Questions tagged [simpsons-rule]
For questions regarding Simpson's rule and its applications.
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Numerical integration: The composite Newton-Cotes formulas, uniqueness and inductive definition for a given order of exactneness
I have a question on Rabinowitz and Davis: Methods of numerical Integration. They start to give a sequence for what they call The (composite) Integration Newton-Cotes formulas. This together with my ...
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some errata? or Not in Calculus book?
I think I found some errata in the book James Stewart Calculus 8th EarlyTran... (Still found not corrected in 9th also).
Chapter 7.7 (Approximate Integration)
Page 522, about Simpson's Rule:
At the ...
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Integral to derive Simpson's Rule error expression
I have this question from an old Numerical Analysis exam:
Let $h>0$ and $f$ be a sufficiently differentiable function. Prove that
\begin{align*}
I:=\frac{1}{6}\int_0^h[f'''(-t)-f'''(t)]t(t-h)^2dt=-\...
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Finding error in Simpson's Integration Rule by Lagrange's Interpolating Polynomial
This was asked here before but I wonder if I can also get the error expression in Simmpson's Rule as follows:
Suppose we want to estimate the integral a function $f$ in the interval $[x_0,x_2]$. We ...
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Proof for the Error Formula for Composite Simpson's Rule
I have the composite Simpson's rule as
$$
\frac{h}{6}(f(a)+f(b)) + \frac{h}{3}(f(a+h)+f(a+2h)+...+f(a+(n-1)h) + \frac{2h}{3}(f(a+\frac{h}{2})+(f(a+3\frac{h}{2})+...+f(a+(2n-1)\frac{h}{2})
$$
My ...
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Quadratic Polynomials and Simpsons rule
Question:
(i). Approximate the integral, $\int_0^1 \frac{1}{1+x^4} dx$, using trapezoidal rule by dividing the interval $[0, 1]$ into 4 intervals of equal length.
(ii). Let $f$, $g$ be quadratic ...
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Simpsons 1/3 rule error using lagrange polynomial
I am deriving the error term of Simpson's $\dfrac{1}{3}$ rule using the approach given in the Book Brian Bradie. It uses the divided difference approach for error calculation. However when i tried it ...
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Error of Simpson's Rule [duplicate]
How do I show that
$$\int_a^bf(x)dx - I_1 = - \frac{h^5}{90}f^{(4)} (\xi)$$
with $\xi \in [a,b]$ and
$$I_1 := \frac{h}{3}(f(a) + 4f\left( \frac{a+b}{2}\right) + f(b))$$
and $h= \frac{b-a}{2}$.
I was ...
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Numerical quadrature with preassigned points
I have been looking for a numerical quadrature that might be possible to pre-assign specific nodes.
For instance, I need to numerically calculate the integral of $f(x)$ in the interval $[a,b]$ but the ...
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Numerical integration - singularity problem
I’m trying to code a numerical integration using Simpson’s rule, the equation is given by Anderson in the book "Fundamentals of Aerodynamics" as
$$ \alpha_i(y_n)= \frac{1}{4\pi V_\infty}\...
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What is wrong with my derivation of Simpson's Rule via Lagrange interpolation? + Is my alternative a better approximation?
I understand that correct derivations exist on this site. I am, however, interested in why my workings are incorrect.
Let $f(x)\approx L(x)=\sum_{n=0}^2\ell_n(x)\cdot f(c_i)$, where $c_i$ are three ...
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Error on Simpson Sum and Trapezoidal sum
I was coding an algorithm to approximate the integral $\int_{0}^{1} f(x) dx\int_{0}^{1} cos(x)+1/10cos(10x)+1/50cos(50x) dx$ as i encountered some interessting graphs. When approximating for $N=2^k$ ...
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How to apply Simpson's rule to f(x)^2*x?
I have points available for $x$ and $f(x)$. To calculate area between $(f(x)^2)\cdot(x)$ and $x$, can I consider $g(x) = (f(x)^2)\cdot(x)$ and apply Simpson's rule on $g(x)$ and $x$?
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Given $n$ subsections, how many parabolas are made in Simpson's rule?
Here's a picture from Stewart's calculus for Simpson's rule
Since a parabola goes through $x_n, x_{n+1}, x_{n+2}$, we would get $\frac{n}{2}$ parabolas? So in this case of $n=6$, then we'd get $3$ ...
user612996
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Simpson’s rule for monomial degree > 4
Does there exist a monic $x^n$, n > 4, for which Simpson’s rule is exact? If not, why?
$$ S(f) = \frac{b-a}{6}f(a) + \frac{2(b-a)}{3}f(\frac{a+b}{2})+ \frac{b-a}{6}f(b)$$
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What is the difference in accuracy in answers from Definite Integration and Simpson's rule?
I have worked out the same equation with the Definite Integration and Simpson's Rule. I now need to comment on the accuracy of these answers. I know Simpson's rule is an approximation, but does that ...
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How many points are necessary to calculate $\int_2^6\ln(x)dx$ using Simpson's rule with error $\le 5*10^{-4}$?
How many points of integration are necessary to calculate the integral
$$\int_2^6\ln(x)dx$$
using Simpson's rule such that the absolute error of integration is lesser or equal to $5*10^{-4}$, knowing ...
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Find the upper bound of the error for the approximation of $\int_{x_1}^{x_2}\cos(x)$
Consider $f(x) = \cos(x)$ and the points $x_0 = \pi/2$, $x_1 = \pi/4$
and $x_2=3\pi/4$. Consider $f(x_i)$ with 5 decimal places. Find an
estimate for the error of integration when you use Simpson's ...
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Use the composite Simpson's rule to approximate how far a tricycle traveled from a table with the time and velocity
Use the composite Simpson's rule to find how far the car traveled
between 15:00 and 15:40 from the following table (Hora/s = Hour/s)
I did
$$I(f)\approx \frac{(b-a)}{3N}[f_0+4f_1+2f_2+4f_3+f_4] \\ =
\...
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Can a Composite Simpson's Rule be applied in this case?
I have been given a table with three values of a function at three different points. In my textbook, the Composite Simpson's Rule has a following form:
$I=\frac h6(f_0+4f_1+2f_2+4f_3+...+2f_{2n-2}+4f_{...
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Numerical Integration - Trapezium and Simpson's rule
I was reading through my professor's notes on Trapezoid and Simpson's rule in Numerical Integration and I was wondering if I get the intuition right.
Given a function $f$ and an interval $[a,b]$ these ...
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Simpson's rule for improper integrals
Let $$F(x)=\int_{-\infty}^x f(t)dt,$$
where $x\in\mathcal{R}$, $f\geq 0$ is complicated (it cannot be integrated analytically).
Can I used the Simpson's rule to approximate this integral, knowing that ...
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Demonstrating equivalency between Runge-Kutta and Simpson's Rule
I am trying to show that a fourth order Runge-Kutta method is equivalent to Simpson's method for approximating an integral over an interval [a,b]. I've read that it is, but I'm unsure how to ...
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How to solve this improper integral using Simpson's rule [closed]
I'm trying to calculate value of $\int_{0}^{\infty}{e^{-2x}(1+e^{-x})}dx$ using Simpson 1/3 rule.
I know the solution is substitution $x$ by $g(t)$ but every $g(t)$ I've tested the result was ...
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Simpson's Rule Question with Example
I'm just learning Simpson's Rule for integral approximation and I have a question.
$
\int _a^b\:f\left(x\right)dx\:approx=\frac{\frac{b-a}{n}}{3}\left[\left(1f\left(x0\right)\right)+4f\left(x1\right)+...
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Evaluating an integral using Simpson's Rule
Below is a problem I made up and did. I would be interested in feed back from the group on the quality of my answer. Does breaking up the integral in two parts make sense?
Problem:
Give an estimate of ...
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What is the best numerical integration method for evaluating probabilities from z scores.
I used Simpson's rule to calculate the area under a standard distribution where $z\gt1$.
This is the integral: $$A=\frac{1}{\sqrt{2\pi}}\int_1^{\infty}e^{\frac{-x^2}{2}}dx$$
I decided to go as far as $...
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Saving nodes in iterative Adaptive Simpson quadrature method (Matlab)
Implementing on Matlab the Simpson adaptive rule to approximate an integral (the following code), I am struggling with saving nodes correctly. I have tried different solutions, but none of them seems ...
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Simpsons rule and follow up question to expressing the integral of a function as $\pi$
Use Simpsons rule with n = 4 to estimate the integral $\int_0^2\sqrt{4-x^2}dx$. Notice that the integral gives you an approximation for ${\pi}$ and therefore demonstrate without evaluating this ...
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Evaluate $\int_{0}^{2}(x^2 + x^{3/2} + x + x^{1/2})dx$ using Simpson's Rule
I'm running into some troubles while trying to evaluate
$$\int_{0}^{2}(x^2 + x^{3/2} + x + x^{1/2})dx$$ using Simpson's Rule
Simpson's Rule states
$$Q(f) = \frac{b-a}{6}(f(a) + 4f(\frac{a+b}{2}) +f(...
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Show that the integral of an odd function over symmetric interval is always $0$ for a cubic function in the Simpson's rule?
I know that the definite integral of a cubic function $f(x)$ over a symmetric interval is $0$. I just need some clarity on why that is.
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Questions about the error in quadrature formulas
everyone, I am new to these subjects and I would like to clarify a subject that is little covered in my textbook.
It tells me that fixed an n, a number of sub-intervals to use to calculate the ...
user432382
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Evaluate $\int_{-1}^{0}(x^4-x^2+2)dx$ using Simpson's Rule
We're asked to evaluate the integral $$\int_{-1}^{0}(x^4-x^2+2)dx$$ using Simpson's Rule.
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How can I estimate error when computing improper integral with a finite interval?
I'm trying to evaluate integrals of the form $$I = \int ^{\infty }_{a} f( x) dx$$
Where $a$ is any real number. In order to estimate this integral, I pick some large positive number $M$ and instead ...
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Simpson's rule — where did the coefficients come from?
I am reading how Simpson's Rule works for numerical integration. So I understand that given the two endpoints $x_0$ and $x_2$, and one intermediate point $x_1$, we can connect these points to make a ...
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Basis of Simpson's rule for numeric integration
Exact Problem
Let $f(x) = x^2$, and let P denote a partition of [a, b].
(b) Use the fact that $f(x)$ is continuous to show that for any $\epsilon > 0$ there exists $\delta > 0$ such that for ...
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Simpson's rule using Taylor polynomial
From a proof of Simpson's rule using Taylor polynomial where $f\in[x_{0},x_{2}]$ and, for
$$x_{1}=x_{0}+h$$
where
$$h=\frac{x_{2}-x_{0}}{2}$$,
it got:
$$\int_{x_{0}}^{x_{2}}f(x)dx\cong2hf(x_{1})+h^{3}...
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Accuracy comparision between composite simpson's rule and $\frac{3}{8}$ simpson's rule
I am integrating the function $f(x)=0.2+
25x-200x^2+675x^3+900x^4+400x^5$, I have to integrate it between $a=0$ to $b=0.8$ with $n=5$ (five segments), I have integrated it by using only $\frac{3}{8}$ ...
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How to use Simpsons Rule Maths for Ship Stability
I did a Civil Engineering course some years ago and from my textbook I had this question. As I am interested in this I have been trying to solve this, but unfortunately I haven't been able to find a ...
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When does the exact value of the integral $\int_{-3}^{3}(x^3\cos(x)+x)\,\mathrm{d}x$ occur when it is solved by the Simpson method?
When solving $$\int_{-3}^{3}(x^3\cos(x)+x)\,\mathrm{d}x$$ by Simpson's method, taking $n$ subintervals, the exact value is obtained if:
$n$ is even.
$n$ is greater than $3$.
$n$ is even ...
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Composite 2D Simpsons Rule with odd intervals
This question is an extension of this question for 2D integration.
The formulation of the problem is based on this page
Basically, the composite Simpson's rule for 2D integration is
$
\iint_R f(x,...
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Composite Simpson's rule vs Trapezoidal on integrating $\int_0^{2\pi}\sin^2x dx$
A simple question comparing both methods for numerical integration for a very specific case. We expect the Simpsons rule to have a smaller error than the trapezoidal method, but if we want to ...
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Simpsons rule problem. why is this equation setup this way?
I am a bit confused as to why a problem in my book is using A(t) instead of D(t) in teh setup for simpsons rule. Why is the integral at the end setup like:
$$\int_0^43200 A(t)dt$$ and not $$\int_0^...
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Estimating with simpson rule
I have a question that is supposed to be very easy in a test:
We approximate $\displaystyle \int_0^1 x^2$ with Simpsonrule, and 5 intervals. Choose solution:
Firstly I don't understand what is meant ...
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Give a function for which Simpson's rule returns an exact value
Let I denote the integral $I = \int_0^{\pi/2}\sqrt{\sin x}dx$ and 4 strips.
Give a function for which Simpson’s rule returns an exact value.
I just entered in the exact values (so $\sin(\pi/2)$ for ...
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How to derive Adam Moulton 2 step implicit method using taylor expansion
I have some confusion on the derivation of multistep method using Taylor expansions. For example, we want to derive the linear 2 step Simpson's rule: My professor first write down the scheme of an ...
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Evaluate $\int_0^\pi \frac {\sin x}{x} \,dx$ using Simpson's $\frac {1}{3}$ rule and $\frac {3}{8}$ rule with $n=6$
Evaluate $\displaystyle\int_0^\pi \frac {\sin x}{x}\,dx$ using Simpson's $\dfrac {1}{3}$ rule and $\dfrac {3}{8}$ rule with $n=6$.
For $n=6$, $h=\dfrac {\pi - 0}{6}=\dfrac {\pi}{6}$.
But, the ...
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Simpson's Rule Via Table
Alright, so I'm a bit stumped on this one. I learned Simpson's rule via my textbook as follows:
$$\frac{h}{3}[y_0 + y_n + 2(y_2+y_4+...+y_{n-2}) + 4(y_1+y_3+...+y_{n-1})]$$
I was given a problem ...
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Am I using the Simpsons rule and Gauss-Legendre method correctly? [closed]
I have the integral here:
Simpsons rule:
Answer 414.11411
Gauss-Legendre method
Here the limits are x+y
I found the answer to be around 0.923
Just wanted to make sure these values are correct.
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Numerical Integration Bounded by Two Singularities
I would like to solve the following definite integral numerically using Simpson's Rule, however it has singularities at both ends. I was told it's possible to perform a simple change of variable in ...
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