Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=\sqrt{x}$ on $LaTeX: \displaystyle (4,8)$ with $LaTeX: \displaystyle n=26$ .

$LaTeX: \displaystyle \Delta x = \frac{ 8 - 4 }{ 26 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 4 + i \frac{2}{13}$ 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. $LaTeX: \displaystyle x_k = 4 + (2k-1)\cdot \frac{2}{13}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =13$ and $LaTeX: \displaystyle x_j = 4 + (2j)\cdot \frac{2}{13}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =12$ . $LaTeX: \displaystyle f(4) +f(8)+4\sum_{k=1}^{13}f\left(\frac{4 k}{13} + \frac{50}{13}\right) + 2\sum_{j=1}^{12}f\left(\frac{4 j}{13} + 4\right)$ . The value is $LaTeX: \displaystyle 4.0857$