Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=\sin{\left(x \right)}$ on $LaTeX: \displaystyle (2,5)$ with $LaTeX: \displaystyle n=24$ .

$LaTeX: \displaystyle \Delta x = \frac{ 5 - 2 }{ 24 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 2 + i \frac{1}{8}$ 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. $LaTeX: \displaystyle x_k = 2 + (2k-1)\cdot \frac{1}{8}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =12$ and $LaTeX: \displaystyle x_j = 2 + (2j)\cdot \frac{1}{8}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =11$ . $LaTeX: \displaystyle f(2) +f(5)+4\sum_{k=1}^{12}f\left(\frac{k}{4} + \frac{15}{8}\right) + 2\sum_{j=1}^{11}f\left(\frac{j}{4} + 2\right)$ . The value is $LaTeX: \displaystyle 3.6701$