Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=\ln{\left(x \right)}$ on $LaTeX: \displaystyle (4,8)$ with $LaTeX: \displaystyle n=40$ .

$LaTeX: \displaystyle \Delta x = \frac{ 8 - 4 }{ 40 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 4 + i \frac{1}{10}$ 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. $LaTeX: \displaystyle x_k = 4 + (2k-1)\cdot \frac{1}{10}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =20$ and $LaTeX: \displaystyle x_j = 4 + (2j)\cdot \frac{1}{10}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =19$ . $LaTeX: \displaystyle f(4) +f(8)+4\sum_{k=1}^{20}f\left(\frac{k}{5} + \frac{39}{10}\right) + 2\sum_{j=1}^{19}f\left(\frac{j}{5} + 4\right)$ . The value is $LaTeX: \displaystyle 4.0619$