# 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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1 vote
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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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### 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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### 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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