Mathematics Stack Exchange
Mathematics Stack Exchange

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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integration of 1/x as a riemann sum

To integrate $x^\alpha$ when $\alpha\neq1$ we subdivide the interval [a,b] by the point of geometric progression: $$a, aq, aq^2, \ldots, aq^{n-1}, aq^n=b$$ where $q=\sqrt[n]{b/a}$. We then only need to evaluate the sum of geometric series. Given the…
Stenio Ramos
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Riemann Sums - Prove $\lim_{n\to\infty} \frac{1}{n} \sqrt[n]{n!} = 1$

This is about a homework I have to do. I don't want the straight answer, just the hint that may help me start on this. To give you context, we worked on series, and we're now studying integrals, linking the two with Riemann sums. Now here is the…
Furrane
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Help! Converting a Riemann Sum to a Definite Integral

Can someone please explain how to convert this into a definite integral in the form $$\lim _{ n\rightarrow \infty }{ \left\{ \ln { \sqrt [ n ]{ \left( n+1 \right) \left( n+2 \right) \left( n+3 \right) ...\left( 2n \right) } -n\ln { \sqrt [ n ]{ n…
ZellAllon
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Showing that $f(x)=1$ if $x=\frac{1}{n}$, $0$ otherwise on [0,1] is Riemann Integrable

I have to show that the following function $f:[0,1]\rightarrow\mathbb{R}$ is Riemann Integrable: $$f(x) = \left\{ \begin{array}{ll} 1 & \mbox{if } x = \frac{1}{n} \\ 0 & \mbox{otherwise} \end{array} \right.$$ For the upper and lower Riemann…
Gehaktmolen
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Double Integral using riemann sums

So there is $R=\left[-1,3\right]\times\left[0,2\right]$. I have to use a Riemann sum with m=4,n=2 to estimate the value of double integral $\int \int(y^2-2x^2)\ \mathrm{d}A$, taking the sample points to be the upper left corners of the rectangles.…
Jermaine Law
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Evaluating this Riemann sum

$$\lim_{x\rightarrow\infty}\sum_{k=1}^{n}\frac{6(k+1)^2}{n^3}\sqrt{1+\frac{2(k+1)^3}{n^3}}$$ How do we determine if it is a Right/Left or midpoint Riemann Sum? How do we find the values a and b? Is $$f(x)=\frac{3x^2}{2}\sqrt{1+\frac{x^3}{4}}$$…
user902961
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Riemann sum limit

Can this limit be solved with Riemann sums?: $$\lim_{n\to\infty}\left( n-\sum_{k=1}^ne^{\frac{k}{n^2}}\right) $$ Tried solving it like this: $$\lim_{n\to\infty}n\left(1-\frac{1}{n}\sum_{k=1}^ne^{\frac{k}{n^2}}\right)$$ and after integrating the…
Lola
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Area under curve, infinite rectangles

I'm trying to calculate the area under the curve of $ y=x^2 $ between $ x=1 $ and $x = 3$ and above $y=0$ using the sum of infinitely many rectangles. So far I've tackled it by first defining the width of every rectangle to be $\Delta x =…
SmhConfused
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Find the area below $y=\sqrt{4-x^2}$ and above the x-axis between $x=0$ and $x=2$

I'm trying to figure out how to go about this, but I'm not entirely sure. Is there a specific way to find the area under a curve or line (and strictly above the x-axis) before doing Riemann sums?
user10984587
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Prove for every Riemann mid sum of $f(x)=x$ of $[0,b]$ is the integral $\int_{0}^{b}f(x)dx$?

I found that Riemann mid sum for arbitrary partition is $S(P, f)=\dfrac{1}{2}\sum_{i=1}^n(x_i^2-x_{i-1}^2)=\dfrac{b^2}{2}$, and now I have to show that Riemann sum approaches this value, i.e. chosen points and partition doesn't affect as long norm…
Law Neutral
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Composition of convex and continuous function

Let $f: [0,1] \rightarrow \mathbb{R}$ be continuous non-negative function and let $g$ be convex function on $[0,\infty)$. Show that $(g \, o \, f )$ is Riemann Integrable on $[0,1]$.
math123
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Riemann Sums for exponential functions

I had a question about Riemann Sums for exponential functions. Function in question is: $e^{-x^2}$ [More commonly known as the Gaussian Integral.] The integral from from 0 to 1 (definite integral with the upper limit 1 and the lower limit 0) Using…
Ali Imtiaz
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Counter-Example for Darboux Sums: "Finer" Partition with Greater Difference.

Let $P = \{p_i\}_{i= 1, n}$, $P' = \{p'_j\}_{j = 1, m}$ be partitions of an interval with max$|p'_j| \le $min$|p_i|$, i.e. all the sub-intervals of $P'$ are at least as short as all the sub-intervals of $P$. Let $f$ be a bounded function on the…
Tom Collinge
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Finding upper bound of Riemann's sum

I have $$f(x) = -x^2 + 2x + 5,\ x \in [1, 3]$$ Since $f(1) = 6$ and $f(3) = 2$, $f$ is decreasing. $$\overline{S_n} = \sum_{i=0}^{n-1} f(\frac{i}{n})\Delta x_i,\ \Delta x_i = \frac{b - a}{n} = \frac{2}{n}$$ $$\overline{S_n} = \sum_{i=0}^{n-1}…
Winter
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Does the tag in Riemann-sums have to be between $x_{i-1}$ and $x_i$?

Context: As I understand this limit, it can be interpreted as a Riemann Sum with tag $c_i=\frac{(2i-1)}{n}$. $$\lim_{n\to\infty}\sum_{i=1}^n \frac{2}{n}\left(1+\frac{2i-1}{n}\right)^\frac{1}{3}$$ Therefore it…
GambitSquared
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