Given, a close curve $C$ and a vector field $F$, how can we find a bound, say $\epsilon$, between the line integral of $F$ and its Riemann approximation with $n$ samples?
$$\vert I(n) - L\vert\le \epsilon$$
where
$$I(n) := \sum_{i=1}^{n-1}F(r(t_i)) \cdot (r(t_i+\delta)-r(t_i))$$
and
$$L := \oint F(r)\cdot dr$$