Use Simpson's rule to find the arclength of the curve $LaTeX: \displaystyle f(x)=x^{2}$ on $LaTeX: \displaystyle (5,7)$ with $LaTeX: \displaystyle n=40$ .

$LaTeX: \displaystyle \Delta x = \frac{ 7 - 5 }{ 40 }$ . $LaTeX: \displaystyle x_i = a +i\Delta x = 5 + i \frac{1}{20}$ 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. $LaTeX: \displaystyle x_k = 5 + (2k-1)\cdot \frac{1}{20}$ for $LaTeX: \displaystyle k=1$ to $LaTeX: \displaystyle k =20$ and $LaTeX: \displaystyle x_j = 5 + (2j)\cdot \frac{1}{20}$ for $LaTeX: \displaystyle j=1$ to $LaTeX: \displaystyle j =19$ . $LaTeX: \displaystyle f(5) +f(7)+4\sum_{k=1}^{20}f\left(\frac{k}{10} + \frac{99}{20}\right) + 2\sum_{j=1}^{19}f\left(\frac{j}{10} + 5\right)$ . The value is $LaTeX: \displaystyle 24.084$