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

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How to switch from continuous to discrete formulation (integral to sum) in a specific case?

in the paper "Economic conditions and the popularity of parties: a survey" Kirchgaessner (1986) transforms a utility function from continuous to discrete. I get the intuition and the meaning,... user avatar george1994

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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 Van Wilder

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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 User

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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 Alex Paschal

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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 Mustafa

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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 carbon tetrachloride

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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 user58955

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$\int_{-2}^xf(t)dt$ 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 Dayne

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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 Clyde Kertzer

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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 moboDawn_φ

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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 Clyde Kertzer

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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 Peverel Shipley

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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 John Doe

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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 leen sakh

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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 user983206
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