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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Riemann Integral, Stepfunctions

I am working on a Problem about Riemann-Integrals. I am working on this Problem for about a week now. But i had no success. I really appreciate it if you would give me some Hints and Ideas to work ...
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$\lim\limits_{n\to\infty}\int_0^2 \frac{nx^{n-1}e^{-x^n}}{1+x}dx=1-\lim\limits_{n\to\infty}\int_0^2\frac{e^{-x^n}}{(1+x)^2}dx$

How can I show $\lim\limits_{n\to\infty}\int_0^2 \frac{nx^{n-1}e^{-x^n}}{1+x}dx=1-\lim\limits_{n\to\infty}\int_0^2\frac{e^{-x^n}}{(1+x)^2}dx$? It is $\frac{d}{dx}x^n=nx^{n-1}$ so I tried to solve it ...
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Trouble understanding a criteria for integrability

Let $f : [0, 1] → \mathbb{R}$ be bounded and $f ≥ 0$. Prove that if the set $\{x\in[0,1]|f(x)\geq \lambda \}$ is finite for every $λ > 0$ then $f$ is integrable. I'm having trouble understanding ...
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Difference between two similar definitions of integrability

Let $f ∈ TF([a, b])$ (set of all step functions) be a bounded,positive function and let $T_U$ be the set of step functions with $u ≤ f$, $\forall u ∈ T_U$ and $T_O$ be the set of step functions with $...
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Integrability based on continuity

Let $f:[a,b]\rightarrow \mathbb{R}$ and $f(x)= \begin{cases} a_n,& \text{if }x= \frac{1}{n}, n\in\mathbb{N} \\ 0 & \text{otherwise} \end{cases}$ where $a_n$ is bounded sequence ...
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A more comprehensive proof [closed]

Let $f:[a,b]\rightarrow \mathbb{R}$ be a riemann integrable function and let $g:[a,b]\rightarrow\mathbb{R}$ be another function such that $f(x)=g(x),\forall x\in [a,b]$. Prove that $g(x)$ is ...
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Integration according to the riemann concept [closed]

Is there a quick way (other than the traditional methods) enables me to know whether a certain function is capable integrable according to the Riemann concept or not?
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Does multiple integral have the second definition?

I have noticed that there are two definitions in Riemann integral(not multiple). Definition 1. For all $\epsilon > 0$, there exists $\delta > 0$ such that for any tagged partition $x_0$, ..., $...
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What happen to $\sum_{a \leq x_1 < .. < x_n \leq b} f(x_i)$ as $n \rightarrow \infty$?

Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function and $[a,b]$ an interval. Is there a way to estimate $$ \sum_{a \leq x_1 < .. < x_n \leq b} f(x_i) $$ as $n$ goes to infinity? If each sum ...
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If $f$ is Riemann integrable on $[a,b]$ then is $f \circ f$ Riemann integrable?

I know the composition of Riemann integrable functions is not necessarily Riemann integrable. But I am not finding any argument how to conclude this for self composition, or how to find a ...
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How to prove that $\int_{-\infty}^{+\infty}\frac{1+x}{1+x^2}dx$ doesn't exist

I'm trying to prove directly that $$\int_{-\infty}^{+\infty}\frac{1+x}{1+x^2}dx$$ doesn't exist: Integrating, \begin{align*}\int_u^v\frac{1+x}{1+x^2}dx&=\arctan x\Big|_u^v+\frac{1}{2}\log(1+x^2)\...
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How do I prove that the integral of f on [-a,a] is equal to two times the integral of f on [0,a]?

The question is this: Let f be an integrable function on [-a,a] (a>0). Assume f is an even function (i.e. f(-x) = f(x)) for all x ∈ [-a,a]. Then use the definition of an integral to prove $\int_{-a}...
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The limits of a definite integral using Mean Value theorem

Show that: $$ -\frac{1}{2} < \int_0^1 \frac {x^3 cos 5x}{2+x^2} dx <\frac{1}{2}. $$ I have started by proving that for any two integrable functions $f, g$ defined on $R[a,b]$ such that $f>g$ ...
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Composition of Riemann-integrable and increasing functions.

This is exercise 7.3.3 from Abbot's Understanding analysis. The section is Integrating functions with discontinuities. I am struggling with this exercise. I can't either come up with any simple ...
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Integration similar to consequences of Riemann lebesgue lemma

Let $f\in C^1[-\pi,\pi]$. Define , for $n\in N$ b_n=$\int_{-\pi}^\pi f(x)sinntdx$. Which of the following statement are true? a) $b_n\to 0$, as $n\to \infty$ b)$nb_n\to 0$, as $n\to \infty$ Please ...
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proving the integral convention $\int_{a}^{b}f(x)dx = \int_{a}^{c}f(x)dx + \int_{c}^{b}f(x)dx$ using the epsilon-delta definition

let $a,b,$ and $c$ be real numbers with $a<c<b$ and function $f$ is integrable in intervals $[a,c]$ and $[c,b]$. Show that the function f is integrable in $[a,b]$ and that $\int_{a}^{b}f(x)dx = \...
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Supremum with respect to an object

In my Real Analysis class I've often seen the Upper Riemann Sum (in the definition of Riemann integrability) defined as: $$\sup_P\{U(f,P)\}$$ This seems to mean the supremum of all possible upper sums ...
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Finding the volume of an $n$-dimensional simplex (recursion)

I want to find the volume of an $n$-dimensional simplex, i.e. determine \begin{align*} \sigma _{n} = \int_{A_{n}}^{} \mathrm{~d}\mu , \quad A_{n}= \{ ( x_{1}, \ldots, x_{n})\in \mathbb{R}^{n} \mid \...
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Find integral of $\sqrt{x}$ using Riemann sum definition

Let $a > 1$ be a real number. Evaluate the definite integral \begin{equation} \int_{1}^{a} \sqrt{x} \,dx \end{equation} from the Riemann sum definition. My approach I know a Riemann sum consists of ...
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$\int_{-2}^xf(t)dt$ for $f(t) = \tiny\begin{cases} -1 \, &t<0 \\ 1 \, &t\ge 0 \end{cases} $, and its limit at $x=0$

Let $f: [-2,2] \to \mathbb R$, $$ f(t) = \begin{cases} -1 \, &t<0 \\ 1 \, &t\ge 0 \end{cases} $$ Define $g: [-2,2] \to \mathbb R$ as: $$g(x) = \int_{-2}^xf(t)dt$$ Plot $g(x)$ and find it'...
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Prove that $\sup_x |\rho(x)-\rho_{\alpha}(x)| \leq |\rho'|_{L^{\infty}(\mathbb{R})} |1_{x>0}-h_{\alpha}|_{L^1}$

Let $\rho(x)$ be a function such that its derivative is bounded and its limit at $x \to -\infty$ exists. We can rewrite $\rho$ like so and define $\rho_{\alpha}$: $$ \rho(x)=\rho(-\infty)+\int_{-\...
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Lebesgue criterion for Riemann integrability.

I am trying to prove Lebesgue's criterion for Riemann integrability which states that: A bounded function $f:[a,b]\to \mathbb R$ is Riemann integrable iff it is continuous $\lambda$-a.e. and if it is ...
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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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larger function improper Riemann integrable implies smaller too

Problem: Let $g$ be a nonnegative and improper Riemann-integrable function on $[a,+\infty)$. Let $f$ be a function which is Riemann integrable on $[a,b]$ for every $b\geq a$ and $|f(x)|\leq g(x)$ ...
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A nice Romanian Olympiad integral (Proof Verification)

The question is, if $f:[1,\infty)\rightarrow [0,1)$ a non-increasing function with $\lim_{x\to \infty}f(x)=0$. Then Show that $\lim_{n\to \infty}[\int_1^2 f^n(x)+\int_2^3 f^{n-1}(x)+......+ \int_{n-1}...
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Help me to find the suitable partition to prove the function is Riemann integrable.

I want to prove the following function $$f(x)=\begin{cases}\frac{1}{q}& x=\frac{p}{q}\in\mathbb Q \ where \ p \ and\ q\ are\ coprime\\ 0&otherwise\end{cases}$$ is Riemann integrable on $[a,b]$ ...
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Example for $\int_a^b f(x)g(x)dx\neq g(a)\int_a^c f(x)dx + g(b) \int_c^b f(x)dx$ for $f\geq 0$ and g a non-monotonic function.

Well we all know about this theorem. Let $f\geq 0$ be a function that is integrable over [a,b] and $g:[a,b]\to \mathbb{R}$ be a monotonic function. Then there is a $c \in [a,b]$ with $$\int_a^b f(x)g(...
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Why is the upper sum of a refinement less than the upper sum of the original partition.

In the lecture notes for my real analysis module, it is proven that: If we have a partition $P$ and $P'$ is a refinement of $P$ then $U(f,P') \le U(f,P)$. But why is this the case? My lecturer used ...
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If $f$ and $g$ are integrable and $f(x)\leq h(x)\leq g(x)$, is $h$ integrable?

Let $f(x)$ and $g(x)$ be Riemann integrable on $[a, b]$ and $f(x) \leq h(x) \leq g(x) \forall x \in [a,b]$. Then prove that $h(x)$ may or may not be Riemann integrable on $[a, b]$.
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Checking for the integrability of $\int_U \ln{\sqrt{x^2+y^2}}$ using exhaustions

The open set to check for integrability is $U = \{(x,y) \in \mathbb{R}^2 | 0<x^2+y^2<4\}$. I've considered a succesion of sets $U_n = \{(x,y) \in \mathbb{R}^2 | \frac{1}{n^2}<x^2+y^2<4\}$, ...
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Continuous Factorial

I am working on a theory of quantum information and am unsure on some of the mathematical formalism I need. I have learned that integration can be thought of as summing up infinitely thin slices. My ...
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Prove $f$ is Riemann integrable by the definition. [duplicate]

I want to prove the following function $$f(x)=\begin{cases}1& x=\frac{1}{n} \ where\ n\in\mathbb N\\ 0&otherwise\end{cases}$$ is Riemann integrable on $[0,1]$ by the integrability definition: ...
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Existence of $c$ such that $\int_{a}^{b} f(x)g(x)dx = f(c)\int_{a}^{b} g(x)$ with g taking on negative values as well.

I am doing a task right now as an excercice for myself. Let $f,g:[a,b] \to \mathbb{R}$ continuous, $g(x)>0 \forall x \in [a,b]$. Then there is $c \in (a,b)$ such that $\int_{a}^{b} f(x)g(x)dx = f(...
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What is the correct integral definition here?

I know $\int_{-\infty}^{\infty}\frac{\sin x}{x}dx=\pi$, but what is the definition of the integral here? $\int_{-\infty}^{\infty}\left|\frac{\sin x}{x}\right|dx$ is not finite, therefore $\frac{\sin x}...
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Let $f:[a,b]\to\mathbb{R}$ be integrable and $\alpha \in \mathbb{R}$. Then $\int_a^b \alpha f=\alpha \int_a^b f$

I need help with the following task (dealing with the Riemann-Integration), I am doing as an exercise for myself right now. Let $f:[a,b]\to\mathbb{R}$ be integrable and $\alpha \in \mathbb{R}$. Then $\...
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Help evaluating a limit.

Using the substitution $u=\frac{1}{x}$, show that $\int_{\frac{1}{2}}^{2} \frac{\ln{x}}{1+x^2} \,dx = 0$. Hence or otherwise, evaluate the limit $\lim_{n\to\infty} \sum_{k=1}^{3n} \frac{1}{2n} \times \...
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Help with the integrability of a function throughout $\mathbb{R}^n$

Given a continuous function $f\colon \mathbb{R}^n \to \mathbb{R}$ such that, for some M, R, p constant values: $$\vert f(x) \vert \leq \frac{M}{\vert\vert x \vert\vert^p}$$ If $\vert\vert x \vert\vert\...
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$f,g:[a,b]\to\mathbb{R}$ integrable. Then f+g is integrable with $\int_a^b (f+g)=\int_a^b f + \int_a^b g$

First of all I know there are many posts where someone asked for the proof of this sentence. But I just need help to understand one step of the first part of the proof, I hope there is someone who ...
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Show that $f$ is Riemann integrable and that $\int_a^b f (x)dx = 0$

Let $k \in \mathbb{R}$, $x_0 \in [a, b]$, and define $f:[a, b] \to \mathbb{R}$ via $$ \begin{cases} 0 & x \ne x_0\\ k & x= x_0 \end{cases} $$ Show that $f$ is Riemann integrable and ...
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$\int_0^1|f(x)|dx=0$ if and only if $f(x)=0$ for all $x\in[0,1]$ [duplicate]

If $f:[0,1]\rightarrow\Bbb R$ is a continuous function and not negative function such that $$ \int_0^1f(x)dx=0 $$ then could exist $x_0\in [0,1]$ such that $f(x_0)$ is not zero? I know this result but ...
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If $f(x)$ is analytic and non-negative, does convergence of $\int_a^{\infty} f(x)\,dx$ imply $\lim_{x\to\infty} f(x)=0$?

It is well known that improper integrals don't have to satisfy $\lim_{x\to\infty} f(x)=0$ in order for $\int_a^{\infty} f(x)\,dx$ to converge, for instance $f(x)=\sin(x^2)$. It is also possible to ...
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Is there any set conditions on $f$ that imply the following integral property?

I have a function, $f(y)$, which is continuous and bounded. I have the integral $$\int^{r}_{-r} f(y)\cdot sgn(y) dy$$ Which has been rewritten in my notes as $$\int^{r}_{0} f(y)\cdot(1) dy + \int^{0}_{...
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Express limit of sum in terms of definite integral

Evaluate the limit by expressing it as a definite integral: \begin{equation} \lim_{n \to \infty} \sum_{k=n+1}^{2n-1} \frac{n}{n^2+k^2} \end{equation} I'm really confused about tackling this. Although ...
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Show that, given the space of Riemann integrable functions, $f \mapsto \sqrt{\int_K f^2}$ satisfies the triangle inequality

The gist of this is that, given $f, g \in J$, where J contains all Riemann integrable functions on a compact rectangle, then $\sqrt{\int_K (f+g)^2}\leq \sqrt{\int_Kf^2} + \sqrt{\int_K g^2}$. Now, I ...
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Area under a parabolic arc using arithmetic substitution

I'm currently studying Calculus from the book "Introduction to Calculus and Analysis" by Richard Courant and Fritz John. I started reading the chapter on Integration, and I stumbled upon the ...
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Definition 6.23 and Theorem 6.17 in Baby Rudin

The definition 6.23 is as follow: Let $f_1,...,f_k$ be real functions on [$a,b$], and let $f$=($f_1,...,f_k$) be the corresponding mapping of [$a,b$] into $R^k$.If $\alpha$ increases monotonically on ...
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If we know that $|f|$ is integrable, and that $f$ is bounded, do we know that $f$ is integrable? [closed]

I am trying to prove or disprove this and this is what I have so far. I am referring to Riemann integrability when I refer to integrability. I am unsure of how to prove this proposed theorem or how to ...
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Bounds of a Riemann Integral

Say we have a function $f:[a,b] \to \mathbb{R}$ and we assume that this function is Riemann integrable. This implies that $f$ is bounded, say by $M$. Then $|f| \le M$. Now consider $F:[a,b] \to \...
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Meaning of Lebesgue integral compared to Riemann integral l

Recently I learned about Lebesgue integral as an alternative to Riemann integral when we want to integrate some classes of functions. In the Riemann integral, in one dimesion, we can interpret it like ...
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$\forall x > 0$, there exists a unique $\xi_x$ s.t.$ \int_{0}^{x}e^{t^2}dt=xe^{\xi_x^2}$; solve $\lim_{x \to \infty} \frac{\xi_x}{x} $

$$proof: \forall x > 0, \text{there exists a unique} \quad \xi_x \quad s.t. \int_{0}^{x}e^{t^2}dt=xe^{\xi_x^2} ; \text{solve} \lim_{x \to \infty} \frac{\xi_x}{x} $$ my thought is to use the First ...
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