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What is the relation between Lagrange interpolation and Simpson's rule to integrate some function with some points $x_{0}, f (x_{0})$ ; ... $x_{n}, f (x_{n})$ ?

ownerweneuf 2022-05-29

Does that make the answer from the Definite Integration a $100$ accurate? or is it just a more accurate estimation?

Wayne Steele 2022-05-23

Evaluate $\int_{- 1}^{0} (x^{4} - x^{2} + 2) d x$ using Simpson's Rule

Elliptic integrals The length of the ellipse
x=acost,y=bsint,0≤t≤2π $x = a \cos t, y = b \sin t, 0 \leq t \leq 2 π$
turns out to be
=4a∫π/201−e2cos2t√dt $= 4 a \int_{0}^{π / 2} \sqrt{1 - e^{2} \cos^{2} t} d t$
where e $e$ is the ellipse's eccentricity. The integral in this formula, called an elliptic integral, is non elementary except when e=0 $e = 0$ or 1 a. Use the Trapezoidal Rule with n=10 $n = 10$ to estimate the length of the ellipse when a=1 $a = 1$ and e=1/2 $e = 1 / 2$ . b. Use the fact that the absolute value of the second derivative of f(t)=1−e2cos2t√ $f (t) = \sqrt{1 - e^{2} \cos^{2} t}$ is less than 1 to find an upper bound for the error in the estimate you obtained in part (a).

dumnealorjavgj 2022-05-13

How to derive the quadrature weights for the trapezoid and the Simpsons rule?
Trapeziod rule: $\int_{a}^{b} f (x)$ $\approx$ $\frac{b - a}{2}$ $[f (a) + f (b)]$
Simpson's rule: $\int_{a}^{b} f (x)$ $\approx$ $\frac{b - a}{6}$ $[f (a) + 4 f ((a + b) / 2) + f (b)]$

Aedan Gonzales 2022-05-10

The function,
$f (x) = e^{x}$
at $11$ equidistant points on the interval $[0, 1]$ .
The question asks whether the trapezoidal rule on $10$ subintervals would give a better approximation than the Simpson rule on $10$ subintervals on the provided function?

Laila Andrews 2022-05-10

Let
$F (x) = \int_{- \infty}^{x} f (t) d t,$
where $x \in R$ , $f \geq 0$ is complicated (it cannot be integrated analytically).
Can we used the Simpson's rule to approximate this integral, knowing that $f (- \infty) = 0$ ?

lifretatox8n 2022-05-10

Show that one extrapolation of the trapezoid rule leads to Simpson's rule.

3c4ar1bzki1u 2022-05-08

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: $R (h) = \frac{16 S (h) - S (a)}{b}$ . What's a and b?

redupticslaz 2022-05-03

Estimate the minimum number of subintervals needed to approx. the integral
$\int_{0}^{1} x d x .$
with an error of magnitude less than $10^{- 4}$ by a) trapezoidal b) Simpson's?

affizupdaftf3opg 2022-04-12

Determine $M$ for Simpson's rule error bound: $2 e^{x} + x^{3}$ where $a = 0$ and $b = 1.6$ .

Jaeden Weaver 2022-04-07

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.

The calculus and analysis, integral calculus, and other concepts may sound like a nightmare to you, yet what about the Simpson's rule problems and answers that can help you with your studies? It may sound unreal but it has received quite a recognition. The equations provided by the Simpsons are not what you may remember from the series but the mass prediction of the Higgs Bosson is an interesting concept to consider when you want to write about Simpson's rule questions for your school assignment or a lab experiment. Take it with a grain of salt though as you research!

Integral Calculus

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