Questions tagged [riemann-sum]
This tag is for questions about Riemann sums and Darboux sums.
1,467 questions
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Work required to fill spherical tank at height
Suppose we have a spherical water tank $24$ feet in diameter, sitting atop a $60$ foot tower. The tank is filled by a hose attached to the bottom of the sphere. How long does it take a $1.5$ ...
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Darboux integral proof (lower and upper integral)
I have to show how $ \underline{I}_D(f) = -\overline{I}_D(-f) $
( where $ \underline{I}_D(f) $ is the Darboux lower integral of the function $ f $ over the set $D $, and $ \overline{I}_D(f) $ is the ...
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Why is Volume the Integral of Area?
Consider a solid $S$ situated on the $x$-axis in the interval $[a,b]$ with cross sectional area $A(x)$. You can find in many calculus textbooks, such as Stewart, that
$$
\text{Volume}(S) = \int_a^b\!A(...
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Using the Definition of the Riemann Integral to Calculate Area of Semi-Circle
I was hoping for some assistance in calculating the area of a unit semi-circle with the definition of the Riemann Integral. Using the function $f(x)=\sqrt{1-x^2}$, with end points $x=1, x=-1,$ I get ...
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Show that for all integers greater or equal to $1$, the alternating harmonic series can be written as $\lim_{n\to \infty}\sum_{k=1}^{n}\frac{1}{k+1}$ [duplicate]
The original question:
Show that $\forall \, n\in \mathbb{Z}, n\geq 1$ the equation below holds
$$\begin{align}
1-\frac12 + \frac13 - \frac14 + \cdots + \frac{1}{2n-1} - \frac{1}{2n} =& \frac{1}{n+...
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Why does a refinement of a tagged partitions make it's Riemann sum more accurate?
I have seen and understood the proof that how refinement increases the accuracy of the Riemann sum when we consider the lower sum $L(P,f)$ and upper sum $U(P,f)$. But I recently saw another definition ...
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Is it possible to have a conditionally convergent Riemann sum?
The standard formula for a Riemann sum of an integral is as follows:
$$\int_a^b f(x) dx = \lim_{n\to\infty} \sum_{i=1}^n f(x_i) \Delta x$$
where $\Delta x = \frac{b-a}{n}$ and $x_i = a+i\Delta x$.
...
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Show that if $f(x) = \int_0^x f(t) \, dt$ and $f(0) = 0$, then $f(x) = 0$ for all $x$ using Riemann sums.
I am trying to show that if $$ f(x) = \int_0^x f(t) \, dt $$ and $ f(0) = 0 $, then $ f(x) = 0 $ for all $ x $ in a neighborhood around $ 0 $.
It is given that $f$ is continuous on $R$.
I want to ...
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Convergence of upper and lower sums with uniform partition
Let $f\ $ be a Darboux-integrable function on $[a,b]$ and $P_n = \{ a, a + \frac{b-a}{n}, ..., a + (n-1)\frac{b-a}{n}, b \} $ a sequence of uniform partitions. Is there a easy way to prove that $\lim_{...
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$f$ integrable implies the existence of a continuous $g(x) \le f(x)$ where the difference of integrals is less than epsilon
Prove that if $f:[a,b]\to\mathbb{R}$ is integrable, $\forall\epsilon\gt0, \exists g:[a,b]\to\mathbb{R}$ such that g is continuous, $g(x)\leq f(x)$ $\forall x\in[a,b]$, and $\int_a^b{[f(x)-g(x)]dx}<\...
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Prove That $\int_{1}^{e}\frac{1}{t}dt=1$ Without the Use of Logarithms
I've been studying Real Analysis by Jay Cummings, and am working through the exercises on integration. The question is as such:
Define a function $L:(0, \infty)\rightarrow\mathbb{R}$ by $L(x)=\int_{1}^...
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Yet another question about Riemann sums
It is well known that if $f:]0,1]\to\mathbb{R}$ is monotonic and if the improper integral $\int_0^1f(t)\,dt$ converges, then :
$$\lim_{n\to+\infty}\frac1n\sum_{k=1}^nf\left(\frac kn\right)=\int_0^1f(t)...
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Asymptotic analysis of the finite product
I have the following product
$$\prod _{n=1}^{\frac{L}{2}} \frac{e^{-\frac{2 \cos \left(\frac{\pi n}{L+1}\right)}{T}}+1}{e^{-\frac{2 \cos \left(\frac{\pi n}{L}\right)}{T}}+1}$$
I am interested in ...
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Definition of Riemann integral using $\displaystyle\lim_{N\to +\infty} \sum_{i=1}^N f(x_i)(x_{i+1} - x_i)$
In this classical mechanics video, professor defines the integral of $f$ over $[a,b]$ as the limit as $N$ tends to infinity of the sum of the rectangles over a partition $\{x_1=a, x_2,..., x_{N-1}, ...
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Are Darboux and Riemann's integral equivalent when using standard partitions?
While studying Analysis, I used many different textbooks: "Understanding Analysis" (Stephen Abbott), "Elements of Real Analysis" (Denlinger), "Calculus" (Michael Spivak).
...
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If we only specify one sequence of partitions in the definition of Riemann-integral.
Update: This question differs from Question about Riemann integrability: do we need to specify that all Riemann sums converge to the same number in the definition? in the following way:
From what I ...
- 1,601
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Spivak calculus and a Riemann sum of a product of functions
I am trying to understand Spivak's development of integration in Calculus, 4th ed. The integral is introduced in Chapter 13, and I find the proofs quite difficult and unfamiliar. The Appendix to the ...
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Riemann sums for polynomials
When solving a programming problem, I needed to calculate Riemann sums for polynomials $\sum_{k=1}^tp(k)$, where $t$ is variable. In search for a solution, I found the Faulhaber's formula $\sum_{k=1}^...
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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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Use a Riemann sum to approximate the integral $\int_{-1}^1\frac{1}{|x|^{2.5}}(e^{-i\pi\omega\cdot x}-1)dx$ in 1d
Consider the function $f:[-1,1]\setminus\{0\}\to \mathbb{R}$ given by $f(x)=\frac{1}{|x|^{2.5}}.$ For dimension $d=1,$ Consider the integral below:
$$\int_{-1}^1\frac{1}{|x|^{2.5}}(e^{-i\pi\omega\cdot ...
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Could we approximate $\int_0^1\frac{1}{x^4}dx$ using a Riemann sum?
We know that in one dimension, the integral $\int_{0}^{1}\left(1/x^{4}\right){\rm d}x$ is not finite. But could we approximate this integral using a Riemann Sum ?.
...
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If every sequence of Riemann sums of a function converges, is the function integrable?
Let $f:[a,b]\to\mathbb{R}$ be a function, and $P_n$ the equidistance partition of $[a,b]$ into $n$ subintervals of an equal length. Let $P_n^\ast$ be the set of sample points from each subinterval of $...
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Riemann Sum vs Quadrature
Is it accurate to say that:
Riemann sum leads to the following approximation methods:
Left sum.
Right sum.
Midpoint sum.
As Riemann sum uses the function value $f(x_i^*) \;$ where $\quad x_{i-1} \...
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Working on details on the Secretary Problem [closed]
I've been trying to follow this proof of the optimal way to solve the secretary problem (ref. https://en.wikipedia.org/wiki/Secretary_problem). Everything is clear to me except where they are ...
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Calculate this limit using integral
Giving $a$, $b$ are two positive numbers and $\alpha$ is a positive integer; calculate the limit:
$$
\lim_{n\to\infty}\left[\dfrac{1}{n^{2\alpha+1}}\sum_{k=1}^{n}\left(k^2+ak+b\right)^{\alpha}\right]
$...
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Function-vector dualism of inner product
I have a question related to the dualism between the inner product of infinite-dimensional vectors and the integral over the product of their corresponding function representations, which is given by
$...
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Conjecture: The $n$th left Riemann sum for $\int_0^1 (x-x^2)^k dx$ is $B(k+1,k+1) + \Theta(n^{-2 \lceil (k+1)/2\rceil})$
While playing around on Wolfram Alpha with integrals of the form $\int_0^1 (x-x^2)^k dx$, where it happens that Wolfram Alpha displays formulas for the $n$th left Riemann sum (using equally spaced ...
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How to calculate the limit of this riemann sum using integrals?
$$
\lim_{n\to \infty} \frac{3}{n} \sum_{k=1}^{n} \sqrt{\frac{n}{n+3(k-1)}}
$$
Assuming this can be written as a Riemann Sum, how can I bring it to an integral?
I'm trying to make it reach the form $\...
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Doubt regarding limits on riemann sums
Find $$\lim_{n\to\infty}\frac{1^p+3^p+\ldots+(2n+1)^p}{n^{p+1}}$$
I found a solution here which goes like this:
By Riemann sums, for any $p>-1$:
$$ \frac{1}{n}\sum_{k=0}^{n}\left(\frac{2k+1}{n}\...
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A question about sum of sequence with fifth powers
$\sum_{r=1}^{p}(4p+3r)^5$
I'm looking for the coefficient of the highest degree term in the formula obtained when this sum is written in terms of $p$. Is there a practical way to do this?
And also ...
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Solving Sequence Using Riemann Sum
I am currently in my second calculus course and my professor asked me to evaluate the limit of a sequence.
$$ b_k = \frac{1}{9k+1} + \frac{1}{9k+2} + \cdots + \frac{1}{20k} $$
We did a similar problem ...
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Is this a Riemann sum?
I have come a cross with a sum that looks like this:
$$\sum_{x\in{\Lambda_N}}\epsilon^2 k(\epsilon x)e^{-i\pi\omega \cdot \epsilon^2 x}\quad \quad\quad\quad(*)$$
Here $x$ takes values in the discrete ...
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Riemann Sum with Supremum [closed]
Suppose we have $f(t, \mathbf{s}): \mathbb{R} \times \mathbb{R}^{g}$ where
$\mathbf{s} \in [0, 1]^{g}$ with $g \in \mathbb{Z}_{+} \cup \infty$,
$t \in \mathbf{t}$ where $\mathbf{t}$ is a set of tagged ...
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Oscillation of an Integrable Function on a Subinterval is smaller than epsilon. Proof check.
Given an integrable function $f : [a,b] \rightarrow \mathbb{R}$, and a subinterval $[c,d] \subseteq [a,b]$ with $c < d$, we aim to prove that for every $\epsilon > 0$, there exists an interval $[...
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Is this Dirichlet type function Riemann Integrable?
\begin{cases}
\cos\left(\frac{\pi}{x}\right) &,\quad \text{if } x \text{ is rational}
\\
0 &,\quad \text{if } x \text{ is irrational}
\end{cases}
Over the interval $[0,1]$
My approach
Solve ...
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Is there a nice closed form of the following function: $f(k) = \lim_{n \rightarrow \infty}\prod^{kn}_{i = 0}\frac{n-2i+1}{n-2i}$
I am tring to find the closed from of the following function:
$$f(k) = \lim_{n \rightarrow \infty}\prod^{kn}_{i = 0}\frac{n-2i+3}{n-2i+2},$$
where $k \in [0,\frac{1}{2}]$
If the numerator is $n-2i+4$ ...
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Evaluate $\lim_{n\rightarrow\infty} \sum_{k=1}^n \frac{1}{k}\tan(\frac{k\pi}{4n + 4})$
My approach:
$\lim_{n\rightarrow\infty} \sum_{k=1}^n \frac{1}{k}\tan(\frac{k\pi}{4n + 4}) = \lim_{n\rightarrow\infty} \sum_{k=1}^{n+1} \frac{1}{k}\frac{n+1}{n+1}\tan(\frac{k\pi}{4n + 4}) = \lim_{n\...
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How to prove every shell is non-overlapping in volume of revolution by "shells"? Does the Riemann sum imply the volume is over-counted?
Why is the shell method not
$$\lim_{n \rightarrow \infty} \sum_{k=1}^n 2\pi\left(\frac{(b-a)}{n}\right)\cdot f\left((k-1)\cdot\frac{(b-a)}{n}\right)\cdot \frac{1}{n} + \pi f\left((k-1)\cdot\frac{(b-a)}...
user1291046
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Evaluate $\int_0^3 x{\sqrt {3-x}}\;dx $ using Riemann sums
So, I was asked to do this integral using the limit method (or the Riemann Sum)
$$\int_0^3 x{\sqrt {3-x}}\;dx $$
And, I do it like this:
$$\int_0^3 x{\sqrt {3-x}}\;dx $$
Firstly, I determine the $\...
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Is $\lim\limits_{n \to\infty}\sum_{i=1}^{l\cdot n}\frac{f\left(\frac{i}{n}\right)}{n}$ a valid definite integral riemann sum? What is it called if so?
I came up with this alternate Riemann sum that correctly gives the value of a definite integral (at least for some simple polynomial and trig functions I tested with wolfram alpha):
$$\lim\limits_{n \...
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The behavior of $\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{n+k+k^\alpha}, \alpha\in \mathbb R$.
As title, I am interested in the value of $$\lim_{n\to \infty}\sum_{k=1}^n\frac{1}{n+k+k^\alpha},$$ where $\alpha\in \mathbb R$. Here is my attempt:
For $\alpha\in (0,1)$, note that $$\sum_{k=1}^n\...
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Limit of $∞.0$ form of an integral and Riemann sum
I was pondering how to solve a limit of the kind $$\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)$$
where k is chosen such that the order ...
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Use Riemann sums to prove $\int_{1}^{b} \frac{1}{\sqrt{x}}dx = 2(\sqrt{b}-1)$ using equal subintervals
This post refers to Question 2 of the review problems at the end of Chapter 6 of George Simmon's Calculus:
Following the general form $$\int_{a}^{b} f(x)dx = \lim \limits_{max \Delta x_k\to0} \sum_{k=...
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Let $f:[0,1]\to \Bbb R$ such that $f(x)=x$ if $x$ be rational $x^2$ if $x$ be irrational. Find $\underline{\int}_0^1 f$ and $\overline{\int}_0^1f.$
A function $f$ is defined on $[0,1]$ by $f(x)=x$ if $x$ be rational $x^2$ if $x$ be irrational. Find $\underline{\int}_0^1 f$ and $\overline{\int}_0^1f.$
The solution given is as follows:
$f$ is ...
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Calculating pretty difficult limit that invloves Riemann sums
Let $S_n = \sum_{k=1}^n\frac{1}{\sqrt{n^2+k^2}}$. Calculate the following limit
$$\lim_{n \to \infty} n\left(n\Big(\ln(1+\sqrt{2})-S_n\Big)-\frac{1}{2\sqrt{2}\,(1+\sqrt{2})}\right).$$
My intuition ...
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How to perform this sum
I encountered this sum
$$S(N,j)= \frac{2 \sqrt{2}h(-1)^j}{N+1}\cdot\sum _{n=1}^{\frac{N}{2}}
\frac{\sin ^2\left(\frac{\pi j n}{N+1}\right)}{\sqrt{2 h^2+\cos \left(\frac{2 \pi n}{N+1}\right)+1}},$$
...
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Convergence of a sum as limit tends to infinity that seems to be harmonic series
I have come across a mathematical problem that is to evaluate the expression:
$$
lim_{n\rightarrow\infty} \left\{\frac{1}{\sqrt{2n-1^2}}+\frac{1}{\sqrt{4n-2^2}}+\frac{1}{\sqrt{6n-3^2}}+...+\frac{1}{\...
- 1,483
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Why is the following claim true in Exercise 7.4, Apostol's Mathematical Analysis?
On this page, a proof of the equivalence of two definitions of Riemann integrals is given by the user Pedro using Apostol's Hint for Exercise 7.4, Mathematical Analysis. However, I still find this ...
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Upper and Lower Integral [duplicate]
Given $F(x): [0,1] \rightarrow \mathbb{R}$. $F(x) = 0, x \in \mathbb{Q}$ and $F(x) \ge \frac{1}{2}, x \not \in \mathbb{Q}$.
Proof/Disproof $F(x)$ Riemann-Integrable!
My Attempt:
Consider that,
$$M_i = ...
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How to prove if $f$ is Darboux integrable then for all $\epsilon > 0$ then $U(f, P_{\epsilon}) - L(f, P_{\epsilon}) < {\epsilon}$ ??
Background:
I am studying Real Analysis (never studied it before) from the book 'Real Analysis' by Jay Cummings. I am at chapter 8 (Integration) when I encounter theorem 8.14 which comes almost right ...
