$9\mathrm{.}2$ Computational Integration: Simpson's method $($By$-$Hand$)$

$6$ pts $($a$)$ Evaluate the integral for the impulse by hand using the Simpson's rule, using the same $n=8$ equally$-$spaced nodes as you did in HW$9\mathrm{.}1.($Hint: for this value of $n$ do you need a combo of $\frac{1}{3}$ and $\frac{3}{8}-$ methods?$)$ And because the Simpson's rule is so accurate, $($I know this is a pain!$)$ you have to do all calculations accurate to at least $6$ decimals $($nearly all the decimals on your calculator$).$

$($b$)$ Using the analytic value of the integral above, calculate the percent $($relative$)$ error in calculating the Impulse for the Simpson's rule, and compare that to the errors in HW$9\mathrm{.}1$ from the Midpoint and Trapezoid rules.

Reminder: Show all your work on paper, especially the values you're using for all the nodes, and do each calculation accurate to at least $6$ decimals!