All Questions
Tagged with integration riemann-integration
1,078
questions
-2
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If $f$ and $g$ coincide almost everywhere on $[a, b]$, then is $\int_a^b f(x) dx = \int_a^b g(x) dx$? [duplicate]
Let $a$ and $b$ be any real numbers such that $a < b$, and let $S$ be a (nonempty) subset of the closed bounded interval $[a, b]$ such that $S$ has measure $0$. Now let $f \colon [a, b] \...
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Does U(f,P,[a,b]) = L(f,P,[a,b]) really imply f is constant
I am currently reading Measure, Integration & Real Analysis by Sheldon Axler, and am working through the practice problems. In particular, I am on this problem right now:
Suppose $f:[a,b]\to\...
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If every sequence of Riemann sums of a function converges, is the function integrable?
Let $f:[a,b]\to\mathbb{R}$ be a function, and $P_n$ the equidistance partition of $[a,b]$ into $n$ subintervals of an equal length. Let $P_n^\ast$ be the set of sample points from each subinterval of $...
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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 ...
- 89
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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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2
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78
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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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Book recommendation for an author that would treat Riemann integrability when $f$ is not necessarily bounded
The two books I have learnt Riemann integrability with (Mathematical Analysis by T. Apostol and PMA by W. Rudin) carefully state at the beginning of their integrability chapter that we work with ...
- 1,231
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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 ...
- 6,620
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Continuous function with a countably infinite number of poles are not improperly Riemann integrable.
How to show that $$f=\frac{1}{\prod_{n=1}^\infty(x-\frac{1}{n})}$$
is not (improperly) Riemann integrable?
Motivation. It is bookwork to show that bounded functions with a finite number of ...
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Is it true or false that $\mathcal R([0,1])=\mathcal R_\alpha([0,1])$, where $\alpha$ is continuous?
Let $\alpha:[0,1]\to\mathbb R$ be a monotonically increasing continuous function. Let $\mathcal R_\alpha([0,1])$ be a set of all functions that are Riemann–Stieltjes integrable on $[0,1]$ with respect ...
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Is this a Riemann sum?
I have come a cross with a sum that looks like this:
$$\sum_{x\in{\Lambda_N}}\epsilon^2 k(\epsilon x)e^{-i\pi\omega \cdot \epsilon^2 x}\quad \quad\quad\quad(*)$$
Here $x$ takes values in the discrete ...
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Hint on examine the integrability?
How to examine the integrability of $f(x,y)=\left\{\!\!\!\begin{array}{c c}{{~x+y,}}&{{x,~y~\mathrm{are~rational},}}\\ {{~x-y,}}&{{\mathrm{other~cases.}}}\end{array}\!\!\right.$ on $[-1, 1]\...
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General form of functions with $\hat{f}(n) = 0$ for all $|n| \geq 2$
How to determine the general form of functions with $\hat{f}(n) = 0$ for all $|n| \geq 2$?
Here $\hat{f}(n)$ denotes the $n$-th Fourier coefficient of Riemann integrable function $f$. It is also ...
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