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
0answers
18 views
Proof of Generalized Riemann Integrability Criterion
Suppose f:[a,b] $\rightarrow$ $\mathbb{R}$ is a bounded function and there is a set Z $\subset$ [a,b] sucht that:
f is continuous at every point x $\notin$Z.
For every $\epsilon$ > 0, the set Z can ...
0
votes
1answer
31 views
Riemann-Stieltjes Integral for Specific Integrators
I'm asked to show that
$\lim_{n\to\infty}\frac{b-a}{n}\sum_{k=1}^n f(a+k\frac{b-a}{n}) = \int_{a}^{b}f(x)dx$.
I tried to come up with a step function that is constant on each of the open intervals $(...
0
votes
0answers
19 views
Riemann Integrable definition in partition distingued $\mathbb{Q}$
A function $f:[a,b]\rightarrow \mathbb{R}$ is Riemann Integrable on $[a,b]$ if
$\exists \thinspace L \in \mathbb{R} : \forall \epsilon >0\thinspace \exists \thinspace \delta >0 :$ if $P$ is any ...
2
votes
1answer
46 views
Function is Riemann integrable in $[a,c]$ and $[c,b]$ then is RI in $[a,b]$
i need to prove that if Riemann integrable in $[a,c]$ and $[c,b]$ then is RI in $[a,b]$, maybe is easy but i can't see it.
Definition of Riemann Integral
A function $f:[a,b]\rightarrow \mathbb{R}...
2
votes
0answers
32 views
Is this proof that $f$ is Riemann integrable correct?
Let $f:[a,b]\rightarrow \mathbb{R}$ be a bounded, non constant function. We know that f is integrable in $[c,b]$ $\forall c \in [a,b]$. Prove that $f$ is Riemann integrable in $[a,b]$.
I have tried ...
1
vote
0answers
44 views
Replacing differentiation with anti-integration?
When thinking about the proof of the differentiability of Taylor series, I noticed that the theorem was proved by using properties of integrals. This got me thinking: To what extent can the role of ...
3
votes
1answer
72 views
Examples of Lebesgue-integrable, but not Riemann-integrable functions
The standard example of this is the characteristic function of the rationals. However this is somewhat pathological as this function is zero almost everywhere. What are other examples that differ from ...
0
votes
1answer
45 views
How to find $f(x)$ if $\int_0^{x^2} (1+t)f'(t)dt=x^4$ and $\int_0^1 f(t)dt=3 $
Knowing that $f:[0,+\infty)\rightarrow \mathbb{R}$ is continuous and derivable, and that:
$\int_0^{x^2} (1+t)f'(t)dt=x^4$;
$\int_0^1 f(t)dt=3 $
Determine $f(x)$.
(Note: This is supposed ...
0
votes
1answer
44 views
Integral inequality $\int_{0}^{1}\left ( \frac{f(x)}{x} \right )^2dx \leq 4\int_{0}^{1}(f'(x))^2dx$
Let $f \in C^1([0;1],\mathbb{R})$ such that $f(0)=0$.
$$\text{Prove that} \qquad \int_{0}^{1}\left ( \frac{f(x)}{x} \right )^2dx \leq 4\int_{0}^{1}(f'(x))^2dx$$
My attempt:
Let $$g(x)=\begin{cases}
...
0
votes
1answer
31 views
Problem with Definition of Riemann Integrable Function
I have difficulty understanding the definition of a Riemann integrable function $f$ on $[a,b]$. I understand the definition of upper sum $U(f,P) = \sum\limits_{i=1}^{n}M_i \Delta x_i$ and lower sum $...
1
vote
2answers
58 views
If $f(x)=\int_a^x f(t)dt$, is $f(x)=0$?
How can I prove this?
Knowing $f:[a,b] \rightarrow \mathbb{R}$ is continuos in $[a,b]$ and
$f(x)=\displaystyle\int _a^x f(t)dt$,
how can I conclude that $f(x)=0$ $\forall x \in [a,b]$? I can't even ...
0
votes
0answers
59 views
Baby Rudin 6.11 and 6.8
Theorem 6.8
If $f$ is continuous on $[a, b]$ then $f \ ; \epsilon \; R(\alpha)$$f$ is Riemann integrable on$ [a, b]$.
$f \epsilon R(\alpha)$ means $f$ is Riemann integrable w.r.t. $\alpha$
...
0
votes
2answers
65 views
Integral inequality $\int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}x|f'(x)|dx$
Let $f \in C^1([0;1],\mathbb{R})$ such that $f(1)=0$.
$$\text{Prove that} \qquad \int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}x|f'(x)|dx$$
My attempt:
\begin{align}
\int_{0}^{1}x|f'(x)|dx = \int_{0}^{1}...
0
votes
1answer
22 views
How to prove that a function is of bounded outside variation?
Let $E$ be the Vitali set (or any other nonmeasurable subset of $[0,1]$ with respect ot Lebesque measure), so $E\subset [0,1]$ is nonmeasurable. Consider the following finction $f\colon [0,1]\to l_{\...
1
vote
1answer
49 views
Two definitions for Riemann integrability of $f:[a,b]\rightarrow X$, where $X$ is a Banach space.
Let $X$ be a Banach space and $f:[a,b]\rightarrow X$ be a function. Consider the following two definitions of Riemann integrability:
Definition 1: there exists $x\in X$ and a sequence of partitions $\...
0
votes
0answers
15 views
Calculating upper Riemann sum. Is it sufficient to only consider “simple” partitions
Let $b > 0$. I'd like to calculate $ \int_{0}^b x^2 dx$ using upper and lower Riemann sums. Since the function is continuous on $[0,b]$ I know that it is integrable so I only need to calculate
$$ ...
0
votes
1answer
40 views
Volume of a tilted cylinder
Suppose I have a tilted cylinder of length l inclined to the horizontal by an angle of $\theta$ then it's volume comes out to be same as that of a straight cylinder of height $l\sin\theta$.
I tried to ...
1
vote
1answer
44 views
Show that if $f \in R_\alpha$,and g increasing and continuous then $ f(g(x)) \in R_{\alpha(g(x))}$
Let $f \in R_\alpha[a,b]$ and $g:[c,d] \rightarrow \mathbb{R}$ continuous and strictly increasing, such that $g(c) = a$ and $g(d) = b$. Prove that $f(g(x)) \in R_{\alpha(g(x))}$.
In calculus I ...
3
votes
1answer
62 views
Question about Riemann integrability: do we need to specify that all Riemann sums converge to the same number in the definition?
Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. Suppose that there is a sequence of partitions $\{P_n\}_{n=1}^\infty$ with mesh tending to $0$, $P_n=\{a=t_0^n<t_1^n<\ldots<t_{r_n}^n=b\}$, ...
0
votes
2answers
19 views
Lower and upper sums of a bounded function
Assume $f:[a,b]\to\mathbb{R}$ is bounded.
Let $P=\{x_0,...,x_N\}$ be a partition of $[a,b]$.
Why, for $1\leq n \leq N$ do we have that
$$\inf\{-f(x),\,\,\,x\in[x_{n-1,x_n}]\}=-\sup\{f(x),\,\,\,x\...
-2
votes
0answers
23 views
Derivative under the integral sign
I have a question about derivative under the integral sign, perhaps someone could help me with this issue. The problem is the following: Let us consider two intervals $I_{x} \in (0, +\infty)$ and $...
2
votes
1answer
36 views
Showing that $f \in R_\alpha[a,b]$ and $\lim_{n\rightarrow \infty}\int_a^b f d\alpha_n = \int_a^b f d\alpha $
Let $\left(\alpha_n \right)_{n\in \mathbb{N}}$ a succesion in $BV[a,b]$ and $f:[a,b] \rightarrow \mathbb{R}$ such that $f \in R_{\alpha_n} [a,b]$.
If $\alpha \in BV[a,b]$ and $V_a^b(\alpha_n - \alpha) ...
2
votes
1answer
45 views
If $f \in BV[0,2\pi], f(0)= f(2\pi)$ Show that $\int_0^{2\pi}f(x)\sin(nx)dx$ exist for each $n$ natural.
This is from Carothers 14.38
If $f \in BV[0,2\pi], f(0)= f(2\pi)$ show that $\int_0^{2\pi}f(x)\sin(nx)dx$ exist for each $n$ natural and $$\left|\int_0^{2\pi}f(x)\sin(nx)dx\right| \leq \frac{V_0^{2\...
1
vote
1answer
69 views
Show that $\sum_{{i = 1}}^{n} f(i) = \lfloor n \rfloor f(n)- \int_{1}^{n}f'(x)\lfloor x\rfloor\, dx$
Where $f$ is a function defined in $\mathbb{R}$ with countinuos derivative in all
$\mathbb{R}$, for each $n\in \mathbb{N}$ and the function $\lfloor x \rfloor$ is the floor function.
I tried using ...
0
votes
2answers
36 views
Definition of integrability
By the definition of Riemann integrals I have to show that:
$\int_{-a}^{a} f(x^2) dx=2\int_{0}^{a} f(x^2) dx$ (given that both integrals exist)
I thought to approach it as:
$\int_{-a}^{a} f(x^2) dx=\...
1
vote
1answer
26 views
Proof that a function is Riemann integrable if for any $\epsilon > 0$ there exists a partition P such that: $U(P, f) − L(P, f) < \epsilon$
I am working my way through the proof of the following:
Let $f$ be bounded on [a,b]. Then $f$ is Riemann integrable if and only if for every $\epsilon$ there is a partition on $[a,b]$ such that: $0 \...
0
votes
1answer
22 views
Sequence of integrals converges to an improper integral proof?
Let $f: (0,1) \rightarrow \mathbb{R}$ be increasing on $(0,1)$. If the improper integral (of second kind) $\int_0^1 f(t)dt$ converges ($f$ is unbounded at $0$ and at $1$), then show that the sequence $...
0
votes
1answer
26 views
trying to proove this limit equality to a specific integral [closed]
can you help me prove this equality?
I tried to use Riemann sums but I haven't succeeded to find something useful.
$$\lim_{n\to\infty} \sum_{k=1}^n f\left(\frac{k}{n}\right)\frac{1}{n}
= \int_0^1f(...
2
votes
0answers
51 views
If $\int_a^b f\, d\alpha $ exists for all continuous $f$ then $\alpha$ is of bounded variation
In this answer the following theorem is proved:
Theorem: Let $\alpha:[a, b] \to\mathbb {R} $ be a function such that the Riemann-Stieltjes integral $$\int_{a} ^{b} f(x) \, d\alpha(x) $$ exists for ...
1
vote
1answer
62 views
Prove that $g(x)=\sqrt{x}$ is not an integral function on $[0,1]$.
Prove that $g(x)=\sqrt{x}$ is not an integral function on $[0,1]$,
i.e. $\nexists f \in R[0,1]$ so that $$\sqrt{x}=\int_0^x f(t) \mathrm{d}t$$
I see that $f$ cannot be continuous, because if it ...
2
votes
0answers
62 views
Why can’t improper integrals be defined directly using Riemann sums?
The standard way to define an improper integral of the form $\int_a^\infty f(t)dt$ is as follows. We first define the Riemann integral $\int_a^xf(t)dt$ for each $x>a$ in the standard way, i.e. ...
6
votes
1answer
98 views
Riemann Integrability of Composite functions.
Function $f$ is continuous and strictly increasing on $[a,b]$ and $g$ is Riemann integrable such that $g \circ f$ is defined. Is $g \circ f$ Riemann integrable?
I was able to show that if $f$ ...
3
votes
0answers
59 views
A new equivalent characterization of Riemann-Integrability
Question:
Given two bounded functions $\,f:[a,b]\to \mathbb R\;$ and $\;\theta:(0,b-a]\to [0,1]$.
Suppose $\,P:a=x_0<x_1<⋯<x_n=b\;$ is a partition of $[a,b]$.
Let $\,Δx_k=x_k−x_{k−1}$ and ...
0
votes
1answer
33 views
Counterexample for this statement about the Riemann integral?
I'm considering a Riemann integral, and trying to work out if the following statement is true: Let $P$ and $P'$ be two partitions such that $\mu (P') \lt \mu (P)$. $\mu(P)$ denotes the mesh of P, the ...
0
votes
1answer
33 views
Laplace Transform: Piecewise Function Integrability and Existence of Laplace Transform
I am trying to decide whether the function
$$f(t) =
\begin{cases}
1, & \text{$t$ is even} \\
0, & \text{$t$ is odd}
\end{cases}$$
has a Laplace transform, or is even integrable in the ...
0
votes
2answers
37 views
Between proper integrals and improper integrals
I just started learning about improper integrals. Many of them are improper because the function evaluates to infinity at some point in their domains, e.g. $f(x)=1/x$ on the domain of $(0,1)$. My ...
4
votes
1answer
24 views
Lebesgue measurable but not Riemann integrable
Every bounded function $f:[a,b]\to\mathbb R$, which is Riemann integrable, it is also Lebesgue integrable.
On the other hand
$$
g(x)=\left\{
\begin{array}{lll}
1 & \text{if} & x\in\mathbb Q\...
1
vote
0answers
64 views
Is this kind of limit defined?
Is there any limit like this
$$\lim_{f(x)\to0} g(x)=0?$$
Defined as follows$$\forall \epsilon>0,\exists \delta>0 | |g(x)|<\epsilon
,\forall x |0<|f(x)|<\delta?$$
Where f ...
4
votes
0answers
110 views
If $f$ is integrable, then $\| f\|$ is also integrable.
As usual, a partition of a compact interval $[a, b]$ is, by definition, an strictly increasing family $\Pi = (t_k)_{k = 0}^m$ ($m \geq 0$) of points in the interval such that $t_0 = a$ and $t_m = b;$ $...
1
vote
0answers
20 views
Riemman-Stieltjes Integration-IVT
I need to show this,
Suppose that $f,g: [a,b] \rightarrow \mathbb{R}$ are continous.
Show that exist $\eta \in (a,b)$ such that
$$g(\eta)\int_a^\eta f(x)dx=f(\eta)\int_\eta^b g(x)dx $$
I defined $...
1
vote
0answers
23 views
Existence of antiderivative of a function [duplicate]
Let $f: \mathbb{R}\rightarrow{\mathbb{R}}$ be given by $f(x)=sen(\frac{1}{x})$ si $x\neq 0$ and $f(x)=c$ si $x=0$, where $c\in [-1,1]$
For which values of c there is an antiderivative?
Proof (...
2
votes
1answer
33 views
limit of Integral of product of convergent functions
Let $f,g$ be real functions that are Riemann-integrable over $[0,t] \, \forall t \in {\mathbb R^+}$. With:
$$\lim_{x \to \infty}(f(x))= p$$
$$\lim_{x \to \infty}(g(x))= q$$
Show that:
$$\lim_{t \to \...
5
votes
1answer
72 views
Integration and a function increasing
Let $g_{1},g_{2}\in\mathcal{R}([a,b])$ (Riemann-integrable) such that $$\int_{a}^{x}{g_{2}(t)dt}\leq\int_{a}^{x}{g_{1}(t)dt}\phantom{a}\text{for each}\phantom{a}x\in [a,b]$$ and $$\int_{a}^{b}{g_{1}(t)...
0
votes
1answer
71 views
Integration and existence of an antiderivative
Let $ f: \mathbb {R} \rightarrow {\mathbb {R}} $ be given by
$$ f (x) = \left \{\begin {matrix}
\sin ({\frac {1} {x}}), & \text {if} \phantom {a} x \neq 0; \\
c, & \text {if} \phantom {a} x ...
2
votes
0answers
42 views
Example of sequence of continuous function which is not Riemann integrable with following property
$\{f_n\}$ is a sequence of continuous functions with following properties:
$0 \leq f_n(x) \leq 1, \forall x \in \Bbb R$
$f_n(x)$ is monotonically decreasing sequnce as $n\to \infty$
The Limiting ...
2
votes
1answer
86 views
Find the limit of $\lim\limits_{n\to\infty}\frac{1}{n^2}\sum\limits_{k=1}^{n}k\arctan{\big(\frac{pk-p+1}{pn}\big)}$
I am required to find the limits of two "siblings" using the same idea, they are:
$$\lim_{n\to\infty}\sin{\left(\frac{1}{pn+r}\right)}\sum_{k=1}^{n}\sin{\left(\frac{2pk-2p+r}{2pn}\right)}$$ with $0&...
1
vote
2answers
75 views
Prove that a function is Riemann Integrable
Let $ f: [0,1] \rightarrow \Bbb R$ defined by:
$f(x)= \sin(\frac{1}{x}) $ if $ x \notin \Bbb Q$ , $ 0 $ if $x \in \Bbb Q$
I have to prove that this function is Riemann integrable in that interval. ...
1
vote
2answers
74 views
Evaluate using Riemann sums $\int_{\pi/2}^{3\pi/2}(4\sin 3x - 3 \cos 4x)dx$
I have to evaluate this integral using Riemann sums. I've already tried:
$$\int_{\pi/2}^{3\pi/2}(4\sin 3x - 3\cos 4x)dx = [\delta x = \frac{\pi}{n}, x_i = \frac{\pi}{2} + \frac{i\pi}{n}] =$$$$= \...
0
votes
0answers
45 views
$\int_a^b f(x)g(x)\ dx=g(a)\int_a^{x_0}f(x)\ dx+g(b)\int_{x_0}^bf(x)dx$
Let $f,g:[a,b]\to\mathbb{R}$ be Riemann integrable functions such that $g$ is monotone. Show that there exists $x_0\in [a,b]$ such that $\int_a^b f(x)g(x)\ dx=g(a)\int_a^{x_0}f(x)\ dx+g(b)\int_{x_0}^...
2
votes
2answers
94 views
Integration and null function [closed]
Let $ f: [a, b] \rightarrow {\mathbb {R}} $ be continuous and such that $$ \int_ {a} ^ {b} {f (x) g (x) dx} = 0 $$ for all $ g: [a, b] \rightarrow {\mathbb {R}} $ continuous with $ g (a) = g (b) = 0 $....
