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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prove that $\int_{a}^{c} f(x) \,dx = \int_{a}^{b} f(x) \,dx + \int_{b}^{c} f(x) \,dx $
I've been working on a proof related to the additivity of Riemann integrals and would greatly appreciate insights and feedback for clarity and correctness of the proof. Because i've never seen a text, ...
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Show that $f(x+1)$ is Riemann integrable [closed]
Let $f$ be a Riemann integrable function in $[a,b]$, show that $f(x+1)$ is Riemann integrable in $[a+1,b+1]$
I tried to use the Riemann criterion but I am not sure if this is the right way, because I ...
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For an arbitrary continuous curve $\gamma:[0,1]\to\mathbb R^n$, does the Riemann-Stieltjes integral $\int_0^1\gamma(t)\wedge d\gamma(t)$ exist?
We consider sums of the form
$$\sum_{i=1}^m\gamma(t_i^*)\wedge\Big(\gamma(t_i)-\gamma(t_{i-1})\Big),$$
where $\gamma:[0,1]\to\mathbb R^n$ is a continuous function, and
$$0=t_0\leq t_1^*\leq t_1\leq ...
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Suppose that for each $a < b$ s.t $a, b \in [0, 1]$ there exists $t_1, t_2 \in [a, b]$ which satisfies $g(t_2) \leq f(t_1)$
Let $f, g$ be Riemann-integrable functions at $[0,1]$.
Suppose that for each $a < b$ s.t $a, b \in [0, 1]$ there exists $t_1, t_2 \in [a, b]$ which satisfies $g(t_2) \leq f(t_1)$.
Prove that $\...
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Combination of function values on subinterval is bounded by $\inf$/$\sup$ of function.
Let $f:[a,b] \to \mathbb{R}$ be a real valued function. And $\mathcal{P} = (x_0, \ldots, x_n)$ be a partition of $[a,b]$. Why is
$$\inf_{x\in[x_{k-1},x_{k}]}f(x) \leq f(x_{k-1})+\dfrac{f(x_k) - f(x_{k-...
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Integral of Cantor set indicator function [closed]
Say I have a function h(x) which is 1 whenever x is in the Cantor set and 0 for all other points in [0,1]. How can I prove this function is Riemann-integrable, and how can I show its integral is 0?
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Why did the author write "Why?"? Any deep reason? ("Calculus Fourth Edition" by Michael Spivak)
THEOREM 6 If $f$ is integrable on $[a,b]$, then for any number $c$, the function $cf$ is integrable on $[a,b]$ and $$\int_a^b cf=c\cdot\int_a^b f.$$
PROOF The proof (which is much easier than that of ...
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Use the definition to find $\int_0^2f(x)dx$ where $f (x )=\cases{ 1,&$x \ne 1$\\2,& $x = 1$}$
Use the definition to find $\int_0^2f(x)dx$ where $f (x )=\cases{ 1,&$x \ne 1$\\2,& $x = 1$}$
I am stuck with this question, we studying Riemann integration, in my mind I can do the integral ...
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If all projections on the axis are integrable, do the iterated integrals exist?
Suppose $f(x,y) : [0,1] \times [0,1] \to \mathbb{R}_{\ge 0}$ is a function, not necessarily continuous, nor necessarily Riemann integrable on $[0,1] \times[0,1]$. By existence of an integral here we ...
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Doubt in some steps in the Nirenberg-Gagliardo-Sobolev inequality.
I understood almost everything. But I got stuck in two steps that seems to be very simple.
The first doubt is in the first inequality, in the beginning of the proof. In the first equality, for ...
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Proof for integral absolute value inequality
Thinking about this
$$\left|\int_a^bf(x)\,\mathrm dx\right|\le\int_a^b|f(x)|\,\mathrm dx.$$
Why is this proof wrong? Be $f$ a continuous and integrable function, with primitive $F$, hence
$$\left|\...
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what is the difference between a Riemann integral and Cauchy limit theorem?
If $f$ is a Riemannian integrable function on $[0,1]$, then $\forall \epsilon >0 \ \ \exists \delta>0 $ s.t. if the norm of the partitions $||P||< \delta$, then $\left|\sum\limits_{k=1} ^n ...
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How to solve this problem involving the definite integral of a convex function?
Let $i, j \in \mathbf{Z}, i<j$. Let $g:[i, j+1] \rightarrow \mathbf{R}$ be a decreasing convex function that satisfies
$$
\begin{gathered}
|g(z)-g(w)| \leq \frac{1}{2}|z-w|, \quad \forall z, w
\end{...
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Is this function equal almost everywhere to a Riemann integrable function?
Let $(a_n)_{n=1}^{\infty}$ be an enumeration of $\mathbb{Q} \cap [0,1]$. Define $B_n := \left\{x \in [0,1] : x \in \left( a_n \pm \frac{1}{2 \cdot 3^n}\right)\right\}$, $C := \bigcup_{n=1}^{\infty}B_n$...
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How to compute $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n \log (1+\frac{2i}{n})$
I was asked this series
$$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n \log \left( 1+\frac{2i}{n} \right) $$
I tried to use Riemann sum
$$
\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n \log \left( 1+\frac{2i}{...
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Darboux integral is an translation invariance
Let $U\subset\mathbb R^n$ be an open set, $T(x)=x+v\,$ is a translation in $\mathbb R^n$ and $f:\,T(U)\longrightarrow\mathbb R$ is Riemannian integrable. I want to show that $\displaystyle\intop_Uf$ ...
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Bounded with projection content zero implies Jordan measurable
So, I am taking a Several Variables Calculus course, and this problem appeared in one of my tests -
Let $\Omega \in \mathbb{R}^3$ be a bounded set, and $\Lambda = \{(x, y) \in \mathbb{R}^2 \mid (x,y,...
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The improper integral evaluation
I need to evaluate the following integral:
$$I=\int\limits_{0}^{1}\frac{1}{x}\frac{\frac{q}{p}\frac{x^{-p}-1}{x^{-q}-1}-1%
}{x^{-q}-1}dx,$$ where $0<p/2<q\leq p$.
It seems not to have a closed-...
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Monotonicity of the integral
If $f(x) \leq g(x)$ for all $x \in [a,b]$ then
$\int_a^b f \leq \int_a^b g.$
What I did i followed Bartle's method but I cannot keep up on what He meant.
This is the uniqueness so that we'll be ...
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Trying to understand what seems like a simple step in the proof of Theorem 6.12(a) of Rudin
Without getting too much into details, we are trying to prove that if $f=f_1+f_2$ on $[a,b]$ and $f_1,f_2$ are Riemann-Stiljetes integrable, then $f$ is Riemann-Stiljetes integrable and
$$\int_{a}^{b} ...
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Change of variable Theorem's proof for linear transformations
Let $U\subset\mathbb R^n$ be a rectifiable, bounded, open subset and $T:\ \mathbb R^n\longrightarrow\mathbb R^n$ be a linear map. Let $f:\ T(U)\longrightarrow\mathbb R$. Suppose $T(U)$ is rectifiable,...
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The Riemann Integral as the limit of a net
It is usually claimed that the Riemann integral can be seen as the limit of a net, but this is not clear to me. I can see that if the Riemann exists, the limit of the net must exist and they must ...
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Prove Riemann integrability if $g(x) = f(x)$
I'm not sure if the following prove is correct. It seems incomplete. Hope you can help me:
Theorem:
Suppose $f:[a,b] \to \mathbb R$ is Riemann integrable. Suppose $g: [a,b] \to \mathbb R$ is a ...
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Interchange Integral and Limit [duplicate]
I have the following question:
Let $(f_n)_n$ be a sequence of continuous functions on $[a,b]$, such that $f_n$ converges pointwise to $f$, where $f$ itself is continuous, does then
$$\int_a^b f(x)dx= \...
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Estimate of sum of elements of a partition of an interval
I'm reading the proof of the sewing lemma and i'm not able to proof a quite straightforward result that is implicetly used in the paper.
Given an interval $\left[s,t\right]$ and a partition $\mathcal{...
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Understanding the Riemann-Integrability of $ 2 \pi$-periodic functions on $( 0,2\pi )$ whose absolute value squared is Riemann-integrable
In my lecture, we are considering the vector space
$$\{ f : \mathbb{R} \to \mathbb{R} \, | \, f(x+2\pi) = f(x) \, \forall x \in \mathbb{R} \text{ and } \| f \| < \infty \}$$
where
$$\| f \| := \...
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Theorem on Riemann integral of monotone functions
I want to prove the following:
Theorem:
Let $(f_n)$ be a sequence of monotone (integrable) functions $f_n : [0,\infty) \rightarrow \mathbb{R}$ (for $t_i \leq t_j$ we have $f_n(t_i) \leq f_n(t_j)$ such ...
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Why do Riemann sums converge?
I know from calculus that Riemann Integrals are Riemann sums. We use such integrals to calculate areas under curves. But why does adding an infinite amount of small terms give a finite result for non ...
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why does $u$ lies in an interval $I_k=[x_{k-1},x_k]$ of $\dot P_1$ in Example 7.1.4, introduction to real analysis by Bartle & Sherbert?
I want to refer that similar question was asked here before (with no answers)
Let $g:[0,3]\to \mathbb{R}$ be defined by $g(x)=2$ for $0\le x\le 1,$ and $g(x)=3$ for $1<x \le3.$
Let $\dot P$ be a ...
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Riemann integrability of a charactersitic function
If $A$ is an open subset of $(0,1),$ then $\chi_A$ is Riemann integrable on $[0,1].$
My attempt: Since $A$ is an open subset of $(0,1),$ we can write $A$ as countable union of disjoint open intervals ...
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Limit of continuous functions is Riemann integrable
Here is an analysis problem I'm stuck on:
Let $f\in C^0([0,1])$ with $f(0)=0$ and $f$ increasing and convex. Define:
$$ f_n(x) = n\big[f(x)-f(x-\tfrac{1}{n})\big] $$
Show:
$f(1-\tfrac{1}{n})\le\...
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Show f is Riemann integrable if L(-f,[a,b])=-L(f,[a,b])
This question is from Sheldon Axler's real analysis book exercise 1b.
Suppose f is a bounded function on [a, b]. Show that f is Riemann integrable on [a,b] iff L(-f,[a,b])=-L(f,[a,b]).
This is how I ...
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Show that an integral doesn't exist by using the limit of a Riemann sum [duplicate]
Given that
$$\ f(x)=\begin{cases} 0, & \text{if $x$ is rational} \\ 1, & \text{if $x$ is irrational}\end{cases}$$
I want to show that the integral
$$\int_0^1 f(x)\,dx$$
does not exist by using ...
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Let $f : [0,1] \to \mathbb R$ satisfy $\sum_1^n |f(t_i) - f(t_{i-1})|^2 < 100$ for any $ \leq t_0 < t_1 < ...< t_n \leq 1$. Prove $f$ is integrable
Suppose $f : [0,1] \to \mathbb R$ be such that $$\displaystyle\sum_{i=1}^n |f(t_i) - f(t_{i-1})|^2 < 100$$ for any $n \in \mathbb N$ and $0 \leq t_0 < \cdots < t_n \leq 1$. Prove that $f$ is ...
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Same integral on every interval of fixed length
Let $f$ be a continuous function defined on some interval $I \subseteq \mathbb R$, and let $\ell$ be a positive real number. It seems pretty intuitive to me that, if $\int_a^b f$ is the same for every ...
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Most general conditions for variable substitution with Riemann integral
This question is motivated by the discussion here:
https://matheducators.stackexchange.com/a/26687/117
Let $g$ be defined and differentiable on an interval containing $[a,b]$ and $f$ be defined on an ...
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Spivak, Calculus 3rd edition. chapter 19. problem 42 c
exercise: prove that $$\lim\limits_{\lambda\rightarrow\pi}\int_0^\pi \sin\left(\left(\lambda+\frac{1}{2}\right)x\right)\left[\frac{2}{x}-\frac{1}{\sin\left(\frac{x}{2}\right)}\right]\, dx=0$$butI ...
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Uniformly convergence of a functions sequence and Riemann sum
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function.
Suppose $\langle h_n \space \space | \space \space n \in \mathbb{N} \rangle$ is a sequence of real functions given by, $\forall x ...
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Riemann integral of a function nonzero at countable points is zero
The problem is as follows.
Let $E = \left\{\frac{1}{n}: n \in \mathbb{N}\right\}$ and let $f: [0,1] \mapsto \mathbb{R}$ be such that $f(x) = 1 \text{ for } x \in E$, zero otherwise. Show that $f$ is ...
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Convergence of Riemann sums with partitions with equidistant points
I am reading Spivak's book "Calculus".
The definition of integrable given there is the following:
$L(f,P)$ and $U(f,P)$ are the lower sum and the upper sum of $f$ for partition $P$.
I ...
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Do there exist $f$ and $g$ continuous functions on $[1,2]$ such that $\int f dg$ does not exist?
Do there exist $f$ and $g$ continuous functions on $[1,2]$ such that $\int f dg$ does not exist ?
I tried with
$f(x)=1$
$g(x) = \begin{cases}
(x-1)\sin\left(\frac{1}{x-1}\right) & \text{if } x \...
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Does the upper bound of lower sums and the lower bound of higher sums depends on the type of intervals (open, closed or semi-open)?
I have consulted many books and notes on internet on the Riemann integral but in none of them have I found whether the upper bound of lower sums and the lower bound of higher sums depends on the type ...
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Calculating Upper and Lower Sums for the Function f(x) = cos²(πx)
Given the function $f:\mathbb{R}\rightarrow\mathbb{R}, f(x)=cos^{2}(\pi x)$
and the partition $P=\left\{ \frac{k}{n}\mid k=0,1,\ldots,2n\right\}$.
How can I calculate the upper sum U(f,P) of the ...
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Relationship between the Riemann integrability of a curve and the integrability of its module.
After some thought, I think the core of the question is: Can it be proved without using measure theory (lebesgue) that if f is continuous and g is riemann integrable then the composition $f\circ g$ is ...
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If $m(B):=\int_Bf(x)dV_g(x)$ for the Riemannian volume form $V_g$ and $f\in L^1(M,dV_g)$, which subset $B$ of $M$ are $m$-measurable?
Let $(M,g)$ be a compact Riemannian manifold and take $f\in L^1(M, dV_g)$ for the Riemannian volume form $V_g$. If we define $m(B):=\int_Bf(x)dV_g(x)$, which sets are $m$-measurable? Heck, maybe the ...
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Is it possible to evaluate $\lim_{n \rightarrow \infty} \frac{1}{H_n} \sum_{v=1}^n 1/v^{m+1}$?
I have read, that it is not possible to define an equal measure on the natural numbers. While taking this literaly, this is true, but when trying to capture what is meant, I came to the following ...
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Why can't we compute this series using Riemann's sum?
Consider a series,
$$S=\sum_{n\ge0}\frac{(-1)^n}{2n+1}$$
Which can also be written as,
$$S=\lim_{n\rightarrow\infty}\sum_{r=0}^{n} \frac{2n}{(4r+1)(4r+3)}\cdot \frac{1}{n}$$
Substituting
$$\frac{r}{n}=...
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Almost everywhere differentiable functions: what can we do? Integration by parts?
Let $A$ be an open measurable set and $A_0$ is a subset of $A$ which has measure zero.
Let $g$ and $f$ be almost everywhere differentiable function defined on $A$.
What is the impact of "almost ...
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how to turn the sum $\lim\limits_{n\to\infty}\left(\prod\limits_{k=0}^n \frac{2n+2k}{2n+2k+1}\right)$ into integral with Riemann sum
I want to solve
$\lim_{n\to\infty}\left(\prod_{k=0}^n \frac{2n+2k}{2n+2k+1}\right)$ with integration
I know that $\lim_{n \to \infty}\frac{b-a}{n} \sum_{k=0}^nf(a+\frac{b-a}{n})=\int_a^b f(x)dx$
so ...
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Is averaging over a ball smooth?
Let $M = (M, g)$ be a Riemannian manifold, and for $f \in L_{loc}^1(M)$ (with respect to the volume measure $V$), write
$$A_{x, r} f : = V(B(x, r))^{-1} \int_{B(x, r)} f \; \mathrm{d} V .$$
My ...
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