= Solution
With independent samples and finite <variance>, <Monte Carlo integration> has a root-mean-square stochastic error proportional to \b[$N^{-1/2}$]. The estimated <standard error> measures that random sampling uncertainty; it is not a deterministic bound.
For a sufficiently smooth one-dimensional <integrand>, the composite <trapezoidal rule> has error
$$
I-T_n=-\frac{(b-a)^3}{12n^2}f''(\xi)
$$
for some $\xi\in(a,b)$, hence \b[$O(n^{-2})$]. It is normally much more efficient in one dimension. The stated continuity assumption alone guarantees convergence, but does not guarantee this second-order rate; bounded second derivatives are an additional assumption.
The dimensional advantage is different: a tensor grid with $N$ total points in $d$ dimensions has only about $N^{1/d}$ points per direction and typically error $O(N^{-2/d})$. The Monte Carlo exponent remains $-1/2$, although its <variance> prefactor can still deteriorate with dimension. This explains why the stochastic method becomes attractive for high-dimensional integrals.
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