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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.

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How to solve the following Riemann sum

Could someone explain step by step how to solve this Riemann sum: $$\frac{1}{n} \sum_{i=0}^n\left(\frac{k}{n}\right)^2\frac{1}{8}\arcsin\frac{k}{n}\frac{1}{2} $$ k/n inside the sum is all squared. I don't know what to do with that 1/2 and 1/8. I…
Lola
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True or false question about riemann

If there exists partition $P$ in interval $[a, b]$, $P = {x_0, x_1, ... x_n}$ s. t. $U(f, P) > 0$, then the upper integral is $>0$ It's part of the three true or false questions, I've done the other 2 but here I can't think of a counterexample, and…
repulsive23
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$\lim_{N \to \infty} \frac{4}{\pi} \sum_{ k > 0, \ k \text{ odd }}^{N} \frac{\sin({\frac{k\pi}{N+1}})}{k} = $ Gibbs konstant

Show that $$\lim_{N \to \infty} \frac{4}{\pi} \sum_{ k > 0, \ k \text{ odd }}^{N} \frac{\sin({\frac{k\pi}{N+1}})}{k} = \frac{2}{\pi}\int_{0}^{\pi} \frac{\sin (x)}{x} dx$$ I thought that I write the left hand side as a Riemann sum. $$\frac{4}{\pi}…
Olba12
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Show that for any $\epsilon$ there exists $n$ so that $U(f,P_n)-L(f,P_n)<\epsilon$

I have the upper and lower Riemann sums of the function $x^2$ on the interval $[1,b]$ with the partition $P_n: x_0=1, x_1=b^{1/n},...x_n=b^{n/n}=b$ for every positive integer $n$. The sums are equal to $U(f,P_n)=b^{(2/n)} (b^{(1/n)}-1) …
Espen van der meeren
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Do definite Riemann integrals have a uniqe Riemann sums?

Motivation for asking this question came from observing number of times some limit sums are turned into integrals. Are there some limit sums that are not Riemann sum yet are equivalent to some integral? Also is it possible to have two different…
jimjim
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Show Riemann sums satisfy $|f(b) - f(a)| * ||P||$

Suppose $f$ is monotonically decreasing on interval [a,b]. For any partition $P = (a = x_0, x_1, ..., x_n = b)$ of [a,b] let $P_L$ denote the tagged partition with tags chosen to be the left endpoints and $P_R$ the tagged partition with tags chose…
Ant
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Estimate the area under $f(x) = \frac{1}{2}x^2 - 1$ from $x = 0$ to $4$ using two right hand rectangles.

Help with beginner college calc plz: I can't upload pictures on my computer (it's a very old desktop) but I'll try and explain what I did. So obviously I started with graphing f(x); (2, 1) (-2, 1) (0, -1) (3, 4) (-3, 4) Then I drew my rectangles.…
user165349
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What are the requirements of a function so that the left Riemann sum equals the right Riemann sum?

My homework question in particular specifies over an interval of [0,1], the function is negative, and the function is decreasing.
Jeremy Downey
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