For questions regarding Simpson's rule and its applications.

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Combining error terms in Simpson's rule

My numerical analysis textbook (Burden and Faires) derives Simpson's rule as $$\begin{align} \int_{x_0}^{x_2}f(x)\,dx&=2hf(x_1)+\frac{h^3}{3}f''(x_1)+\frac{h^5}{60}f^{(4)}(\xi_1) ...
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27 views

Simpson's Rule approximation

I'm very lost with this problem. How large should $n$ be to guarantee that the Simpson's Rule approximation on the Integral ...
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36 views

Method check: numerically calculate 1D integral of a 3D function

I have a function $f(r)$ where $r=\sqrt{x^{2}+y^{2}+z^{2}}$, $\forall x,y,z \geq 0$. I know the values of the function at many points, essentially I have a table of values with $r$ and the ...
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Some doubts on Simpsons Rule by the Method of Undetermined Coefficients

There is this note about Quadratic Interpolation by Simpsons Rule that I don't quite understand how to get the LHS. Simpsons Rule by the Method of Undetermined Coefficients We seek an approximation ...
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explain why there is an observed rate of convergence

Using your knowledge and theorems explain why there is an observed rate of convergence when using the composite simpsons rule?
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35 views

Find the 'rough' error bound to the composite simpson rule

Provide a rough error bound for the following composite simpsons rule. I am aware that the upper bound is $f$ to the forth derivative evaluated at some $t$ in the open interval ...
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Numerical integration in Matlab (Simpson's rule)

A cubic polynomial is given by $$y=\frac{x^3}{6RL}$$ with $R$ and $L$ being constants. Use Matlab and numerical methods to find $x_l$ so that $$L=\int^{x_l}_0 \sqrt{1+(y')^2} dx$$ when $R=200$ and ...
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37 views

Numerical Integration error for simpson's rule through taylor series

I am looking at the derivation of the simpson's rule as well as an error analysis in my textbook and I am slightly confused over two things. Firstly, in the derivations, the text uses a taylor ...
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24 views

Richardson Extrapolation for Quintic Integration

In the picture above, what exactly is the question asking for? I know that the error in simpson's rule is to the order of h^5. Thus doubling the length increases error by 32x and so should I ...
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53 views

Sign of the error in Simpson's rule

Let $f : [a,b] \to \mathbb{R}$ be a $C^\infty$ function. The Riemann integral $I = \int_a^b f(x)\,dx$ can be approximated by using Simpson's rule: $$I \approx S = \frac{b-a}{6} \left[ f(a) + 4 ...
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Integral with Simpson's method not converging

I'm trying to use Simpson's rule to integrate the following function in a program: $$\int_{z_a}^{z_b}\frac{Cf(z)}{(C^2 - f(z)^2)^{3/2}}\,dz$$ where $C$ is a constant and $f(z)$ are interpolated ...
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A mathematical explanation of the Simpson's Paradox?

In general, Simpson's Paradox occurs because situation such as following occurs for some arbitrary events $A,B,$ and $C$: $P( A | B , C) < P(A| B^c,C)$ $P( A | B , C^c ) < P(A| B^c,C^c)$ But, ...
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Evaluating integral using Simpson's rule, error in book?

This is from Morris Kline's "An intuitive and Physical Approach" calc book, chapter 9 last section, page 264 problem 4. The problem states: Given the following data on a function $y = f(x)$: x: 0, ...
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Find an approximate value of R 2 1 x −2 dx using composite Simpson’s rule with h = 0.25. Giv

Find an approximate value of $\int _1^2 x^{-2} dx$ using composite Simpson's rule with h = 0.25. Give a bound on the error. Then calculate the exact value of the integration and compute the exact ...
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2answers
58 views

What value of $N$ to use in Simpson's rule to reach desired accuracy?

I need to calculate Simpsons rule for the integral of $$\frac{e^x-1}{\sin x}$$ from $0$ to $\pi/2$ with minimum number of intervals $N$ up to $10^{-6}$ accuracy. Wolfram alpha seems to be giving me a ...
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1answer
35 views

Simpsons rule Help [closed]

∫x/sin(x) from [0,pi/2], n = 2, using Simpsons Rule. public static double simpsonsRuleFunction1(double valueN, double valueA, double valueB, double valueDx) { ...
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1answer
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Simpson's Rule for IVP. Truncation Error proof

Edit: replaced all c's with y's as the c just denotes replacing a series of coupled linear equations Ay with uncoupled equations $\Lambda c$ no biggie. Im working through the lecture notes for a ...
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140 views

Confusion about order of accuracy of Simpson's rule

Suppose I have a function $f(x)$ which is to be integrated in the interval $[a,b]$ using step size $h$. Also, $b = a+2n\cdot h$ and $n \in \mathbb{N}$ I am unable to understand how exactly does the ...
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1answer
67 views

Simpson's rule with precision?

I have an integral: $$\int_0^1sinx^2dx$$ Task is to solve this integral using Simpson's rule with precision $\frac{1}{2}10^{-4}$. I am not sure how should I do that. I have this formula for ...
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1answer
65 views

Reasons for different answers when finding area using Simpsons rule and numerical integration? [closed]

I have a function $\sqrt{x^4(x+4)}$ to be integrated from 0 up to -4. Using Simpson's will give me 19.02 but using normal numerical methods giving me -19.5 ! What's the reason behind this difference ...
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Simpson's Rule for Double Integrals

Simpson's Rule for double integrals: $$\int_a^b\int_c^df(x,y) dx dy$$ is given by $$S_{mn}=\frac{(b-a)(d-c)}{9mn} \sum_{i,j=0,0}^{m,n} W_{i+1,j+1} f(x_i,y_j) $$ where: $$W= \begin{pmatrix} ...
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54 views

Modified version of Simpson Rule

I'm supposed to use some different version of Simpson's Rule in my Numerical Methods homework to compute some areas, considering the non-uniform spacing case . Namely, I've got two equal length ...
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387 views

Simpson's rule to estimate distance traveled given velocity at certain points

Problem: A boat drives a steady course with a variable speed for 4 hours. The speed is registered at regular intervals in meters per second. The registration shows $2.4, 4.4, 7.6, 8.4, 8.6, 7.9, ...
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Series expansion as a means of 'proving' Simpson's Rule?

I've been working out questions regarding Newton Raphson and Simpson's Rule, whilst they're fairly easy to execute, the latter seems to boggle my mind a little bit more in terms of what the examiner's ...
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70 views

Trapezoidal Error is lower than Simpson Error, Find some condition? [closed]

I find a problem that have no idea for it. in calculating $ \int^{1}_{0} (x^6-mx^5)dx $ we know Trapezoidal Error is lower than Simpson Error. what is the range of $m$? Solution: $\frac {217}{210} ...
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82 views

find the upperbound for margin of error when estimating area using trapzoid method

I am trying to find the margin of error upper limit when estimating the area under the function sin(x) with 10 partitions on bounds 0 to pi. I am using the trapezoid method and I can't seem to figure ...
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144 views

Simpson's 3/8 rule formula

I am trying to work with Simpson's 3/8 rule, but I wanted to double check my formula: $$I(f) = \int_a^bf(x) dx \ \approxeq \ \frac{3h}{8}\left(f(a) \ + \ 3f\left(\frac{a+b}{3}\right) \ + \ ...
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1answer
205 views

Trapezoidal and simpson rule question here?

The trapezoidal rule applied on $ \int_0^{2} [f(x)] dx$ gives the value 5 and the Midpoint rule gives the value 4. What value does Simpson's rule give? So we have that T=f(0)+f(2). The Simpson's ...
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174 views

Simpson's 3/8 Rule

When deriving Simpson's 1/3 Rule, I used a second order polynomial $P(x) = Ax^2 + Bx + C$, and integrated over the region $[-h,h]$ Integrating gave me: $ \ \dfrac{h}{3}(2Ah^2 +6C)$ I evaluated ...
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Find the approximate area using Simpson's Rule

Find the approximate area of the shaded figure shown using Simpson's rule. Each of the equidistant parallel chords is measured from the base to a point on the curve. All units are expressed in km. ...
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1answer
81 views

Find upper limit of normal distribution integration

Considering the normal distribution with standard deviation equals to 0.9 and mean 2.1: $$ P(X\leq a) = \frac{1}{0.9\sqrt{2\pi}}\int_{-\infty}^{a} e^{-\frac12\frac{(x-2.1)^2}{0.9^2}}\,dx $$ I must ...
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1answer
231 views

Simpson's rule over cubic splines

I'm helping a friend of mine to do her homework, but i need help understanding some results (sorry but i took numeric methods class a looooong time ago) So, the task is to fit a cubic spline over ...
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65 views

Quadrature obtained from Simpson's rule, and its order of error

Express $Q$ as a weighted combination of the five function values $f(a)$ through $f(e)$ and establish that its order is six. (See section 6.2.) This is from Numerical Methods by Moler, ...
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Trying to re-write Simpson's Rule: mistake?

Pre-Question (edited): Thanks Arthur Orignal Problem: The standard form of Simpson's Rule requires an even value of n so that you can make a series of parabolas Parabola 1 has area ...
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How large should n be to guarantee that the Simpson's Rule approximation on the Integral (from 0 to 1) 19e^x^2 dx is accurate to within 0.0001?

I'm very lost on the following problem and will appreciate your help very much. How large should n be to guarantee that the Simpson's Rule approximation on the Integral (from 0 to 1) 19e^x^2 dx is ...
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124 views

internal rectangle area intersected by a circle

I need to compute the internal rectangle area intersected by a circle like (the blue area) on these 3 examples: I know every vertex (x,y) coordinate and then their distance from circle center but ...
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1answer
83 views

Help me integrate this function using Simpson's rule

I have a question: compute $$\int_0^1 \frac{\sin(x)}{x}\,dx$$ for $n=10$ divisions. I got the value $0.9127$ but I think its a bit too high.
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40 views

Newton-cotes formulas help

I am having a hard time understanding how to use this formula. If given the following problem: Compute ∫ sin x dx using Simpson's rule with 3 points in the range 0 to Pi/2. Do I have to take the ...
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Simpson's rule error rate for N-dimension

I'm doing a project that involves numerical method, but I'm not too familiar on calculus. I'm using Simpson's rule to integrate n-dimension gaussian, I was able to get the integration result for ...
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1answer
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Do I have the right formula for the Composite Simpson's 3/8 Rule?

Let $n$ be the number of segments, which is a multiple of 3. And let $h$ be the width of each of these segments, where $h=\dfrac{(b-a)}{n}$. So the formula I have is that the integral $I$ for a ...
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1answer
170 views

Doubling Number of Nodes In Composite Simpson's Rule

Let n be even. Show how the composite Simpson rule with 2n equally spaced nodes can be computed from the case of n equally spaced nodes with a minimum amount of additional work. I've been working on ...
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How do I get an answer of $14$ using simpsons rule for $\frac{152e}{180n^4}<.0001$

I must have the algebra wrong somewhere but here is the original equation: $$\frac{152e}{180n^4}<.0001$$ If I then multiply like this: $$152e<.0001(180)n^4$$ Which then gives: $$152e < ...
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577 views

Simpson's Error Bound Estimation

The problem: I need to use Error Bound to find n (least) to the $10^{-9}$ in approximating the integral of 5e^x^2 from 0 to 1 I'm using $$Error(Sn) \le \frac{k(b-a)^5}{180N^4}$$ I found the 4th ...
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Use Error Bound to Find Least Possible Value of N

I greatly appreciate it if someone could help me with this problem: Use the Error Bound to find the least possible value of $N$ for which $Error(S_N)\le 1 \times 10^{-9}$ in approximating the ...
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156 views

Compute multiple Rectangles area intersect by a circle

I've a need to compute the area of single elements (dice) of a matrix like this: http://i.stack.imgur.com/EKVSz.jpg The matrix is composed by 'c' columns and 'r' rows and every element/rectangle has ...
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1answer
78 views

Does Simpsons rule still apply when a < 0?

I am currently working on an assignment where I have to find the answer to the following integral using Simpsons rule:$\int x+1$ (MIN = -1 MAX = 3), I choose to have 6 intervals. I now start ...
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215 views

Error bound for $\cos(x^3)$ under Mn, Tn and Sn

In this problem we will approximate the integral of $\cos(x^3)$ over the interval $[0, 2]$. (a) Write an expression for MN, TN and SN with $N = 4$. (b) For each of the approximations determine an N so ...
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223 views

How to use undefined value in Composite Simpson's Rule

I have to use the Composite Simpson's Rule to approximate the integral $\int_0^1 t^2\cdot sin(\frac{1}{t}) dt$. I've used the Composite Simpson's Rule, but when I work through the algorithm, one step ...
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2k views

Length of bezier curve with Simpson's rule

I wish to approximate the length of a Bézier curve in a 2D plane. I could do this iteratively – and may – but I want to experiment using Simpson's rule. I am not sure how to set this up and could use ...
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220 views

numerical integration for N datapoints

I understand why Simpson's Rule is better than the trapezoidal rule for 3 datapoints (because under the assumption that the function is smooth, a parabolic approximation is going to be better than a ...