# 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 ...
1 vote
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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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### 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 ...
1 vote
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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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### 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 $$\int_{1}^{a} \sqrt{x} \,dx$$ 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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### 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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