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Questions tagged [riemann-sum]

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

The Riemann sum is an approximation that is calculated by dividing the region you are working in into shapes. These shapes form a smaller region (similar to the one you are measuring) and then calculating the area of these smaller shapes. Then you add all these small areas together to give the approximation.

It was considered the foundation of integration until the introduction of the much more rigorous Lebesgue integral in 1904.

1443 questions
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Evaluate $\lim_{N \to \infty}$ $\sum_{k=1}^N$ $k^3 \over N^4$ (Riemann sum)

Given, $\sum_{k=1}^N $$k^3 \over N^4$, I have to evaluate $\lim_{N \to \infty}$ $\sum_{k=1}^N$ $k^3 \over N^4$ with the help of the Riemann sum. Approach Assume $a_N$ $:=$ $1 \over N$, $x_k$ $:=$ $k \over N$, $f(x_k)$ $:=$ $k^3 \over N^4$. Then, I…
Julian
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Is Riemann Sum increasing if $f$ is continuous, decreasing and concave?

$$f: [0,1] \rightarrow \Bbb{R} \;\; \text{(the set of real numbers).}$$ $$f(0) = 1\;, \;\;f(1) = 0$$ $$f \;\text{is continuous, decreasing and concave.}$$ $$ S_n = \sum_{i=1}^n f\left(\frac{i}{n}\right)\frac{1}{n}.$$ Question: Is the sequence…
acamath 72
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Sum of $\sum_{k=1}^n\sin{k\theta}$

I have to calculate the sum of this series $$ \sum_{k=1}^n\sin{k\theta} $$ I tried solving it like this $$ \sum_{k=1}^n\sin{k\theta} = \operatorname{Im}\sum_{k=1}^ne^{(i\theta)k} $$ I recognized it as a geometric sum, but to get the sum I first had…
JamMaster
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Proving that a function is Riemann non-integrable

Let $f\colon[a,b]\to\mathbb{R}$ be a bounded function, and let $\mathcal{P}_{n}$ be a sequence of partitions (not necessarily evenly spaced) with mesh $m(\mathcal{P}_{n}) \to 0$ as $n\to \infty$. Then…
user2850514
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Riemann integral

For a function $f$ to be Riemann integrable on an interval $[a, b]$ does $f$ have to be continuous for all $x \in [a, b]$? Also does this function have to be vertically bounded?
user2850514
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If $f$ is continuous on $I$, prove $f$ is Darboux integrable on $I$.

We are given: Suppose f is a real-valued function defined on the closed and bounded interval $I = [a, b] \subset \Bbb R$. And on the previous problem we proved that for each $\varepsilon > 0$, if there exists a subdivision $\delta$ of $I$ such that…
Tacosi
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Is this a valid property for Riemann Sums

$R$ is a refinement of $Q$ then $$U(R, f) − L(R, f) ≤ U(Q, f) − L(Q, f)$$
Rahul Silva
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Convert Riemann Sum into definite integral

I have the equation $$\lim_{n\to\infty}\frac{\left(r-1\right)}{n}\sum_{j=r}^{n}\frac{n}{j-1}\left(\frac{1}{n}\right)$$ which is converted to a definite integral using x as the limit of (r-1)/n, t for (i-1)/n and dt for 1/n $${x\int _{x}^{1}{\frac…
Justin
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Diagram for the Riemann Sum

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 the right Riemann sum, but it was the wrong…
user30200
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Reimann Sums rectangle height location

My math teacher was teaching us Riemann Sums a few days back, and how if you estimate the height of the rectangle to be at the upper/lower, as n approaches infinity, the area becomes exact. But what I don't understand is why uppers/lowers matter at…
Shreyas Shridharan
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The Area under the curve $\sqrt{x}$ from $1\le x\le 4$ using no integrals but using riemann sums

I tried to solve this problem by using summation with about $1000$ rectangles and then kept adding more until about $100000$ rectangles. I got closer and closer to $\frac{14}{3}$ and my closest answer was $4.66668166$ . Is it safe to assume that if…
weareallin
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Finding a limit using Riemann sum

In the interval [0,1] I have to find the limit of a Riemann sum $$\lim _{n\to \infty }\sum _{i=1}^n\left(\frac{i^4}{n^5}+\frac{i}{n^2}\right)$$ so far I have this $$\lim _{n\to \infty }\sum…
Titor
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Calculate performance of deleting each first element from vector, calculate sum

My friend told me that deletion of all the elements from vector in big O notation is quadratic performance is worst case (big O notation). The worst case requires to delete always the first object, and all other elements are shifted to the left by…
Oleg
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Conversion of summation to integral

Consider an integer with $N$ digits (in decimal notation, without leading zeroes) $D_1, D_2, D_3, \ldots , D_N$. Here, $D_1$ is the most significant digit and $D_N$ the least significant. The weight of this integer is defined as $$\sum_{i=2}^N (D_i…
Java
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Calculate $L(f,P)$ and $U(f,P)$ according to the darboux definition of them

Calculate $L(f,P)$ and $U(f,P)$ for the following: $$f(x) = 2x, x \in [0,1];~ P = \{0,\frac{1}{4},\frac{1}{2},1\}.$$ By definition I know that $$L(f,P) = \sum_{i=1}^{n} m_i(x_{i}-x_{i-1}) \text{ and } U(f,P) = \sum_{i=1}^{n} M_i(x_{i} -…
user349557
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