Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=\frac{1}{x}$ on $LaTeX: \displaystyle (1,3)$ with $LaTeX: \displaystyle n=22$ .

$LaTeX: \displaystyle \Delta x = \frac{ 3 - 1 }{ 22 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 1 + i \frac{1}{11}$ 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. $LaTeX: \displaystyle x_k = 1 + (2k-1)\cdot \frac{1}{11}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =11$ and $LaTeX: \displaystyle x_j = 1 + (2j)\cdot \frac{1}{11}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =10$ . $LaTeX: \displaystyle f(1) +f(3)+4\sum_{k=1}^{11}f\left(\frac{2 k}{11} + \frac{10}{11}\right) + 2\sum_{j=1}^{10}f\left(\frac{2 j}{11} + 1\right)$ . The value is $LaTeX: \displaystyle 2.1466$