Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=\sin{\left(x \right)}$ on $LaTeX: \displaystyle (1,8)$ with $LaTeX: \displaystyle n=30$ .

$LaTeX: \displaystyle \Delta x = \frac{ 8 - 1 }{ 30 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 1 + i \frac{7}{30}$ 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. $LaTeX: \displaystyle x_k = 1 + (2k-1)\cdot \frac{7}{30}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =15$ and $LaTeX: \displaystyle x_j = 1 + (2j)\cdot \frac{7}{30}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =14$ . $LaTeX: \displaystyle f(1) +f(8)+4\sum_{k=1}^{15}f\left(\frac{7 k}{15} + \frac{23}{30}\right) + 2\sum_{j=1}^{14}f\left(\frac{7 j}{15} + 1\right)$ . The value is $LaTeX: \displaystyle 8.3855$