Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=\ln{\left(x \right)}$ on $LaTeX: \displaystyle (2,4)$ with $LaTeX: \displaystyle n=36$ .

$LaTeX: \displaystyle \Delta x = \frac{ 4 - 2 }{ 36 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 2 + i \frac{1}{18}$ 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. $LaTeX: \displaystyle x_k = 2 + (2k-1)\cdot \frac{1}{18}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =18$ and $LaTeX: \displaystyle x_j = 2 + (2j)\cdot \frac{1}{18}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =17$ . $LaTeX: \displaystyle f(2) +f(4)+4\sum_{k=1}^{18}f\left(\frac{k}{9} + \frac{35}{18}\right) + 2\sum_{j=1}^{17}f\left(\frac{j}{9} + 2\right)$ . The value is $LaTeX: \displaystyle 2.1208$