In the picture above, what exactly is the question asking for? I know that the error in simpson's rule is to the order of h^5.
Thus doubling the length increases error by 32x and so should I divide the first equation by 32 and subtract it from the second one?
What exactly would this achieve and how does this help me determine the integral?
Thanks for any help or advice given!
This is a badly formulated problem since the quantity computed is $\sim 4h·f(x_0)$, thus itself depending on $h$. The computation probably stands as example for the integration over a fixed interval with refinements by doubling the number of sub-intervals, i.e., so that $2^nh=(b-a)$. Thus even as the local error is $O(h^5)$, the global error is $O((b-a)h^4)$. Then the extrapolation formula for $S_h=C·h^4+O(h^6)$ is $$ 2^4·S_h-S_{2h}=15·S_*+O(h^6) $$ which gives coefficient sequence $$ \frac{2h}{3·15}·\Bigl([8,32,16,32,8]-[1,0,4,0,1]\Bigr) = \frac{2h}{45}·[7,32,12,32,7]. $$