# Questions tagged [riemann-sum]

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

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### Showing integrability of f+g and additivity of the Darboux integral

I am currently working on the following question from Measure, Integration & Real Analysis by Sheldon Axler: Suppose $f,g:[a,b]\to\mathbb{R}$ are Riemann (Darboux) integrable on $[a,b]$. Prove ...
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### Conjecture: The $n$th left Riemann sum for $\int_0^1 (x-x^2)^k dx$ is $B(k+1,k+1) + \Theta(n^{-2 \lceil (k+1)/2\rceil})$

While playing around on Wolfram Alpha with integrals of the form $\int_0^1 (x-x^2)^k dx$, where it happens that Wolfram Alpha displays formulas for the $n$th left Riemann sum (using equally spaced ...
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### Limit of $∞.0$ form of an integral and Riemann sum

I was pondering how to solve a limit of the kind $$\lim_{n \to ∞} n^k \left(\dfrac{1}{n}\sum_{k=0}^{n-1} f \left(\dfrac{k}{n} \right) -\int_0^1 f(x)dx \right)$$ where k is chosen such that the order ...
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### Why is the following claim true in Exercise 7.4, Apostol's Mathematical Analysis?

On this page, a proof of the equivalence of two definitions of Riemann integrals is given by the user Pedro using Apostol's Hint for Exercise 7.4, Mathematical Analysis. However, I still find this ...
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### Doubt about Rudin exercise 6.3 a

Define three functions $B_1, B_2, B_3$ as follows: $B_j(x) = 0$ if $x < 0$, $B_j(x) = 1$ if $x > O$ for $j = 1, 2, 3$; and $B_1(0) = 0, B_2(0) =1, B_3(0) = \frac{1}{2}$. Let $f$ be a bounded ...
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### prove that $\int_{a}^{c} f(x) \,dx = \int_{a}^{b} f(x) \,dx + \int_{b}^{c} f(x) \,dx$

I've been working on a proof related to the additivity of Riemann integrals and would greatly appreciate insights and feedback for clarity and correctness of the proof. Because i've never seen a text, ...
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### Where does the differential come from when passing from a discrete sum to a continuous case, for instance for the case of the inner product?

I'm trying to develop some intuition behind the inner product. I understand for two n-dimensional vectors we can compute the dot product as: $\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i$ So ...
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### Calculate $\int_0^1 x^2~dx$ only by using upper-/lower Darboux sums

Let be $f:[0,1]\to\mathbb{R}$ with $f(x)=x^2$. We know that $f$ is Riemann-integrable. Calculate $\int\limits_0^1 x^2~dx$ only by using upper-/lower Darboux sums. Let be $P$ a partition of $[0,1]$. ...
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### Infimum of the Upper Sums [closed]

Let $P$ be the partition of the interval $[a,b]$. If $P$ is divided into two partitions $P_1$ and $P_2$ such that $P_1$ covers the interval $[a,c]$ and $P_2$ covers the interval $[c,b]$, then \$U(P, f, ...