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.
2
votes
2answers
67 views
Is a function with limits Riemann integrable?
Suppose
$f
:
[
0, 1
]
→
R$
and
$\lim\limits_{x
\rightarrow
c}
f
(
x
)$
exists for all
$c
\in
[
a
,
b
]$. Show that $f$
is Riemann integrable on
$[
a
,
b
]$.
I can show this $f$ is bounded on $[a,b]$ ...
0
votes
1answer
25 views
Riemann integral is f integrable on I?
P={$0,(2-\frac{1}{n}), (2+\frac{1}{n}), 4$} and interval I=[0,4]
Is f integrable on I?
I have found that $U(f,P)$=$8+\frac{2}{n}$ and $L(f,P)$=$8-\frac{2}{n}$
So does this mean that we get bounds ...
1
vote
1answer
41 views
Integrating continuous function of two variables by one of the variables, do I get continuous function?
Let $f(x,y)$ be a function continuous in $x$ such that $g(x) = \int_a^b f(x,y) dy$ exists for every $x$. Is $g(x)$ necessarily continuous? I am especially interested in the Riemann and Lebesgue ...
0
votes
2answers
56 views
How to show Riemann integrability
Is the function $$f(x) = \begin{cases} \frac{x^2}{2}+4 &, x\ge0 \\
\> \frac{-x^2}{2}+2 &, x<0.\end{cases}$$ Riemann integrable in the interval$[-1,2]$? Does there ...
3
votes
2answers
75 views
Integrable function of which the antiderivative is not equal to the integral
Is there a Riemann-integrable function $f: [a,b]\to\mathbb{R}$ such that $f$ has an antiderivative which is not equal to $t \mapsto \int_a^t f(x)dx +c$ for any $c\in\mathbb{R}$?
-1
votes
0answers
43 views
Is there $\int_ {0} ^ {1} g\,dx$?
Definition: A partition $ P $ of $ [a, b] $ is a finite set $ \{x_{0}, ..., x_ {n} \} \subseteq {[a, b]} $ such that $ a = x_ {0} <x_ {1} <... <x_ {n-1} <x_ {n} = b $. The norm of a $ P $ ...
0
votes
1answer
33 views
Show that there is a sequence $(P_n)$ of partitions of $[0,1]$ such that $||P_n||\to0$ & $\lim_{n\to\infty} S(g,P_n)$ for the $g(x)$ defined.
Let $g\colon[0,1]\rightarrow \mathbb{R}$, be defined as $g(x) = 0$ if $x \in \mathbb{Q}$ and $g(x)=1/x$ if $x \not\in\mathbb{Q}$. Show that there exists a sequence $(P_n)$ of partitions of $[0,1]$ ...
-2
votes
0answers
32 views
For what type of functions do various types of Riemann sum approximations give exact answers? [closed]
For what kind of functions do each of the following approximation methods give exact answers: Mn, Ln, Rn, Tn, and Sn?
Mn would be a midpoint Riemann sum, Ln would be a lefthand Riemann sum, etc.
0
votes
1answer
23 views
How to prove that Riemann integrable function is measurable?
If a function $f: A \rightarrow \mathbb{R}$ on a bounded set $A$ is Riemann integrable, how do you prove that $f$ is measurable?
From partitions by $n$-dimensional intervals, I can make simple ...
1
vote
1answer
42 views
$ f\in\mathcal {R} _ {\alpha} ([a, b])$ and its equivalences
Definition: A partition $ P $ of $ [a, b] $ is a finite set $ \{x_{0}, ..., x_ {n} \} \subseteq {[a, b]} $ such that $ a = x_ {0} <x_ {1} <... <x_ {n-1} <x_ {n} = b $. The norm of a $ P $ ...
0
votes
1answer
30 views
If $f:A\subset \mathbb{R}^n\to\mathbb{R}$ bounded, $f(x)\geq 0$, for all $\epsilon>0$ exist P s.t. $S(f,P)<\epsilon$ then $f$ integrable, $\int_A f=0$
Problem: Given $f:A\subset \mathbb{R}^n\to\mathbb{R}$ a bounded function, $f(x)\geq 0$. Prove that if for all $\epsilon>0$ exist P partition such that $S(f,P)<\epsilon$ then $f$ is integrable ...
0
votes
1answer
16 views
Validity Check: Every function is integrable on its domain, regardless to continuity and differentiablitty.
F(x) is a function before I attempt to integrate such a function over a certain interval, what I do is check if the interval is a subset of the function's domain, if not then its not integrable.
And ...
0
votes
1answer
19 views
Uniform convergence on compact subset and integration.
Let $(a,b)$ be a open interval and $f_n$ be a sequence of continuous functions on $(a,b)$. Given $f_n$ is uniformly convergent to $0$-function on any compact subset of $(a,b)$. I want to show that:
$...
4
votes
2answers
62 views
Why is it necessary to split the definite integral of a piecewise function into a sum
The second fundamental theorem of calculus (Newton-Leibniz) tells us that:
If $f$ is a real-valued function on a closed interval $[a, b]$ and $F$ is an antiderivative of $f$ in $[a,b]$ s.t. $F'(x)=f(...
4
votes
0answers
65 views
Arc length of a Polar curve as a Riemann sum
Suppose we have a curve in polar plane satisfying the equation $r=f(\theta)$ with $\theta\in[a,b].$ To find the area enclosed by this curve in this range of $\theta$ using Riemann integrals, we ...
0
votes
0answers
41 views
Reference request: Relation between Riemann and Lebesgue integral
I'm looking for a reference that gives the relations between Riemann and Lebesgue integrals.
For example, it must contain facts like:
(1) Let $f: [a,b] \to \mathbb{R}$ be a map. Then
$f$ Riemann-...
0
votes
1answer
76 views
How to do this problem without using infinitesimal?
A rod of linear charge density a of length h, What will be the
electric field at an axial point at a distance x from end of the
rod (the end at which the origin is chosen for defining charge
...
1
vote
0answers
46 views
derivative exist almost everywhere
I am working on a problem finding an error of an example that uses integration by parts.
The condition of the example says that $f'$ and $g'$ exist almost everywhere.
If $f'$ and $g'$ exist almost ...
0
votes
0answers
22 views
Two-dimension Integral proof question
So I have two assumptions as follows:
Let $S$ be a bounded subset of $\mathbb{R^2}$.
For every $\epsilon>0$ there exists rectangles $R_1, ..., R_L$, s.t. the set $S \subset \cup_{i=1}^{L} R_i$ ...
1
vote
1answer
31 views
Connection between u-substitution change of limit and Riemann sum
I worked with integration by $u$ substitution for a bit now, but I still have a hard time rationalizing why we change the limits of the integral when doing $u$ substitution (I know we don't have to ...
0
votes
1answer
17 views
Understanding the Lebesgue criterion for Riemann-integrability
The Lebesgue criterion for Riemann-integrability of a function $f \colon \mathcal{D} \subseteq \mathbb{R} \longrightarrow \mathbb{R}$ states that a function is Riemann-integrable in a compact $\...
0
votes
0answers
49 views
What is the probability of a function being Riemann Integrable?
Suppose we are given a set $F =\{f|\ f:\mathbb{R} \to \mathbb{R}\}$, i.e. an arbitrary set (not sure about its countablility) of real-valued functions. What is the probability, that a function $f_k$, ...
1
vote
2answers
116 views
Prove that the interval [a, b] does not have measure zero
I want to prove that $[0,1]$ does not have measure zero but the book says $``$Explain why the following observation is not a solution to the problem: Every open interval that contains $[a,b]$ has ...
0
votes
0answers
18 views
Upper and lower Riemann sum of Partitions depending on $n$
I am currently self studying Riemann integrability and got to a part that states that if partitions P and Q are such that Q is a refinement of P then $$L (f,P) \leq U (f,Q) \leq U (f,Q) \leq U (f,P)$$ ...
1
vote
1answer
29 views
Riemann-integration problem
Here is the exercise
Let $f:[a,b]\rightarrow \mathbb{R}$ be Riemann-integrable. Prove that $f^+$, $f^-$ and $|f|$ are also Riemann-integrable, when
$$f^+=\begin{cases}
f(x) & f(x)\...
0
votes
1answer
27 views
Does $\implies wf(y)<\epsilon$ , mean $|f(y)-f(x)|<\epsilon?$
Right here, the poster claimed that for $\epsilon>0\;\text{and}\;x\in I,$ $$|y-x|<\delta\implies wf(y)<\epsilon$$
which implies that the $f$ is continuous at $x.$
Question: Does $\implies wf(...
2
votes
1answer
77 views
Let $f:[a,b]\to\Bbb{R}$ be a bounded function which is continuous except at a point $x_0\in [a,b]$. Then, $f$ is Riemann integrable over $[a,b]$.
Let $f:[a,b]\to\Bbb{R}$ be a bounded function which is continuous except at a point $x_0\in [a,b]$. Then, $f$ is Riemann integrable over $[a,b]$.
My Proof
Consider the restriction
\begin{align} g:[...
2
votes
2answers
85 views
If $f$ is a derivative then is $|f|$ also a derivative?
If $f$ is a derivative then, is $|f|$ also a derivative?
If $f$ is Riemann integrable then it's true. But, if it's not the case,then is it true?
2
votes
1answer
34 views
Prove that this function is Riemann Integrable on $[0,1]$ [duplicate]
Let $E:=\{\frac{1}{n}: n\in \mathbb{N}\}$. Define $f$ on $[0,1]$ by
$f(x)=\begin{cases}1 ~~~~~~~~\text{if $x\in E$}\\ 0~~~~~~~~\text{if $x\notin E$ }\end{cases}$.
Show that f is Riemann ...
0
votes
1answer
33 views
Riemann-integral for a function that has infinitely many discontinuity points
The question is following:
Let $f:[0,1]\rightarrow \mathbb{R}.$
$f(x)=x,$ if $x=1/n, n\in\mathbb{N}$
$f(x)=0,$ otherwise.
Is $f$ Riemann-integrable? If it is, what is its value?
I know that the ...
1
vote
1answer
33 views
Show that $f(x)=0,\;0\leq x<1/2,\; f(x)=1,\;1/2\leq x\leq 1$ is Riemann integrable over $[0,1]$ and find its value.
I want to prove that \begin{align} f:[0&,1]\to \Bbb{R}\\&x\mapsto \begin{cases}0,&0\leq x<1/2,\\\\ 1,&1/2\leq x\leq 1. \end{cases}\end{align}
is Riemann integrable over $[0,1]$ ...
4
votes
1answer
69 views
On anti-derivative of functions [closed]
Let $f,g,h: \mathbb R \to \mathbb R$ be differentiable functions.
(1) Does there necessarily exist a differentiable function $F: \mathbb R \to \mathbb R $ such that $F'=\max \{f' ,g' \}$ ?
(2) Does ...
0
votes
1answer
21 views
Bounded function with finitely many discontinuities is integrable $\overset{?}{\Rightarrow}$ density of continuous distribution function is not unique
The density function of the distribution function of a continuous random variable is not uniquely defined.
A new density function can be obtained by changing the value of the function at finite ...
0
votes
0answers
28 views
Riemann Integral of discontinuous function
So I know you can integrate some discontinuous functions when the function is discontinuous at a finite number of points. So you can integrate for example this function on a interval [-1,3]:
$$ f(x):=...
1
vote
1answer
29 views
Logical mistake in a proof of Dirichlet's test
While writing a proof of Dirichlet's Test, [ in which, a required condition is that $\displaystyle{\int_a^B\phi(x)dx}$ is bounded $\forall B>a]$, I made a logical mistake, which I noticed upon ...
1
vote
0answers
22 views
Clarification required for the proof of Riemann Lebesgue Lemma
Riemann Lebesgue Lemma Proof.
In the proof given above, I came across an unfamiliar notation $g= \sum m_i \chi_{[x_i-1,x_i]}$. What does the $\chi$ mean here? Is it the characteristic function of the ...
1
vote
1answer
44 views
Prove that if $f$ is continuous at $x_0\in [a,b]$ and $f(x_0)\neq 0,$ then $\sup\limits_{P}L(|f|,P)>0$
Can you help me check if this proof is correct? If not, kindly provide a better proof
Prove that if $f$ is continuous at $x_0\in [a,b]$ and $f(x_0)\neq 0,$ then $\sup\limits_{P}L(|f|,P)>0$
...
0
votes
1answer
48 views
A case where Lebesgue integrable implies Riemann integrable
Let $I$ an interval on $\mathbb{R}$ such as $I=(a,b)$, with $a$ or $b$ could be equal to infinity.
And we have $f\in \mathcal{L}^1(I,\mathcal{B}(I), \lambda)$, then do we have always
$$\int_{(a,b)}...
1
vote
1answer
31 views
Are there any numerical integration methods that do not involve rectangles or polynomial approximations?
For the Riemann integral, are there any methods of numerical integration that do not involve rectangles or approximating the area with a polynomial function? I am aware of the trapezoidal rule, but I ...
3
votes
2answers
306 views
When is a Lebesgue integrable function a Riemann integrable function?
When is a Lebesgue integrable function a Riemann integrable function ?
And if we have $f\in \mathcal{L}^1([0,1],\lambda)$, does it implies that $f$ is Riemann integrable, and why ?
1
vote
0answers
23 views
Let $f$ an a.e continious and bounded function, is $f$ Riemann integrable function? [closed]
If we have an almost evrywhere continious and bounded function $f$ on $[0,1]$, can we deduce that $f$ is Reimann integrable ?
1
vote
2answers
45 views
Right Riemann sum Error bound proof
How do we prove the right Riemann sum error bound?
In wikipedia (https://en.wikipedia.org/wiki/Riemann_sum#Right_Riemann_sum) they have mentioned the following bound, but no proof.
$$\left | \int_a^bf(...
3
votes
0answers
54 views
Proof of second Fundamental theorem of calculus
Is there any other proof of this?
Second fundamental Theorem of Calculus: If $f$ is differentiable on $[a,b]$ and $f'$ is integrable on $[a,b]$, then
$$\int^{b}_a f'(t)dt=f(b)-f(a)$$
My proof
...
0
votes
1answer
40 views
Prove that a Jordan measurable set of measure 0 is Riemann integrable
My problem is: I have a set $A$ which is Jordan measurable, and I have a function $f:A\rightarrow\mathbb{R}$. I need to prove that if the measure of A is $0$, then $f$ is Riemann integrable and ...
3
votes
2answers
84 views
Proof of first Fundamental theorem of calculus
Can you please, check if my proof is correct?
Suppose that $f:[a,b]\to \Bbb{R}$ is continuous and $F(x)=\int^{x}_{a}f(t)dt$, then $F\in C^{1}[a,b]$ and
$$\dfrac{d}{dx}\int^{x}_{a}f(t)dt:=F'(x)=...
0
votes
1answer
54 views
Disprove: If $f$ is Riemann integrable on $ [a, b]$, then $f$ is continuous on $[a, b]$.
Do you have any other function that serves as a counterexample to this statement?
If $f$ is Riemann integrable on $ [a, b]$, then $f$ is continuous on $[a, b]$.
My counter-examples
Consider the ...
2
votes
2answers
193 views
Example where The Lebesgue Integral is Better
What is an example that involves a fuction on an interval of the real numbers where the Lebesgue integral is better than the Riemann integral.
By better, it probably means that the Lebesgue ...
4
votes
4answers
94 views
Find $\lim\limits_{n\rightarrow\infty}\int\limits_0^1f(x^n)dx$
Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function.
Show that for each $\epsilon\in(0,1)$, $\lim\limits_{n\rightarrow\infty}\int\limits_0^{1-\epsilon}f(x^n)dx=(1-\epsilon)f(0)$
...
0
votes
2answers
17 views
Riemann Integral: prove $\{U(P,f):P$ is a partition of $[a,b]\}$ is bounded below
[Defining the Riemann Integral: ]
We consider a partition, $P=\{a=x_0<x_1<...<x_n=b\}$ of $[a,b]$ and a bounded function $f:[a,b]\rightarrow\mathbb{R}$.
Next, we define-
$$M_i=\sup\{f(x)|x\in[...
2
votes
1answer
47 views
$\underset{n\rightarrow \infty}{\lim}\int_{0}^{b_{n}}f_n(x)dx=\int_0^1f(x)dx$
Let $(f_n)_{n\in\mathbb{N}}$ ne a sequence of continuous functions on
$[0,1]\rightarrow\mathbb{R}$ that converges uniformly to
$f:[0,1]\rightarrow\mathbb{R}$. Let $(b_n)_{n\in\mathbb{R}}$ be an
...
