Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=\ln{\left(x \right)}$ on $LaTeX: \displaystyle (3,6)$ with $LaTeX: \displaystyle n=40$ .

$LaTeX: \displaystyle \Delta x = \frac{ 6 - 3 }{ 40 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 3 + i \frac{3}{40}$ 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. $LaTeX: \displaystyle x_k = 3 + (2k-1)\cdot \frac{3}{40}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =20$ and $LaTeX: \displaystyle x_j = 3 + (2j)\cdot \frac{3}{40}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =19$ . $LaTeX: \displaystyle f(3) +f(6)+4\sum_{k=1}^{20}f\left(\frac{3 k}{20} + \frac{117}{40}\right) + 2\sum_{j=1}^{19}f\left(\frac{3 j}{20} + 3\right)$ . The value is $LaTeX: \displaystyle 3.082$