Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=\sin{\left(x \right)}$ on $LaTeX: \displaystyle (2,4)$ with $LaTeX: \displaystyle n=22$ .

$LaTeX: \displaystyle \Delta x = \frac{ 4 - 2 }{ 22 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 2 + i \frac{1}{11}$ Using the 1,4,2,...,2,4,1 pattern the sum can be written as $LaTeX: \displaystyle x_i$ can be written split into the even and odd terms. $LaTeX: \displaystyle x_k = 2 + (2k-1)\cdot \frac{1}{11}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =11$ and $LaTeX: \displaystyle x_j = 2 + (2j)\cdot \frac{1}{11}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =10$ . $LaTeX: \displaystyle f(2) +f(4)+4\sum_{k=1}^{11}f\left(\frac{2 k}{11} + \frac{21}{11}\right) + 2\sum_{j=1}^{10}f\left(\frac{2 j}{11} + 2\right)$ . The value is $LaTeX: \displaystyle 2.6149$