Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=\frac{1}{x}$ on $LaTeX: \displaystyle (2,8)$ with $LaTeX: \displaystyle n=20$ .

$LaTeX: \displaystyle \Delta x = \frac{ 8 - 2 }{ 20 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 2 + i \frac{3}{10}$ 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. $LaTeX: \displaystyle x_k = 2 + (2k-1)\cdot \frac{3}{10}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =10$ and $LaTeX: \displaystyle x_j = 2 + (2j)\cdot \frac{3}{10}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =9$ . $LaTeX: \displaystyle f(2) +f(8)+4\sum_{k=1}^{10}f\left(\frac{3 k}{5} + \frac{17}{10}\right) + 2\sum_{j=1}^{9}f\left(\frac{3 j}{5} + 2\right)$ . The value is $LaTeX: \displaystyle 6.0204$