step function is of the form $$f=a_11_{[a,t_1]}+\sum\limits_{i=2}^n a_{i} 1_{(t_{i-1},t_{i}]}$$ i.e. in simple words $f(x)=a_1$ for all $x\in [a,t_1]$ and $f(x)=a_{i}$ for ...
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1answer
44 views

Show that $L(f)=\sup\{L(P,f):P \in \mathcal P_{c}\}$,where $c\in \left(a,b\right)$

Assume $\mathcal P_{[a,b]}$ is the set of all partitions of $[a,b]$ and $\mathcal P_{c}$ is the set of all partitions of $[a,b]$ containing $c$,where $c\in \left(a,b\right)$,if $f:I \to \mathbb R$ ...
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1answer
46 views

Show that $m\le\left(\frac{1}{b-a}\int_{a}^{b}f^{2}\left(x\right)dx\right)^{\frac{1}{2}}\le M$

Assume $f$ is a real-valued function which is integrable over the interval $I=[a,b]$ and for every $x \in [a,b]$ we have that $0\le m \le f(x) \le M$,show the following inequality does hold: $$m\le\...
4
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2answers
124 views

Is it inuitively correct to say that $\lim_{n \to \infty} \frac{x}{n} = \mathrm{d}x$?

We know that from the Riemann sum that $\Delta x = \mathrm{d}x$ where $n \to \infty$ and $\Delta x = \dfrac{b - a}{n}$. If, however, $x$ represents some length of interval, can we also say that $\...
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2answers
49 views

Can one say that $\int_{r_{a}}^{r_{b}} \mathrm{d}\vec{r}$ is the infinite sum of infinitesimal magnitudes, $\mathrm{d}r$, with direction, $\hat{r}$?

Say, for example, that you had a situation where you needed to some up an infinite number of infinitesimally small portions of some vector, $\vec{r}$. This vector, $\vec{r}$, can be then defined as \...
3
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1answer
60 views

Prove that the lower Riemann integral of $f$ on $[a, b]$

Prove that the lower Riemann integral of $f$ on $[a, b]$ is $\sup\{\int_a^b\psi: \psi \text{ is a step function on } [a, b],\text{ such that } \psi\leq f \text{ on }[a, b]\}$. The definitions that I ...

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