Questions tagged [simpsons-rule]

For questions regarding Simpson's rule and its applications.

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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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1answer
44 views

Simpson's 3/8 method [closed]

What are the advantages and disadvantages of simpson's $\frac {3}{8}$ rule. Also what are the applications of this rule.
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5 views

How to choose interval count for simpsons rule?

Is there some ways to compute interval count for [a; b] according to some error $\epsilon$? Or maybe according to function properties and interval size?
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18 views

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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1answer
39 views

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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2answers
53 views

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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1answer
36 views

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 ...
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1answer
40 views

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 indefinite 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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2answers
59 views

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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32 views

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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approximate error term for Simpson's Rule

With the approximate error term for Simpson's rule $|Es|$ $\leq$ $\frac{(b-a)^5}{180n^4}$ whereas $f^{iv}(x)$ $\leq$ $K$ for $a$ $\leq$$b$ what size $n$ is needed to have an approximation of the ...
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1answer
59 views

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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118 views

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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277 views

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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1answer
29 views

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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1answer
224 views

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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445 views

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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1answer
17 views

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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1answer
29 views

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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1answer
99 views

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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273 views

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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1answer
88 views

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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40 views

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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1answer
66 views

How do I calculate $\int_1^3$ $x \ln (x)\ dx$ with a given accuracy, using Simpson's rule?

How do I calculate $\int_1^3 x \ln (x)\ dx$ with a given accuracy, in this case $10^{-4}$ using Simpson's rule? The problem I encounter is that the fourth derivative is $=2/x^3$. So how do I go ...
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1answer
54 views

Simpson's Rule over [-3,3]

To Find $I = \int_{-3}^3 \sin x^4 dx$ using the Simpson's (Parabolic) rule with $n=6$ intervals. $h=\frac{b-a}{n}=\frac{3-(-3)}{6}=1$ \begin{array}{|c|c|c|}\hline x\rightarrow&x_0&x_1&...
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132 views

Simpson vs. trapezoidal rule for numerically integrating $\cos{x}\cosh{x}$ in range 0 to $\pi$?

I have to numerically calculate many integrals similar to this: $$\int_0^\pi \cosh{\left(\frac{a_1\cos{x}+a_2\cos{2x}+a_3\cos{3x}+\ldots}{10}\right)}\cos{jx}\cos{kx}\space dx$$ Right now I am using ...
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41 views

Integrating using Simpson's Rule

Suppose there are functions $$ g(x) = \frac{(2 \cdot \lfloor x\rfloor)}{(3\cdot x - \lfloor x \rfloor)} \tag{1}$$ and $$ f(x) =\frac{\mid g(x)\mid}{ g(x)} \tag{2} $$ Now how do we do the definite ...
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802 views

What is the order of the midpoint rule?

If the Trapezoidal-Rule has the order $n=1$, and Simpson's has order $n=2$, what is the order $n$ of the midpoint rule? And if the weights of the Trapeziumrule are ($1/2, 1/2$) and those of the ...
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1answer
237 views

Numerical solution of generalized Fresnel integral

We need to find an approximate solution for the generalized Fresnel integral: $\int_0^S \cos(as+\frac{bs^2}{2}+\frac{cs^3}{3}+\frac{ds^4}{4})ds$ Our approach is to use the Simpsons rule: $\int_a^bf(...
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1answer
171 views

Modified simpson's rule for varying step sizes

I'm reading the paper A Modified Simpson's Rule and Fortran Subroutine for Cumulative Numerical Integration of a Function Defined by Data Points by L.V. Blake, and it notes, on page 8 a further ...
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2answers
530 views

Implementation of cumulative simpson method

I am working on a program which uses cumulative integration methods to solve differential equations, and I want to confirm that my implementation of the Simpson method is correct, as I could not find ...
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1answer
610 views

Richardson extrapolation Simpson's rule

"Assume that S(h) is equivalent to the (composite) Simpson's rule where h is the size of the step. Correct use of Richardson's extrapolation gives the formula: $R(h) = \frac {16S(h)-S(a)} {b}$. What's ...
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152 views

Use Simpson's rule to estimate the error of integral $\int_1^5 \ln(x) \mathrm dx$

In my homework I'm requested to estimate the error of the definite integral: $$\int_1^5 \ln(x) \mathrm dx$$ I am also given the formula: $$|E_n|<= \frac {(K(b-a)^5)}{180N^4}$$ Where K is an ...
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220 views

How to approximate the following improper integral by using Simpson's rule

I am trying to approximate the following integral by using Simpson's Rule with $n=6$ $$I = \int_{1}^{\infty} \frac{\sin(x)}{x^4} dx$$ The textbook I am using for Numerical Analysis (Burden, Faires 9-...
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2answers
171 views

Understanding Example of Simpson's Rule

Using figure 8, let $x_0 = -h, x_1=0, x_2=h$ So the area under the curve is exemplified below: $$\int_{-h}^h(Ax^2+Bx+C)dx=2\int_0^h(Ax^2+C)dx$$ $$=2\left[A\frac{x^3}{3}+Cx\right]_0^h$$ $$=2\left(A\...
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How to find Simpson's rule

Can anyone help me find my mistake? $$I = \int_0^2 f(x) ~ dx, \qquad f(x)=\frac {3^x}{x+1}$$ I got $f(0)=1$ $f(0)=1$ $f(1/2)=\frac{2(3^{1/2})}3$ $f(1)=3/2$ $f(3/2)=\frac{6(3^{1/2})}5$ $f(2)=3$ ...
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1answer
392 views

Approximating optimal step size for numerical integrator

I am trying to find a formula to approximate the ideal step size for the Trapezoid and Simpson's rules. As an example, consider the finite difference formula $$g(x,h) = \frac{f(x+h)-f(x-h)}{2h}$$ ...
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1answer
70 views

Step in proof: error Simpson's rule for $f\in C^4$

They claim later on in the proof that $G’(0)=0$. I don’t see this. I would say that we have $$ G(t)=\int_0^tF(\tau)d\tau-\int_0^{-t}F(\tau)d\tau-t/3[F(-t)+4F(0)+F(t)], $$ which yields $$ G’(0)=F(0)-F(...
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1answer
874 views

Error Expectations for Composite Simpson's Rule

I have written a program that implements the composite Simpson's rule for integrating functions over the interval $[0,1]$. In checking the program for correctness, I test the routine on the integral $\...
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1answer
241 views

How large should $n$ be to guarantee that the Simpson's Rule approximation on $\int_0^1 9e^{x^2} dx$ is accurate to within $0.0001$?

Please help I am really struggling with this problem. I have been working on it and trying to look up how to do it but nothing is making sense. Thank you!
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1answer
102 views

Approximate derivative in midpoint rule error with notation of Big O

Error of midpoint rule is $E_m = \frac{f''(c)}{24}(b-a)^3$, where $c\in (a, b) $. I made research and I found out that i can approximate it by $E_m = \frac{f''(a)}{24}(b-a)^3 + O((b-a)^4)$, but i don'...
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45 views

On Newton-Cotes functions for higher derivates

Newton-Cotes formulas simply have a usage for approximating definite integrals. One can get more information on here about Newton-Cotes formulas. For instance, Simpson's rule gives $$\int_{x_1}^{x_3} ...
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1answer
130 views

Quadrature rule for spline interpolation

Consider an integrable function $f$ on $[-1,1]$. We denote $\left(x_j\right)_{-N}^{N}$ the equally spaced grid on $[-1,1]$, and wish to compute the integral $I = \int\limits_{-1}^{1} f(x) \, dx$ using ...
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54 views

Integration of a function approximated by a nth order polynomial

I've been playing with Simpson's rule and a thing came up to my mind. The rectangular rule is a 0th order polynomial approximation of integration. The trapezoidal rule is 1st. Simpson's rule is 2nd. ...
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1answer
248 views

How can Simpson Rule use curves to find the Integral?

If Integration is about finding the Area under the curve, how can the Simpson Rule use curved Parabolas to find Area under the slice? I understand the Trapezoid rule because it uses triangles which ...
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1answer
178 views

Numerical Integration: How do you numerically integrate functions which include derivatives within the integrand?

I am given the problem to find $B(y)$ by solving the following integral numerically: $$B(y) = \int_{z=0}^{\infty}\frac{1}{\sigma (y,z)}\frac{\partial^2 A(y,z)}{\partial z^2}+\frac{\partial A(y,z)}{\...
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1answer
144 views

Simson's Rule but Intervals Aren't the Same

I'm currently working on some homework, and ran into a question that has stumped me. We're working on Simpson's rule, which is easy enough, but then I got to a problem where the intervals aren't the ...