For questions regarding Simpson's rule and its applications.

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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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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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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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63 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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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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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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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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108 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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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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129 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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51 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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714 views

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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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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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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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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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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357 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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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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64 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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200 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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191 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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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 ...