Questions tagged [riemann-sum]

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

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Riemann sum calculations

After writing an integral as a limit of a Riemann sum, how do we actually calculate the integral? It seems that generally, we're in some form that isn't simplified. For example, take $$\int_0^3e^xdx=e^...
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On trigonometric Riemann sums

Question Use Riemann sums to evaluate the integral $$\int_0^{\frac {\pi} {3}} \sin x\ \mathrm{d}x\ .$$ Hint Use the fact that $$\sum_{i = 1}^n \sin (i\theta) = \frac {\sin \frac {(n + 1)\theta} {2} \...
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Is this Integration From First Principles notation standard outside of A Level Maths?

I teach A Level Maths in the UK. We are required to do some 'introduction' to integral from first principles as part of the specification (link, page 25 is the interesting part). In a previous exam ...
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Finding bacteria population using right Riemann sum

Find the number of bacteria after $12$ hours if the population increases at a rate of $f(t)=e^{0.15t^2}$ million bacteria per hour, using a right-hand sum with $\Delta t=4$. So I made the following ...
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Proving the Montonicity Properties of Lower and Upper Sums

Here's what I'm trying to prove. Let $f: [a,b] \to \mathbb{R}$ be a bounded function. Let $P$ and $Q$ be partitions of $[a,b]$ such that $P \subseteq Q$ ($Q$ is a refinement of $P$). Then: $$L(f,P) \...
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1answer
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How to evaluate Riemann Sums where x is an exponent.

I just need to know how I would go about evaluating this: $$ \sum_{t=0}^{n-1} v^t\frac{(1-v^t)}{i} = \frac{1}{i}\sum_{t=0}^{n-1} v^t(1-v^t) $$ $$ =\frac{1}{i}\sum_{t=0}^{n-1} v^t - \frac{1}{i}\...
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Approximation of the sum of a series $S(t)=-\frac{2}{\pi t} \cos(\frac{\pi t}{2}) \sum_{m\ odd}^{\infty}\frac{m^2\alpha_m}{t^2-m^2}$ as $t\to +\infty$

The function S(t) has the following infinite series form: \begin{align} S(t) &=\frac 2\pi \int_0^{\pi/2} dx \sin(tx){\sum_{m\ odd}^{\infty} \alpha_m \cos[m(\frac \pi2-x)]}\\ &=-\frac{2}{\pi t} ...
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Piecewise Gauss-Legendre quadrature order of convergence

Given a definite integral $\int_a^bf$, If we increase the number of nodes and weights of the Gaussian quadrature, we would get closer to the exact integral. But another way to get more exact is to ...
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How to compute infinite limits of summations (Riemann sums) when there are i's in the denominator?

I'm really lost here. I'm trying to use a right Riemann sum to compute: $$\int_0^1{\frac{x}{x^4+2x^2+1}dx}$$ Eventually I get here: $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n\frac{n^3i}{n^4+2n^2i^2+i^...
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Are all definite integrals calcuable by method of Riemann sums?

I'm Googling but not finding an answer. Are all calcuable definite integrals solvable using the method of Riemann sums? It seems like they should be. However, I have come across one that I can solve ...
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On Riemann sums for negative exponents

Let $0 < a < b$. Use Riemann sums to compute $$ \int_{a}^{b}x^{-2}\ \mathrm{d}x\ . $$ So far, I have gotten to the step where \begin{align} \int_{a}^{b} x^{-2}\ \mathrm{d}x & = \lim_{n \to \...
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Real analysis: if two Riemann integrable functions have equivalent lower sums for any partition, their integrals are the same on any subinterval.

I have to prove that if $f,g : [a,b]\longrightarrow \mathbb{R}$ are Riemann integrable functions such that $\underline{S}(f,P)=\underline{S}(g,P)$ for any partition $P$ of $[a,b]$, then $\int\limits_{...
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1answer
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Evaluating a definite integral as the limit of a Riemann sum

I'm having trouble evaluating the following problem using the limit of a Riemann sum: $\int_1^4x^2-4x+2dx$ Using $\lim_{n->\infty}\sum_{i=1}^n f(x_i)\Delta x$ where $a=1$ and $b=4$, $\Delta x = \...
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Finding an integral by evaluating a Riemann Sum

Below is a problem I did. I believe I got the right answer. However, I am not convinced my method is correct. I am hoping somebody here can verify that my solution is correct or tell me where I am ...
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Concentration inequality for $L^2$ functions

Let $\mathcal{X} \subset \mathbb{R}^d$ be a compact domain and $\mu$ be a finite measure defined on $\mathcal{X}$. For simplicity, we can assume that $\mathcal{X} = [0,1]^d$ and $\mu$ is the Lebesgue ...
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Q: Hoping that my logic is correct for testing integrability of a piecewise function

Consider the piecewise function $f$ bounded on $[-1,1]$ given by $$f(x):= \begin{cases} 1 \text{ if } x>0\\ 0 \text{ if } x\leq0\\ \end{cases} $$ Let $P=\left\{-1,-\frac{3}{4},-\frac{1}{2},-\...
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Evaluate this Logarithmic Sum

$\sum_{k=2}^n \log(\frac{k}{k+1})$ So far I have: $$\sum_{k=2}^n\log(k) -\log(k+1) \\ \text{Looking at this I can see that this may be a telescoping series.} \\ \sum_{k=2}^n\log(k) -\log(k+1) = [\log(...
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Quick Limit as Riemann Sum Question

So I was given the following question: "The sum $\sum_{k=1}^n h'(\frac{15}{n}(k-1))\frac{15}{n}$ is a left Riemann sum with $n$ sub intervals of equal length. The limit of this sum as $n$ goes to ...
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Exercise 25, Chapter 24 of Spivak's Calculus 3rd Edition

Theorem: Let $\{f_n\}$ be sequence of integrable functions on interval $I=[a,b]$ and $f$ be the uniform limit of $\{f_n\}$ on the interval, then prove that $f$ is integrable and $\int_a^b f=\lim_{n\to ...
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Diagram for the Riemann Sum [closed]

What is the answer for this? Not the evaluation part but the graph of the Riemann Sum. I originally thought it was the first/top left graph because I thought 'right endpoints' in the question meant ...
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Riemann sum doesn't agree with definite integral?

I'm given the following problem: Calculate the net area between $f(x) = x^3+8$ and the $x$ axis on the interval $[-1,1]$. I do so by finding the Riemann sum, then taking a limit. I've audited this ...
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What Are the $x_i^*,y_j^*$ in the Riemann Sum Definition of the Double Integral?

the double integral $$\iint \limits_{[a,b] \times [c,d]} f(x,y) \, dxdy$$ can be represented with a Riemann Sum as $$\lim_{(\Delta x , \Delta y) \to (0,0)} \sum_{i=1}^n\sum_{j=1}^mf(x_i^*,y_j^*)\Delta ...
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Is the sequence $\sum_{k=0}^n\frac{1}{e^k+\sqrt{k}+1} $ convergent or divergent?

I have been asked to find the limit of the following sequence $$\sum_{k=0}^n\frac{1}{e^k+\sqrt{k}+1}$$ when $ n $ goes to infinity. I tried to write it as a Riemann sum but the exponential term makes ...
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Evaluate : $\lim_{n\to \infty} \sum_{r=0}^{n} \frac{\binom nr}{(r+4)n^r}$

Question Evaluate : $$\lim_{n\to \infty} \sum_{r=0}^{n} \frac{\binom nr}{(r+4)n^r}$$ I have come across a similar question: Evalute $ \lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+...
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The Riemann sum of a divergent integral.

I know that $$ \int_0^1 \dfrac{dx}{x}=+\infty. $$ But when I develop its Riemann sum (with tags of the right), I get $$ \int_0^1 \dfrac{dx}{x}=\lim_{n\to+\infty}\dfrac{1}{n} \sum_{k=1}^n \dfrac{1}{k}=...
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Relating a geometric limit deriving the arc length of a polar curve to a Riemann sum

So recently one of my friends came to me with this problem You start with a circle of radius 1 which is divided into n subdivisions. As you go counterclockwise, you plot a point which is +(1/n) ...
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How does the riemann sum suffice for arbitrary regions?

On the ProofWiki page for the divergence theorem: It suffices to prove the theorem for rectangular prisms; the Riemann-sum nature of the triple integral then guarantees the theorem for arbitrary ...
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How I can evaluate $\int_{x_{i-1}}^{x_i}(x-x_{i-1})(x_i-x)dx $ with $ h=-x_{i-1}+x_i$?

My friend sent me the following problem for help , let $ h=-x_{i-1}+x_i$ , I have tried to evaluate this $\int_{x_{i-1}}^{x_i}(x-x_{i-1})(x_i-x)dx $ using Darboux integral which uses Darboux sum upper ...
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$f$ is 1-periodic prove $\lim_{n \to \infty} \int_{0}^{1} \sin^2(πx)f(nx)\,dx = \frac{1}{2} \int_{0}^{1} f(x)\,dx$

I came across this very nice Question so thought of sharing it! Let $f(x)$ be a continuous function $f:R \to R$ with period $1$. Prove that $\displaystyle \lim_{n \to \infty} \int_{0}^{1} \sin^2(\pi ...
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Can $\sum_{κ=0}^n \sin^3 (θ_2 + κdθ)$ be turned into a Riemann sum?

While trying to find the magnetic field over a solenoid I ended up with two answers that through desmos look to be approximately the same: $$\frac{1}{α}\sum_{κ=0}^n \sin^3 (θ_2 + κdθ)$$ where $α$ the ...
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1answer
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Equivalence of three definitions of Riemann integral for improper integrals.

I know three equivalent definitions for the Riemann integral. Let $f:[a,b] \to \mathbb R$ a bounded function. We say $f$ is Riemann integrable with integral $I$ if either of the three following ...
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How would I go about calculating $\int_{0}^{b}x^{3}$ using the Riemann Sum?

I was working on a question from Chapter 13 of Spivak's Calculus (Pg 266). It asked us to calculate $\int_{0}^{b}x^{3}$ using the Darboux sums. I was able to do this successfully, but then I got to ...
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1answer
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$u_n(x)= \prod_{k=1}^{n} \big(1 +\frac{x}{n} f(\frac{k}{n}) \big)$ : asymptotic

$f$ is continuous from $[0,1]$ to $\mathbb{R}$, $ f \ne 0$ $x \in [-A,A]$ , $A$ is fixed $u_n(x)= \prod_{k=1}^{n} \big(1 +\frac{x}{n} f(\frac{k}{n}) \big)$ $L=\int_{0}^{1} f(z) dz$ The goal of the ...
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129 views

Suppose $f:[0,1]\to\Bbb R$ is a continuous function and $\int_0^xf=\int_x^1f$ for all $x\in[0,1]$. Show that $f(x)=0$ for every $x\in[0,1]$.

Suppose $f : [0, 1] \to \mathbb{R}$ is a continuous function with the property that $$\int_{0}^{x}f = \int_{x}^{1}f $$ for all $x \in [0, 1]$. Show that $f(x) = 0$ for every $x \in [0, 1]$. I am new ...
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1answer
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Proving that a piecewise function is Darboux integrable on $[0,2]$ assistance

Let $f:[0,2] \rightarrow \mathbb{R}$ be given by $$f(x) = \left\{ \begin{array}\ 10,\quad \ 0\leq x < 1, \\ 100,\quad \ x = 1, \\ -5,\quad \ 1 < x \leq 2. \\ ...
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About a limit regarding Riemann sums [duplicate]

Let $f:[a,b] \to \mathbb{R}$ be a differentiable function such that $f'$ is Riemann-integrable. Let $A_n = \frac{b-a}{n} \cdot \sum\limits_{k=0}^{n-1} { f \left( a+k \frac{b-a}{n} \right)} \, , \...
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1answer
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Can someone please explain this integral property of odd functions [duplicate]

I came across this integral in a text $$\int_{-1}^{1} \frac{cosx}{e^{\frac{1}{x}}+1}.$$ This was the approach used in the text: Let $$g(x)= \int_{-1}^{1} \frac{\cos x}{e^{\frac{1}{x}}+1}$$ Then $g(x)$ ...
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1answer
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Definite integrals (Riemann sums) [closed]

can anyone please help with th is Saw the problem from schaums outline advanced calculus Converting that to the limit of a sum I got: The sum I got My question is how did that translate to The sum I ...
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1answer
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series for Riemann sum

Let $ f:[0,1] \rightarrow \mathbb{R}$ of class $C^{3}$. Show that: $$\frac{1}{n}\sum ^{n-1}_{k=0} f\Bigl(\frac{k}{n}\Bigr) =\int ^{1}_{0} f(t)\, dt-\frac{1}{2n}\int ^{1}_{0} f'( t)\, dt+\ \frac{1}{...
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Evaluate the given limit by recognizing it as a Riemann sum; question regarding interval of integration

Problem Find the limit : $\lim_{n\to \infty}\sqrt[n]{(1+1/n)(1+2/n)\cdot...\cdot(1+1/n)}$ which is same as problem solved here Evaluate the limit by first recognizing the sum as a Riemann Sum for a ...
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1answer
28 views

Showing that a function is Darboux Integrable using definition of integral

I have been given a function $f:[0, 4]→ \mathbb{R}$ such that $f(x) = 0$ for all $x \ne 2$ and $f(2) = 2$, and told to show that $f$ is Darboux integrable in $[0,4]$. However, I don't understand how a ...
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What is $\lim_{N\to\infty}\frac{-2}{\pi}\sum_{n=1}^N \frac{(-1)^n}{n} \sin(n\frac{N\pi}{N+1})$?

Here is what I have so far: $$\lim_{N\to\infty} f_N \left(\frac{N\pi}{N+1}\right)$$ $$f_N (x) = \frac{-2}{\pi}\sum_{n=1}^N \frac{(-1)^n}{n} \sin(nx)$$ $$\implies \lim_{N\to\infty} f_N \left( \frac{N\...
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1answer
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An argument in approximating Riemann-type sums: assuming without loss of generality a given partition is contained in all partitions.

I am trying to understand a simple argument from the proof of 17.6 below. So the proof proceeds by taking as $\Delta$ a common refinement of the partitions for the step processes $\tau$ and $c$. Then ...
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Convert $\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{4}{n}\sqrt{2+\frac{4k}{n}}$ into a definite integral.

$$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{4}{n}\sqrt{2+\frac{4k}{n}}$$ So going through this, I found out $a=2$ and $\Delta x = 4$, which got me $ \int_2^6 \sqrt{2+\frac {4k} n} dx $. If this is ...
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3answers
239 views

Show that $\lim\limits_{N\rightarrow\infty}\sum\limits_{n=1}^N\frac{1}{N+n}=\int\limits_1^2 \frac{dx}{x}=\ln(2)$

Show that: $$\lim\limits_{N\rightarrow\infty}\sum\limits_{n=1}^N\frac{1}{N+n}=\int\limits_1^2 \frac{dx}{x}=\ln(2)$$ My attempt: We build a Riemann sum with: $1=x_0<x_1<...<x_{N-1}<x_N=2$ $...
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2answers
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The closed form solution to the equation with sum of exponential functions

I have a function $\begin{equation}f(x)=\sum_{k=1}^{n}\left(\frac{1}{n}+\frac{ax}{k}\right)e^{-a(\frac{nx}{k}-b)}\end{equation}$, where $a,b$ are both positive constants, $n$ is a positive integer. $x^...
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3answers
127 views

Express the limit of a Riemann sum as a definite integral

$$\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^n\left(\frac{12}{n}+\frac{8i}{n^2}\right)$$ I'm having trouble understanding how to find my limits of integration here. If I factor out $\frac{1}{n}$...
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1answer
41 views

Riemann integrability criteria

Thinking back about limits and the original definition of the limit I thought that the Reimann integral (for some bound function $f$ in $[a,b]$) could be defined using limit-like definition. I found ...
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1answer
68 views

limit of a sum of a sequence [closed]

Compute the following limit or prove that it doesn't exist: $$ \lim _{n \rightarrow \infty}\left[\frac{1}{n} \sum_{k=1}^{n} \sin \left(\frac{\pi k}{2 n}\right)\right] $$ Solution Note that the ...
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2answers
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A question about general Riemann sums

My Calculus textbook spent a good chunk of the chapter deriving Left/Right Riemann sums, only to ditch them for general Riemann sums, which they never bother to derive or illustrate. I'm fairly ...

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