Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=\sqrt{x}$ on $LaTeX: \displaystyle (5,10)$ with $LaTeX: \displaystyle n=22$ .

$LaTeX: \displaystyle \Delta x = \frac{ 10 - 5 }{ 22 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 5 + i \frac{5}{22}$ 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. $LaTeX: \displaystyle x_k = 5 + (2k-1)\cdot \frac{5}{22}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =11$ and $LaTeX: \displaystyle x_j = 5 + (2j)\cdot \frac{5}{22}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =10$ . $LaTeX: \displaystyle f(5) +f(10)+4\sum_{k=1}^{11}f\left(\frac{5 k}{11} + \frac{105}{22}\right) + 2\sum_{j=1}^{10}f\left(\frac{5 j}{11} + 5\right)$ . The value is $LaTeX: \displaystyle 5.0859$