Consider a smooth function $f$ defined in the interval $[0,a]$. Let us assume that any order of its derivative is bounded by a constant $C$. It is only evaluated at $N+1$ fixed points, $ka/N$ for $0\le k \le N$. Let $h = a / N$. I want to numerically compute the integration $\int_0^a f(x) \mathrm{d}x$.
If I apply the Newton–Cotes formulas (or the stable version) of order $d$ (by which I mean the $h^d$ error is cancelled) with $n$ steps, the error should be of order $\mathcal{O}(h^{d+1}\cdot N/n) = \mathcal{O} (a h^d / n)$. Naively thinking, to minimize the error, I should take $d$ to be as large as possible. In this case, it is the largest $d$ such that the minimum number of steps $n_d$ to obtain this $d$-th order formula is not larger than $N$.
Is this correct? And if it true, is there a formula between $n_d$ and $d$?