Questions tagged [riemann-sum]

This tag is for questions about Riemann sums and Darboux sums.

Filter by
Sorted by
Tagged with
0 votes
0 answers
11 views

Riemann Sum Problem Explanation f(x)=mx on left endpoints using xk

I am learning Riemann when I encountered this question and its solution. Question A curve f(x)=mx in closed interval [a,b] where m>0 and a>=0. Calculate riemann sum of f(x) using xk as left ...
user avatar
1 vote
1 answer
33 views

Using Riemann sums to approximate the second antiderivative

I’m currently working on a coding project where I’m given the the net force acting on an object at any time $t$ (meaning I essentially have its acceleration). I know the object’s current position and ...
user avatar
  • 111
0 votes
1 answer
22 views

Converting Riemann Sum to Definite Integral with Unequal $\Delta x$ Values

How can I convert this Riemann sum to a definite integral? $$\lim_\limits{n\to\infty}\sum_{i=1}^n\pi\biggl(1.6875+\frac{.75775i}{n}\biggl)^2\frac{1.625}{n}$$ I'm confused because the usual definition ...
user avatar
-1 votes
1 answer
42 views

Let $a_n$ be a series and $n_k$ be a Permutation, Prove that if $\lim_{n\rightarrow \infty}a_n = a$ ifff $\lim_{k\rightarrow \infty}a_{n_k} = a$ [closed]

Let $a_n$ be a series and $n_k$ be a Permutation, Prove that if $\lim_{n\rightarrow \infty}a_n = a$ if and only if $\lim_{k\rightarrow \infty}a_{n_k} = a$ At first, looking is really "looks like&...
user avatar
  • 111
0 votes
1 answer
33 views

The sum of integers from a to b represented as the area under a curve

I was trying to find out how to represent the sum of integers between two integers $a$ and $b$ as the area under a curve and this is the equation I came up with: $$\int_{-a}^bx+\frac{1}{2}dx$$ or $$\...
user avatar
3 votes
2 answers
95 views

Find integral of $\sqrt{x}$ using Riemann sum definition

Let $a > 1$ be a real number. Evaluate the definite integral \begin{equation} \int_{1}^{a} \sqrt{x} \,dx \end{equation} from the Riemann sum definition. My approach I know a Riemann sum consists of ...
user avatar
16 votes
1 answer
256 views

approximation of integral of $|\cos x|^p$

Let $p\in [1,2)$. Let $$ \beta = \frac{1}{2\pi}\int_0^{2\pi} |\cos x|^p\, dx = \frac{\Gamma(\frac{p+1}{2})}{\sqrt{\pi}\Gamma(1+\frac{p}{2})}. $$ Consider the following approximation to the integral ...
user avatar
  • 1,079
1 vote
1 answer
80 views

$\int_{-2}^xf(t)dt$ for $f(t) = \tiny\begin{cases} -1 \, &t<0 \\ 1 \, &t\ge 0 \end{cases} $, and its limit at $x=0$

Let $f: [-2,2] \to \mathbb R$, $$ f(t) = \begin{cases} -1 \, &t<0 \\ 1 \, &t\ge 0 \end{cases} $$ Define $g: [-2,2] \to \mathbb R$ as: $$g(x) = \int_{-2}^xf(t)dt$$ Plot $g(x)$ and find it'...
user avatar
  • 131
1 vote
2 answers
54 views

Is $f(x)=(\sin (1/x))^4$ Riemann integrable on $(0,1]$?

I have shown that $f(x)=\sin(1/x)$ is Riemann integrable on $(0,1]$, but I am wondering if $f(x)=(\sin (1/x))^4$ is Riemann integrable on $(0,1]$? It isn't hard to show that $\sin(1/x)$ is Riemann ...
user avatar
0 votes
1 answer
44 views

Continuous Factorial

I am working on a theory of quantum information and am unsure on some of the mathematical formalism I need. I have learned that integration can be thought of as summing up infinitely thin slices. My ...
user avatar
3 votes
4 answers
348 views

Does there exist such a Riemann integrable function?

Does there exist a Riemann integrable function $f: [a,b] \to \mathbb{R}$ that satisfies the following three criteria? $f(x) \geq 0$ for all $x \in [a,b]$ There exists an infinite set $E \subset [a, b]...
user avatar
3 votes
1 answer
56 views

How to prove that $g(x)=x^2$ is integrable on $[2,5]$ using regular partitions?

So I've been trying to prove that $g(x)=x^2$ is integrable on the interval $[2,5]$ using regular partitions and the theorem that a function is integrable if $$\lim_{n\to\infty}(U(f,P_n)-L(f,P_n)) = 0.$...
user avatar
1 vote
1 answer
32 views

So.. what exactly is Partition of an Interval

I have been researching about Partition of an Interval, and I'm quite confused. Some articles(Peoples) say Partition of $[a,b]$ is a finite sequence of $ a = x_0 < x_1 < \cdot\cdot\cdot < x_n=...
user avatar
0 votes
0 answers
16 views

do the upper and lower darboux sums of a function change depending on the norm(mesh) of the partition?

if we have two partitions of the interval [0,1] p1 and p2 so that the norm of p1 is greater than the norm of p2, then does that mean that U(f,p1) > U(f,p2) ?
user avatar
1 vote
1 answer
40 views

Finding the limits while changing limit of an infinite sum into integral.

I was solving the following question. Find the following limit. $$\lim_{n\to \infty}\dfrac1n \left(\dfrac{1}{1 + \sin\left(\dfrac{\pi}{2n}\right)} + \dfrac{1}{1 + \sin\left(\dfrac{2\pi}{2n}\right)} + ...
user avatar
0 votes
2 answers
75 views

Express limit of sum in terms of definite integral

Evaluate the limit by expressing it as a definite integral: \begin{equation} \lim_{n \to \infty} \sum_{k=n+1}^{2n-1} \frac{n}{n^2+k^2} \end{equation} I'm really confused about tackling this. Although ...
user avatar
-1 votes
1 answer
28 views

Representing the area of a circle, with radius $1$ as the sum of inscribed circumferences $\lim_{n\to\infty} \sum_{k=1}^n \frac{k}{n}\cdot 2\pi$ [closed]

I saw that a similar question had been asked, but mine is specifically about what is wrong with this sum representation. So, if we were to imagine a circle of radius 1, that radius can be divided into ...
user avatar
0 votes
0 answers
79 views

Area under a parabolic arc using arithmetic substitution

I'm currently studying Calculus from the book "Introduction to Calculus and Analysis" by Richard Courant and Fritz John. I started reading the chapter on Integration, and I stumbled upon the ...
user avatar
1 vote
0 answers
28 views

Proof a piecewise function is darboux integrable on $[-1,3]$

Given $$g(x)=\begin{cases} -2 &-1\leq x\lt 0\\ 1 & x=0 \\ 2 & 0 \lt x \leq 3 \end{cases} $$ I need to prove $g$ is Darboux integrable on $[-1,3]$. When I read Bartle's book, I feel like ...
user avatar
  • 55
0 votes
0 answers
30 views

Which stepsize to choose for Riemann integral approximation via a sum over all grid points?

I want to numerically calculate some integral over some function $f$ which I can sample at grid points $x_i$ from the interval $[a,b]$. For simplicity, let's say I want to sample $N$ equidistant ...
user avatar
4 votes
1 answer
111 views

How to take limits of 'almost Riemann' sums like $\lim_{n \to \infty} \sum_{k=0}^n \frac{1}{n} \cos (a \pi k \log(n)/n)$

How can I solve limits of sums that are 'almost' Riemann, but can't be written in the typical form (i.e, $\lim_{n \to \infty} \sum_{k=0}^n \frac{1}{n} f(k/n)$ which we can rewrite as an integral $\...
user avatar
0 votes
0 answers
26 views

Help with Proof of theorem about Riemann Integrability Criteria

To prove the Riemann Integrability Criteria: A bounded function $f$ on the interval [$a, b$] is Riemann integrable iff given an $\epsilon$ > 0 we may determine a positive number $\delta$ so that $\...
user avatar
  • 137
-1 votes
1 answer
38 views

Proof that a function is Riemann integrable by using specific hint. [closed]

I am reading the book A garden of integrals and I came across this exercise: Suppose that $f(x)=x^2, 0\leq x \leq 1.$ Show that $f$ is Riemann integrable on [$0,1$]. Hint: $\sum_P(x_k^2-x_{k-1}^2)\...
user avatar
  • 137
0 votes
0 answers
24 views

Limit of partial Harmonic Sum as a bounded integral

Can anyone explain why $(b,a)=(1,0)$ where $b-a=1$. $\lim_{n\to \infty}(\frac{1}{n+1} +\frac{1}{n+2} +...+\frac{1}{n+n})=\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n \frac{1}{1+i/n}=\int_a^b \frac{dx}{1+...
user avatar
1 vote
1 answer
38 views

Distance between two Riemann-integrable functions [closed]

I have $RI[a,b]$ as the set of all Riemann Integrable functions and $d_1(h_\alpha, h_\beta)$ where $$h_\alpha(t)= \begin{cases} \alpha & t\in\{a,b\} \\ 0 & a<t<b \end{cases}$$ Through ...
user avatar
0 votes
1 answer
66 views

Regarding Theorem 1.9, Conway, complex Analysis

I have a doubt in equation (1.11) in Chapter 4, Section 1 of J.B Conway’s, functions of one complex variable. I know that the estimation of the integral in equation 1.10, comes from the previous ...
user avatar
0 votes
0 answers
238 views

Riemann sum approximation answer

Use the Riemann sum with $n=2$ rectangles to approximate $ \int_{0}^{1} \frac1{x^4+1} \,dx $. Round to $3$ decimal places. Options are 0.867 0.721 0.971 1.441 When I take right Riemann rectangle ...
user avatar
0 votes
1 answer
46 views

Find $\int_{-1}^{2}(5x^3+7x^2-9x+4)dx$ by Riemann sums

I want to determine this integral $\int_{-1}^{2}(5x^3+7x^2-9x+4)dx$ by Riemann sums. Clearly we can take $$\Delta x=\frac{2-(-1)}{n}$$ And $$x_i=-1+\frac{i3}{n}$$ Then $$\int_{-1}^{2}(5x^3+7x^2- 9x+4)...
user avatar
  • 631
0 votes
0 answers
79 views

Riemann sum simplifying for the integral of an exponential function

I am studying for a calculus class(using Stewart) and this question came up in a list of exercises (from my university): Question $\rightarrow$ Find $\int_{a}^{b}e^xdx$ I reached $$\lim_{n\to\infty}...
user avatar
0 votes
0 answers
59 views

Change of variables for the Riemann sum?

Let $f :[0,1] \to \mathbb{R}$ be a Riemann integrable function (or any smooth function will be okay as well). Then, I am a bit confused about limit of the sum of the following form: \begin{equation} \...
user avatar
  • 5,360
8 votes
1 answer
144 views

When would we want to use uneven subintervals in a Riemann integral?

The formal definition of a Riemann Integral is written such that you can have uneven subintervals and it still works. Why do we need to generalize to the case of uneven subintervals? Why not insist ...
user avatar
  • 853
1 vote
0 answers
45 views

Convolution integral over non-uniform grids

Statement of the problem: The convolution of two functions is defined by: \begin{eqnarray} (f*g)(y)=\int_{-\infty}^{+\infty}dx f(x)g(y-x), \end{eqnarray} for functions $f(x),g(x)$ defined over the ...
user avatar
-3 votes
2 answers
94 views

Conventionally $\int_2^0 f(x)dx:=-\int_0^2 f(x)dx$ whereas $\sum_{i=2}^0 := 0$ Why are definite integrals and series treated differently?

Integrals are defined in terms of series, so why is the treatment of definite integrals different than the treatment of series when the lower limit is greater than the upper limit. For a definite ...
user avatar
  • 1,059
0 votes
1 answer
54 views

$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f(\frac{k+\theta}{n})=\int_{0}^{1}f(x)dx$ where $\theta\in(0,1)$?

Let $f\colon [0,1]\to\mathbb{R}$ be a continuous (or Riemann-integrable) function. As we already know, the next equation holds: $$ \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f\left(\frac{k}{n}\right)=...
user avatar
0 votes
1 answer
39 views

$\lim_{n \to \infty} {\frac{1}{n^2}\sum_{k=0}^{n}{\frac{1}{\ln{(1 + \frac{(n+k)\sqrt{n^2+k^2}}{n^3})}}}}$

How can I solve $$\lim_{n \to \infty} {\frac{1}{n^2}\sum_{k=0}^{n}{\frac{1}{\ln{(1 + \frac{(n+k)\sqrt{n^2+k^2}}{n^3})}}}}$$ It looks like a Riemann limit to me, but I'm not able to get it to a final ...
user avatar
  • 1,856
0 votes
1 answer
66 views

error bound for midpoint rule - exact error

I noticed that I get the exact error, using midpoint rule error bound formula, but with $f''(\frac{b-a}{2})$ for $K$, i.e. : $E_m \leq $ $\frac{K(b-a)^3} {24n^2}$ $E_{m2} = $ $\frac{f''(\frac{b-a}{2} )...
user avatar
  • 105
1 vote
1 answer
58 views

Derivation of integral version of arc length formula

I apologize in advance if this question is too vague, and I am happy to try to clarify anything. I'd like to show that one can go from the definition of the length of a curve (as the least upper bound ...
user avatar
  • 355
0 votes
1 answer
56 views

Uniqueness of Riemann Sum to Definite Integral

My child recently was asked a question in Calculus I on the conversion Riemann sums to their equivalent definite integrals. This got me wondering whether this conversion is in general unique. Given, ...
user avatar
  • 110
1 vote
2 answers
70 views

Showing that surface area is equivalent to $\int_{S}\|\partial_u\phi\times\partial_v\phi\|dudv$, and is there MVT for bijections: $\Bbb R\to\Bbb R^2$?

$\newcommand{\d}{\,\mathrm{d}}$It can be shown that arclength, considered as a sum of increasingly fine partitions of the graph, approaches the integral formulation. However, I have only ever seen the ...
user avatar
  • 8,561
0 votes
1 answer
48 views

Question on Riemann Integrals

The upper Riemann integral is defined as $\int_a^{-b}f(x)dx = \inf U(P,f)$, and the lower Riemann integral is defined as $\int_{-a}^{b}f(x)dx = \sup L(P,f)$, where $U$ and $L$ denote the upper and ...
user avatar
6 votes
1 answer
148 views

Is the set of the upper Riemann sums convex for an arbitrary bounded function $f(x)$?

If $f$ is bounded on $[a,b]$ and $P= (x_0,x_1.\cdots,x_n)$ is a partition of $[a,b]$, let $M_j = \sup_{x_{j-1}\le x \le x_j}\text{$f(x)$}$. The upper Riemann sums of $f$ over $P$ is $S(P)= \sum_{j=1}^{...
user avatar
  • 1,129
0 votes
0 answers
75 views

The Riemann integral of a non-negative function

Suppose $g : [−1,1] \to\mathbb R$ is Riemann integrable on $[−1,1], g(x) ≥ 0$ for all $x ∈ [−1,1]$, and $g(0) > 0$. Does it follow that $\int_{-1}^{1} g > 0$? I tried proving that it is true by ...
user avatar
  • 191
0 votes
0 answers
126 views

Validity of the Riemann series theorem

A classic example of the Riemann series theorem use the alternating harmonic series $\sum_{n=1}^k \frac{(-1)^{n+1}}{n}$ that converges to $\ln(2)$ when $k \rightarrow \infty $ and rearrange it to $\...
user avatar
  • 109
3 votes
1 answer
84 views

Integral-sum conversion in the proof that the $\chi^2$ statistic tends to the $\chi^2$ distribution

I cite the derivation on Wikipedia, here. The focus of that section of the article is to show that the $\chi_p^2$ statistic is asymptotically equivalent to the $\chi^2$ distribution. Let $n$ be the ...
user avatar
  • 8,561
0 votes
0 answers
60 views

Question about darboux integral sums and inequality.

I am working through part of my book which deals with area of curvilinear trapezoid. I am getting this inequality $s\leq P_1 \leq P_2 \leq S$ where $s=\sum_{n=1}^n m_i \Delta x_i$ $S=\sum_{n=1}^n M_i \...
user avatar
  • 772
1 vote
1 answer
39 views

Double integral of two continuous functions

Let $R$ an elemental rectangle in $\mathbb{R}^{2}$ and $g, f: R \rightarrow \mathbb{R}$ continuous functions. Show that $$ \lim _{d(P) \rightarrow 0} \sum_{i, j} f\left(u_{i j}\right) g\left(v_{i j}\...
user avatar
  • 58
0 votes
0 answers
15 views

General Riemann Sum difference when subintervals are even

The function is $k^2/2$ on $[-2, 2]$ I'm trying to find $U_n - L_n$ when the number of subintervals is even. In general, $|overestimate - underestimate| = |f(x_n) - f(x_o)|\Delta x$. This is true if ...
user avatar
  • 19
1 vote
0 answers
41 views

Show Riemann-integrability of discontinuous function

Let's assume that $f:[a,b]\to\mathbb{R}$ is bounded, i.e. $\Vert f\Vert_{\infty}\leq M$ and that is has countable infinitely many points where it is discontinuous. Show that $f$ is Riemann-integrable ...
user avatar
  • 3,061
0 votes
1 answer
60 views

Expected Value Explanation

A textbook I'm reading from has the following text explaining the concept of Expected Value of a continuous random variable: Suppose the weight of a bullock on a cattle ranch is described by the ...
user avatar
1 vote
1 answer
97 views

Prove that a bounded $f$ is integrable if $I_0 := \lim_{n\to\infty}L(f,P_n) = \lim_{n\to\infty}U(f,P_n)$

Prove that a bounded function $f$ is integrable on $[0,1]$ if $$I_0 := \lim_{n\to\infty}L(f,P_n) = \lim_{n\to\infty}U(f,P_n),$$ in which case $\int_0^1f(x)dx$ equals $I_0$. Refer here. I suspect that ...
user avatar
  • 746

1
2 3 4 5
27