Recent questions in Simpson's Rule

Integral CalculusAnswered question

Julian Clements 2023-01-30

A= {1, 4, 9, 16, 25, 36, 49, 64, 81, 100} write the sets using rule method

Integral CalculusAnswered question

bandikizaui 2022-07-15

Which rule for numerical integration is more accurate, the Trapezoidal Rule or the Simpson Rule?

Integral CalculusAnswered question

letumsnemesislh 2022-07-10

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?

Integral CalculusAnswered question

Wisniewool 2022-07-05

Use both trapezoid and Simpson's rule to find ${\int}_{0}^{\mathrm{\infty}}{e}^{-x}dx$ starting with $h=2$ where $h$ is the length of subintervals $[{x}_{i},{x}_{i+1}]$.

Integral CalculusAnswered question

Waldronjw 2022-07-03

Problem applying Simpson's rule

$I={\int}_{0}^{2}{\displaystyle \frac{1}{x+4}}dx$

with $n=4$.

$I={\int}_{0}^{2}{\displaystyle \frac{1}{x+4}}dx$

with $n=4$.

Integral CalculusAnswered question

Esmeralda Lane 2022-07-02

What difference Between Simpsons Rule and $3/8$ rule?

Integral CalculusAnswered question

Lorena Beard 2022-07-01

Given the following data on $y=f(x)$,

$\overline{)\begin{array}{cc}\text{x}& \text{y}\\ 0& 32\\ 1& 38\\ 2& 29\\ 3& 33\\ 4& 42\\ 5& 44\\ 6& 38\end{array}}$

Calculate approximately ${\int}_{0}^{6}f(x)dx$.

$\overline{)\begin{array}{cc}\text{x}& \text{y}\\ 0& 32\\ 1& 38\\ 2& 29\\ 3& 33\\ 4& 42\\ 5& 44\\ 6& 38\end{array}}$

Calculate approximately ${\int}_{0}^{6}f(x)dx$.

Integral CalculusAnswered question

abbracciopj 2022-06-30

Evaluate the following using Simpson's rule (by using 2 strips).

${\int}_{1.6}^{1}\frac{sin2t}{t}dt$

${\int}_{1.6}^{1}\frac{sin2t}{t}dt$

Integral CalculusAnswered question

Eden Solomon 2022-06-27

Does there exist a monic ${x}^{n}$, $n>4$, for which Simpson’s rule is exact? If not, why?

$S(f)=\frac{b-a}{6}f(a)+\frac{2(b-a)}{3}f(\frac{a+b}{2})+\frac{b-a}{6}f(b)$

$S(f)=\frac{b-a}{6}f(a)+\frac{2(b-a)}{3}f(\frac{a+b}{2})+\frac{b-a}{6}f(b)$

Integral CalculusAnswered question

Sattelhofsk 2022-06-26

Is there any mathematical proof that I can always get a accurate value using composite simpson's rule rather than only using $\frac{3}{8}$ simpson's rule? First of all is my claim correct that we yield a better accuracy always with composite simpson's rule?

Integral CalculusAnswered question

Semaj Christian 2022-06-26

Derive Simpson's rule with error term by using

${\int}_{{x}_{0}}^{{x}_{2}}f(x)\phantom{\rule{thinmathspace}{0ex}}dx={a}_{0}f({x}_{0})+{a}_{1}f({x}_{1})+{a}_{2}f({x}_{2})+k{f}^{(4)}(\xi )$

Find ${a}_{0}$, ${a}_{1}$, and ${a}_{2}$ from the fact that Simpson's rule is exact for ${x}^{n}$ when $n=1$, $2$, and $3$. Then find k by applying the integration formula with $f(x)={x}^{4}$.

We certainly get the correct coefficients, but I don't see how this tells us we can combine ${\xi}_{1}$ and ${\xi}_{2}$

${\int}_{{x}_{0}}^{{x}_{2}}f(x)\phantom{\rule{thinmathspace}{0ex}}dx={a}_{0}f({x}_{0})+{a}_{1}f({x}_{1})+{a}_{2}f({x}_{2})+k{f}^{(4)}(\xi )$

Find ${a}_{0}$, ${a}_{1}$, and ${a}_{2}$ from the fact that Simpson's rule is exact for ${x}^{n}$ when $n=1$, $2$, and $3$. Then find k by applying the integration formula with $f(x)={x}^{4}$.

We certainly get the correct coefficients, but I don't see how this tells us we can combine ${\xi}_{1}$ and ${\xi}_{2}$

Integral CalculusAnswered question

xonycutieoxl1 2022-06-21

I've got two equal length vectors $x,y$ representing the pairs $({x}_{i},f({x}_{i}))$ and the components of the $x$ array aren't equally spaced.

Is there some modified version of Simpson's Rule that fits my purposes ?

Is there some modified version of Simpson's Rule that fits my purposes ?

Integral CalculusAnswered question

telegrafyx 2022-06-21

Relation between Simpson's Rule, Trapezoid Rule and Midpoint Rule.

How $2n$ numbers come out at the left side while only $n$ numbers at the right side.

How $2n$ numbers come out at the left side while only $n$ numbers at the right side.

Integral CalculusAnswered question

migongoniwt 2022-06-18

Prove that the simple Simpson’s rule

${\int}_{a}^{b}f(x)dx\approx \frac{(b-a)}{6}[f(a)+4f\left(\frac{a+b}{2}\right)+f(b)]$

is exact for all cubic polynomials.

${\int}_{a}^{b}f(x)dx\approx \frac{(b-a)}{6}[f(a)+4f\left(\frac{a+b}{2}\right)+f(b)]$

is exact for all cubic polynomials.

Integral CalculusAnswered question

Alannah Short 2022-06-14

The error bound formulas for trapezoidal rule and simpson's rule say that:

Error Bound for the Trapezoid Rule: Suppose that $\begin{array}{l}\text{Error Bound for the Trapezoid Rule: Suppose that}|{f}^{\mathrm{\prime}\mathrm{\prime}}(x)|\le k\text{for some}k\in \mathbb{R}\text{where}\\ a\le x\le b.\text{Then}\\ \phantom{\rule{2em}{0ex}}\left|{E}_{T}\right|\le k\frac{(b-a{)}^{3}}{12{n}^{2}}\\ \text{Error Bound for Simpson's Rule: Suppose that}|{f}^{(4)}(x)|\le k\text{for some}k\in \mathbb{R}\text{where}\\ a\le x\le b.\text{Then}\\ \phantom{\rule{2em}{0ex}}\left|{E}_{S}\right|\le k\frac{(b-a{)}^{5}}{180{n}^{4}}\end{array}$

Using these formulas, is it possible to find functions where Trapezoid Rule is more accurate than Simpson's rule?

Error Bound for the Trapezoid Rule: Suppose that $\begin{array}{l}\text{Error Bound for the Trapezoid Rule: Suppose that}|{f}^{\mathrm{\prime}\mathrm{\prime}}(x)|\le k\text{for some}k\in \mathbb{R}\text{where}\\ a\le x\le b.\text{Then}\\ \phantom{\rule{2em}{0ex}}\left|{E}_{T}\right|\le k\frac{(b-a{)}^{3}}{12{n}^{2}}\\ \text{Error Bound for Simpson's Rule: Suppose that}|{f}^{(4)}(x)|\le k\text{for some}k\in \mathbb{R}\text{where}\\ a\le x\le b.\text{Then}\\ \phantom{\rule{2em}{0ex}}\left|{E}_{S}\right|\le k\frac{(b-a{)}^{5}}{180{n}^{4}}\end{array}$

Using these formulas, is it possible to find functions where Trapezoid Rule is more accurate than Simpson's rule?

Integral CalculusAnswered question

Ezekiel Yoder 2022-06-13

Prove: Let$S(n)$ and $T(n)$ be the approximations of a function using n intervals by using Simpson's rule and the Trapezoid rule respectfully.

$S(2n)=\frac{4T(2n)-T(n)}{3}$

$S(2n)=\frac{4T(2n)-T(n)}{3}$

Integral CalculusAnswered question

boloman0z 2022-06-13

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:

${\int}_{{x}_{0}}^{{x}_{2}}f(x)dx\cong 2hf({x}_{1})+{h}^{3}\frac{{f}^{\u2033}({x}_{1})}{3}+{h}^{5}\frac{{f}^{(4)}(\xi )}{60}$

and then, it changed ${f}^{\u2033}({x}_{1})$ by

$\frac{f({x}_{0})-2f({x}_{1})+f({x}_{2})}{{h}^{2}}$

Where it came?

${x}_{1}={x}_{0}+h$

where

$h=\frac{{x}_{2}-{x}_{0}}{2}$

it got:

${\int}_{{x}_{0}}^{{x}_{2}}f(x)dx\cong 2hf({x}_{1})+{h}^{3}\frac{{f}^{\u2033}({x}_{1})}{3}+{h}^{5}\frac{{f}^{(4)}(\xi )}{60}$

and then, it changed ${f}^{\u2033}({x}_{1})$ by

$\frac{f({x}_{0})-2f({x}_{1})+f({x}_{2})}{{h}^{2}}$

Where it came?

Integral CalculusAnswered question

Jasmin Pineda 2022-06-12

$\frac{(b-a{)}^{5}{f}^{(4)}(c)}{2880{n}^{4}}$

for a $c\in [a,b]$, if the function has a continuous fourth derivative.

Is this for any $c$ in the interval, or just a unique one?

for a $c\in [a,b]$, if the function has a continuous fourth derivative.

Is this for any $c$ in the interval, or just a unique one?

Integral CalculusAnswered question

Tananiajtac2 2022-06-03

Approximate ${\int}_{0}^{1}\phantom{\rule{mediummathspace}{0ex}}\sqrt{2-{x}^{2}}dx$ using the trapezoidal and simpson's rule for $4$ intervals.

Now we can determine the simpson rule is

$\frac{h}{3}{\textstyle (}f({x}_{0})+4f({x}_{1})+2f({x}_{2})+4f({x}_{3})+f({x}_{4}){\textstyle )}$

and the trapezoidal rule is

$\frac{h}{2}{\textstyle (}f({x}_{0})+2f({x}_{1})+2f({x}_{2})+2f({x}_{3})+f({x}_{4}){\textstyle )}$

and $h=\frac{b-a}{n}$ which I assume is $\frac{1-0}{4}$

but how we add it all together?

Now we can determine the simpson rule is

$\frac{h}{3}{\textstyle (}f({x}_{0})+4f({x}_{1})+2f({x}_{2})+4f({x}_{3})+f({x}_{4}){\textstyle )}$

and the trapezoidal rule is

$\frac{h}{2}{\textstyle (}f({x}_{0})+2f({x}_{1})+2f({x}_{2})+2f({x}_{3})+f({x}_{4}){\textstyle )}$

and $h=\frac{b-a}{n}$ which I assume is $\frac{1-0}{4}$

but how we add it all together?

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