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 $n=4$.

Waldronjw
2022-07-03
Answered

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 $n=4$.

You can still ask an expert for help

Rafael Dillon

Answered 2022-07-04
Author has **15** answers

The factor should be

$\frac{(b-a)}{3n}$

with $b-a=2$ and $n=4$ you get $1/6$ and not $1/3$. One half of $0.8$ is $0.4$ which is close to the exact value.

You can quickly check the correctness of the factor by using the constant function $f=1$.

$\frac{(b-a)}{3n}$

with $b-a=2$ and $n=4$ you get $1/6$ and not $1/3$. One half of $0.8$ is $0.4$ which is close to the exact value.

You can quickly check the correctness of the factor by using the constant function $f=1$.

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

asked 2022-07-02

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

asked 2022-06-01

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

asked 2022-06-15

Calculate:

${\int}_{-14}^{-8}ydx$

${\int}_{-14}^{-8}ydx$

asked 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 ?

asked 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.

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

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

asked 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?