Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=x^{2}$ on $LaTeX: \displaystyle (3,8)$ with $LaTeX: \displaystyle n=36$ .

$LaTeX: \displaystyle \Delta x = \frac{ 8 - 3 }{ 36 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 3 + i \frac{5}{36}$ 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. $LaTeX: \displaystyle x_k = 3 + (2k-1)\cdot \frac{5}{36}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =18$ and $LaTeX: \displaystyle x_j = 3 + (2j)\cdot \frac{5}{36}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =17$ . $LaTeX: \displaystyle f(3) +f(8)+4\sum_{k=1}^{18}f\left(\frac{5 k}{18} + \frac{103}{36}\right) + 2\sum_{j=1}^{17}f\left(\frac{5 j}{18} + 3\right)$ . The value is $LaTeX: \displaystyle 55.244$