 # Get help with Simpson's rule questions

Recent questions in Simpson's Rule migongoniwt 2022-06-18 Answered

### Prove that the simple Simpson’s rule${\int }_{a}^{b}f\left(x\right)dx\approx \frac{\left(b-a\right)}{6}\left[f\left(a\right)+4f\left(\frac{a+b}{2}\right)+f\left(b\right)\right]$is exact for all cubic polynomials. Dale Tate 2022-06-15 Answered

### Calculate:${\int }_{-14}^{-8}ydx$ Alannah Short 2022-06-14 Answered

### The error bound formulas for trapezoidal rule and simpson's rule say that:Error Bound for the Trapezoid Rule: Suppose that Using these formulas, is it possible to find functions where Trapezoid Rule is more accurate than Simpson's rule? Ezekiel Yoder 2022-06-13 Answered

### Prove: Let$S\left(n\right)$ and $T\left(n\right)$ be the approximations of a function using n intervals by using Simpson's rule and the Trapezoid rule respectfully.$S\left(2n\right)=\frac{4T\left(2n\right)-T\left(n\right)}{3}$ boloman0z 2022-06-13 Answered

### From a proof of Simpson's rule using Taylor polynomial where $f\in \left[{x}_{0},{x}_{2}\right]$ and, for${x}_{1}={x}_{0}+h$where$h=\frac{{x}_{2}-{x}_{0}}{2}$it got:${\int }_{{x}_{0}}^{{x}_{2}}f\left(x\right)dx\cong 2hf\left({x}_{1}\right)+{h}^{3}\frac{{f}^{″}\left({x}_{1}\right)}{3}+{h}^{5}\frac{{f}^{\left(4\right)}\left(\xi \right)}{60}$and then, it changed ${f}^{″}\left({x}_{1}\right)$ by$\frac{f\left({x}_{0}\right)-2f\left({x}_{1}\right)+f\left({x}_{2}\right)}{{h}^{2}}$Where it came? Jasmin Pineda 2022-06-12 Answered

### $\frac{\left(b-a{\right)}^{5}{f}^{\left(4\right)}\left(c\right)}{2880{n}^{4}}$for a $c\in \left[a,b\right]$, if the function has a continuous fourth derivative.Is this for any $c$ in the interval, or just a unique one? Tananiajtac2 2022-06-03 Answered

### 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$\frac{h}{3}\left(f\left({x}_{0}\right)+4f\left({x}_{1}\right)+2f\left({x}_{2}\right)+4f\left({x}_{3}\right)+f\left({x}_{4}\right)\right)$and the trapezoidal rule is$\frac{h}{2}\left(f\left({x}_{0}\right)+2f\left({x}_{1}\right)+2f\left({x}_{2}\right)+2f\left({x}_{3}\right)+f\left({x}_{4}\right)\right)$and $h=\frac{b-a}{n}$ which I assume is $\frac{1-0}{4}$but how we add it all together? Imani Bentley 2022-06-01 Answered

### What is the relation between Lagrange interpolation and Simpson's rule to integrate some function with some points ${x}_{0},f\left({x}_{0}\right)$; ... ${x}_{n},f\left({x}_{n}\right)$ ? ownerweneuf 2022-05-29 Answered

### 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 Answered

2022-05-17

### Elliptic integrals The length of the ellipsex=acost,y=bsint,0≤t≤2πturns out to be=4a∫π/201−e2cos2t√dt$=4a{\int }_{0}^{\pi /2}\sqrt{1-{e}^{2}{\mathrm{cos}}^{2}t}dt$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√$f\left(t\right)=\sqrt{1-{e}^{2}{\mathrm{cos}}^{2}t}$ is less than 1 to find an upper bound for the error in the estimate you obtained in part (a). dumnealorjavgj 2022-05-13 Answered

### How to derive the quadrature weights for the trapezoid and the Simpsons rule?Trapeziod rule:$\underset{a}{\overset{b}{\int }}f\left(x\right)$$\approx$$\frac{b-a}{2}$$\left[f\left(a\right)+f\left(b\right)\right]$Simpson's rule:$\underset{a}{\overset{b}{\int }}f\left(x\right)$$\approx$$\frac{b-a}{6}$$\left[f\left(a\right)+4f\left(\left(a+b\right)/2\right)+f\left(b\right)\right]$ Aedan Gonzales 2022-05-10 Answered

### The function,$f\left(x\right)={e}^{x}$at $11$ equidistant points on the interval $\left[0,1\right]$.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 Answered

### Let$F\left(x\right)={\int }_{-\mathrm{\infty }}^{x}f\left(t\right)dt,$where $x\in \mathcal{R}$, $f\ge 0$ is complicated (it cannot be integrated analytically).Can we used the Simpson's rule to approximate this integral, knowing that $f\left(-\mathrm{\infty }\right)=0$? lifretatox8n 2022-05-10 Answered

### Show that one extrapolation of the trapezoid rule leads to Simpson's rule. 3c4ar1bzki1u 2022-05-08 Answered

### Assume that $S\left(h\right)$ 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: $R\left(h\right)=\frac{16S\left(h\right)-S\left(a\right)}{b}$. What's a and b? redupticslaz 2022-05-03 Answered

### Estimate the minimum number of subintervals needed to approx. the integral${\int }_{0}^{1}x\phantom{\rule{thinmathspace}{0ex}}dx.$with an error of magnitude less than ${10}^{-4}$ by a) trapezoidal b) Simpson's? affizupdaftf3opg 2022-04-12 Answered

### 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 Answered