# Questions tagged [riemann-sum]

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

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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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### 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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### 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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### 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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### 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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### $u_n(x)= \prod_{k=1}^{n} \big(1 +\frac{x}{n} f(\frac{k}{n}) \big)$ : 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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### 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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### 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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### 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 ...