Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
26 views

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, ...
Lucas Alland's user avatar
0 votes
0 answers
40 views

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) ...
Christoph Mark's user avatar
-2 votes
1 answer
47 views

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] \...
Saaqib Mahmood's user avatar
0 votes
0 answers
55 views

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\...
Alice's user avatar
  • 508
0 votes
1 answer
56 views

Help with the proof of this statement

Let $ f : [a, b] \rightarrow \mathbb{R} $ be a bounded function. Then for every $\epsilon > 0$, there exists $\delta > 0$such that for any partition $D$ of the interval $[a, b]$ with norm $\nu(D)...
Binky McSquigglebottom's user avatar
0 votes
0 answers
51 views

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 $...
ashpool's user avatar
  • 7,006
2 votes
1 answer
43 views

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 ...
Debu's user avatar
  • 1
1 vote
0 answers
29 views

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 ...
Richard's user avatar
  • 89
1 vote
0 answers
102 views

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 ...
natitati's user avatar
  • 429
0 votes
1 answer
33 views

Show that $\int_0^{\frac{\pi}{2}}\frac{x^m}{\sin^n x}dx$ is convergent if and only if $n < 1 + m.$

Show that $\int_0^{\frac{\pi}{2}}\frac{x^m}{\sin^n x}dx$ is convergent if and only if $n < 1 + m.$ The solution given in the book is as follows: Let the given integral be $\int_0^{\frac{\pi}{2}}f(x)...
Thomas Finley's user avatar
1 vote
1 answer
109 views

Examine the convergence of $\int_0^1x^{n-1}\log x dx.$

Examine the convergence of $\int_0^1x^{n-1}\log x dx.$ The solution given is as follows: $0$ is the only point of infinite discontinuity of the integrand. Let us examine the convergence of $\int_0^{\...
Thomas Finley's user avatar
1 vote
0 answers
100 views

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 ...
MathMister's user avatar
5 votes
1 answer
93 views

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 ...
Thomas Finley's user avatar
0 votes
1 answer
69 views

Riemann integrability for step function

Here is the problem: Fix $c\in\mathbb{R}$ and define $g:[0,2]\to\mathbb{R}$ by $$g(x)=\begin{cases}2 &\text{if } 0\le x<1\\c &\text{if } x=1\\ 1&\text{if } 1< x\le 2.\\\end{cases}$$ ...
Sym Sym's user avatar
0 votes
0 answers
119 views

$f(x) \ge 0$ 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 ...
user13's user avatar
  • 1,689
-4 votes
2 answers
78 views

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 $\int_{R^2}\; \exp (-x^2-y^2)\,dxdy = π$ .
Murilo 's user avatar
1 vote
0 answers
59 views

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 ...
Renato lorentz's user avatar
0 votes
0 answers
17 views

Function-vector dualism of inner product

I have a question related to the dualism between the inner product of infinite-dimensional vectors and the integral over the product of their corresponding function representations, which is given by $...
Richard Schömig's user avatar
0 votes
1 answer
23 views

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 ...
niobium's user avatar
  • 1,231
0 votes
2 answers
40 views

Lebesgue-Vitali theorem and the characteristic function of $\mathbb{Q}$

The Lebesgue-Vitali theorem tell us that almost everywhere continuous functions on a closed interval are Riemann Integrable. I understand that the characteristic function of $\mathbb{Q}$ defined on $[...
Kham Bodrogi's user avatar
2 votes
0 answers
86 views

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 ...
pie's user avatar
  • 6,581
1 vote
2 answers
51 views

Doubt regarding Riemann Integrable problems

I was trying to learn Riemann integration. So after that I wanted to practice some questions and got this. Given a function, $f(x) = \sin{x}\ \ \text{if}\ \ x = \frac{1}{n}$ and $f(x) = \cos{x} \ \...
Tina's user avatar
  • 23
0 votes
0 answers
27 views

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 ...
YH Chow's user avatar
1 vote
0 answers
57 views

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 ...
John Davies's user avatar
2 votes
3 answers
154 views

Does the following always hold in the definite integral case?

On an interval $[a,b] \subset \mathbb{R}$ suppose we are computing the integral $$\int_{a}^{b}f(g(x)) \mathrm{d}x.$$ If this integral is finite, then what we can say about $g(x)$? Is $g(x)$ also ...
Andyale's user avatar
  • 191
0 votes
1 answer
79 views

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 ...
Chang's user avatar
  • 339
1 vote
1 answer
84 views

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]\...
Andrews's user avatar
  • 123
1 vote
0 answers
46 views

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 ...
SiegAndy's user avatar
0 votes
0 answers
25 views

Oscillation of an Integrable Function on a Subinterval is smaller than epsilon. Proof check.

Given an integrable function $f : [a,b] \rightarrow \mathbb{R}$, and a subinterval $[c,d] \subseteq [a,b]$ with $c < d$, we aim to prove that for every $\epsilon > 0$, there exists an interval $[...
mpavlov23's user avatar
  • 123
0 votes
0 answers
42 views

Is this Dirichlet type function Riemann Integrable?

\begin{cases} \cos\left(\frac{\pi}{x}\right) &,\quad \text{if } x \text{ is rational} \\ 0 &,\quad \text{if } x \text{ is irrational} \end{cases} Over the interval $[0,1]$ My approach Solve ...
Theorist's user avatar
0 votes
0 answers
19 views

The usage of theorem for $\sum\limits_T\omega_i\Delta x_i.$

To determine a function $f$ on $[a,b]$ whether Riemann integral or not, we always use theorem: If function $f$ on $[a,b]$ bounded and $\forall \varepsilon>0,\exists \delta>0,$forall partition $T$...
Daeree's user avatar
  • 25
1 vote
0 answers
32 views

Riemann Integration and Supremum

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 ...
SiegAndy's user avatar
3 votes
2 answers
51 views

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 ...
schneiderlog's user avatar
1 vote
1 answer
68 views

Oscillatory integral and Riemann integral

Consider the summation, with parameter $a \ge 0$ and non-negative integer $M = 1, 2, 3...$ $$S (a, M) = \sum_{m=1}^M \frac{2}{1+M} \sin\left( \frac{\pi m}{M+1}\right) \sin\left( \frac{\pi m M}{M+1} \...
Nigel1's user avatar
  • 655
0 votes
1 answer
74 views

Prove $\int_{1}^{x} \frac{dt}{t} \leq \sum_{n \leq x} \frac{1}{n} \leq 1 + \int_{1}^{x} \frac{dt}{t}$

Let $x \geq 1$. I wish to show, by interpreting the sum as a Riemann sum, that $$\log(x) = \int_{1}^{x} \frac{dt}{t} \leq \sum_{n \leq x} \frac{1}{n} \leq 1 + \int_{1}^{x} \frac{dt}{t}.$$ Certainly, $...
V. Elizabeth's user avatar
0 votes
0 answers
62 views

Let $f : [a,b] \times [c,d] \to \mathbb{R} $ be a continuous function. Let $F(t,y) = \int_{a}^{t} f(x,y) dx$ be a function. Show that F is continuous.

Let $f : [a,b] \times [c,d] \to \mathbb{R} $ be a continuous function. Let $F(t,y) = \int_{a}^{t} f(x,y) dx$ be a function. Show that F is continuous. I attempted to prove the above but I am not sure ...
a22's user avatar
  • 87
1 vote
1 answer
46 views

Theorem 7.32 in Apostol's MATHEMATICAL ANALYSIS, 2nd ed: How to establish differentiability?

Here is Theorem 7.32 in the book Mathematical Analysis - A Modern Approach to Advanced Calculus by Tom M. Apostol, 2nd edition: Let $\alpha$ be of bounded variation on $[a, b]$ and assume that $f \in ...
Saaqib Mahmood's user avatar
1 vote
0 answers
63 views

Determine if Thomae's function analog is Riemann integrable

I've proven Thomae's function $$ f(x)=\begin{cases}\frac{1}{q},&\text{if }x=\frac{p}{q}\in\mathbb{Q}\text{ in lowest terms, }x\neq0;\\0,&\text{ otherwise}.\end{cases} $$ is Riemann integrable ...
Alex Byard's user avatar
0 votes
1 answer
61 views

How to find exact Riemann-Sum?

I have been given this rather simple looking assignment, which is confusing me a lot. Given is: $f:[0,5] \rightarrow \mathbb{R}$ where $f(x)=2x+3$. The first thing I had to do, was to determine the ...
N G's user avatar
  • 31
2 votes
2 answers
83 views

Riemann integrable function continuity points

$f$ is Riemann integrable on $[a,b]$, $\int_a^bf(x)^2>0$. Prove that there is a point of continuity $x$ on $[a,b]$ that $f(x) \neq 0$. I provided the following: almost every point of $[a,b]$ is a ...
mathemaniac's user avatar
1 vote
0 answers
60 views

Integrability of an open and convex subset $M$ of $\mathbb{R}^n$ with non-empty interior and compact closure (e.g. $(0,1)\times(0,1))$

I will follow the definition of the volume of a Riemmanian manifold from page 422 of Introduction to Smooth Manifolds Let $M$ be an open and convex subset of $\mathbb{R}^n$ with non-empty interior and ...
Sub Mat's user avatar
  • 11
1 vote
0 answers
100 views

Let $f$: $[a,b]\to \mathbb{R}$ bounded and Riemann-Stieltjes integrable for every monotone increasing function $\alpha$, is $f$ continous?

My attempt: Let $\epsilon$>0, as $f\in R_{\alpha}[a,b]$ there is a partition $P_{\epsilon}$ of $[a,b]$ such that $U(P_{\epsilon},f,\alpha)-L(P_{\epsilon},f,\alpha)<\epsilon$. To prove the ...
JuanFerRp's user avatar
0 votes
1 answer
43 views

Are there other kind of primitives, other than $\int_a ^x f(t)dt +k$ ? For example for non-continuous functions.

The second fundamental theorem of Calculus says if a function is continuous (hence, Riemann integrable) then $\int_a ^x f(t) dt$ is a primitive of $f$. Ok this is for continuous functions. However ...
niobium's user avatar
  • 1,231
1 vote
1 answer
50 views

How to prove that a function is integratable using the Riemann condition?

Specifically, From reading a some texts, it states that the Riemann condition for showing that the function is integratable is by showing that there is an $\epsilon > 0$ and condition $s \leq f \...
thewhale's user avatar
1 vote
1 answer
123 views

Testing Riemann integrability of a function that is discontinuous at all rational points.

Prove that the function $f$ from $[a,b]$ to $\mathbb{R}$ defined by $$f(x) =\begin{cases} \frac{1}{q^2}, & \text{when }x = \frac{p}{q} \\ \frac{1}{q^3}, & \text{when } x=\sqrt{\frac{p}{q}} \...
Sasikuttan's user avatar
4 votes
1 answer
200 views

Show that the piecewise function is not Riemann integrable.

Verify whether the function $f$ from $[0,2]$ to $\mathbb{R}$ defined by $$f(x) =\begin{cases} x+x^2, & x\in[0,2]\cap\mathbb{Q} \\ x^2 + x^3, & x\in[0,2]\setminus\mathbb Q \end{cases}$$ is ...
Sasikuttan's user avatar
0 votes
0 answers
46 views

No Riemann-integrable function has the harmonic series has Fourier coefficients

This is a question that was [already asked on this site][1] but got no satisfactory answer. I would like to rephrase and show my own attempt. So the point is, consider the sequence $a_k = \begin{cases}...
confusedTurtle's user avatar
0 votes
0 answers
26 views

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 ...
Alessandro Vagni's user avatar
1 vote
1 answer
54 views

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 $g$ ...
gon arencibia's user avatar
0 votes
0 answers
20 views

From the view of fourier series and the completion of space of Riemann integrable functions, how to understand the limitations of Riemann integration

Recently I am studying real analysis, when I read about the introduction part of stein's 《real analysis》, I get really confused about the first example he gave in page xvi. From where we can see ...
叶波特's user avatar

1
2 3 4 5
53