# 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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### When does the Fundamental Theorem of Calculus Part 2 (Evaluation Theorem) Hold Across Unbounded Regions of Functions

The second part of the fundamental theorem of calculus is generally stated: If $F^\prime(x) = f(x)$ on $(a,b)$ and $f$ is Reimann integrable on $(a,b)$ then $\int_a^b f(x) dx = F(b)-F(a)$. Inherently, ...
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### Can you give an example of a measurable set which is not Borel and whose closure is a null-set?

If $N$ is a measurable set which is not Borel and whose closure is a null-set, then $1_N$ is Riemann-integrable (as it is constant $=0$ and continuous on $\overline{N}^c$ and hence almost everywhere) ...
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### Can there exist a function $F$ which is differentiable on $[a,b]$ but $F'$ is non-riemann integrable function on $[a,b]$

Question Can there exist a function $F$ which is differentiable on $[a,b]$ but $F'$ is non-riemann integrable function on $[a,b]$. There is a similar question here, but the construction done is not ...
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### Multiple integral with laplace operator

My question comes from the academic paper called Stable solutions of $-\Delta u= f(u)$ in $\mathbb{R}^N$ by L. Dupaigne and A.Farina. I don't understand how to go from the first expression with one ...
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### Help with a Calculus 2 question about integrals and using the Fundamental theorem of Calculus

I've been given this homework assignment: Let $a\geq 0$ and let $g(x)$ be a continuous function on $[a, \infty)$ such that for all $x\geq a$: $$\left|\int_{a}^{x}g(t)dt \right|\leq C$$ For some ...
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### On the absolute continuity of a particular functions.

We consider $f\in L^1[a,b]$, where $F$ is a finite interval of $\mathbb{R}$. The function $F\colon [a,b]\to\mathbb{R}$ defined as $$F(x):=\int_{[a,x]}f\;d\lambda\quad(x\in [a,b])$$ it is called ...
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### Show that $\int_0^1\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}}$ is convergent.

Show that $\int_0^1\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}} \,dx$ is convergent. The points $0$ and $1$ are the only points of infinite discontinuities of $\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}}.$ The integral ...
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### Riemann integrability for step function

Here is the problem: Fix $c\in\mathbb{R}$ and define $g:[0,2]\to\mathbb{R}$ by $$g(x)=\begin{cases}2 &\text{if } 0\le x<1\\c &\text{if } x=1\\ 1&\text{if } 1< x\le 2.\\\end{cases}$$ ...
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### $f(x) \ge 0$ for all $x \in [a, b]$ and $\int_a^b f = 0$. Prove that $f(x) = 0$ for all $x \in [a, b]$.

Suppose that $f$ is continuous on $[a, b]$, that $f(x) \ge 0$ for all $x \in [a, b]$ and that $\int_a^b f = 0$. Prove that $f(x) = 0$ for all $x \in [a, b]$. My attempt: Let $\dot{\Pi}$ be a tagged ...
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### How can I calculate $\int_{R^2}\; \exp (-x^2-xy-y^2)\,dxdy$ this Riemann Integral. [closed]

How can I calculate $\int_{R^2}\; \exp (-x^2-xy-y^2)\,dxdy$ this Riemann Integral. Knowing that $\int_{R^2}\; \exp (-x^2-y^2)\,dxdy = π$ .
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### Is this function riemann integrable in $[0,1]\times[0,1]$?

Let $f[0,1]\times[0,1]\to [0,1]$ given by $$f(x,y):=\begin{cases} 0& x\notin Q\\ 0& x=0\ x\in Q,y\in R-Q\\ \dfrac{1}{q}& x\in Q, y=\dfrac{p}{q}, \gcd(p,q)=1 \end{cases}$$ Is $f$ Riemann ...
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### How to evaluate the general limit of $\lim\limits_{n \to \infty }n\left(\sum\limits_{k=1}^n\frac{1}{n}f\left(\frac k n\right)-\int_0^1f(x)dx\right)$? [duplicate]

I encountered this problem If $f$ is a is continuously differentiable on $[0,1]$, find $$\lim_{n \to \infty }n\left(\sum_{k=1}^n \frac{1}{n}f\left(\frac k n\right)-\int_0^1f(x)dx\right)$$ My attempt ...
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### Riemann Sum with Supremum [closed]

Suppose we have $f(t, \mathbf{s}): \mathbb{R} \times \mathbb{R}^{g}$ where $\mathbf{s} \in [0, 1]^{g}$ with $g \in \mathbb{Z}_{+} \cup \infty$, $t \in \mathbf{t}$ where $\mathbf{t}$ is a set of tagged ...
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### Examples function integrable except dense set

i've done lebesgue vitali theorem and i'm looking for some examples of Riemann integrable function that is discontinous in a dense set. I'm thinking about caratheristic function but i cannot prove ...
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### Give an example of a non-Riemann integrable function $g$ which is equal to a Riemann integrable function $f$ almost everywhere in $[a, b]$ [closed]
So I am having problems facing this exercise to find an example of such function. Let $f\in R([a, b])$. Can we find a function $g\colon [a,b]\to \mathbb R$ such that $f=g$ almost everywhere and $g$ ...