Let's assume we are given an $n$-dimensional smooth curve $\gamma:[a,b] \rightarrow \mathbb{R}^n$ and $N$- sampled points $\{x_1,...,x_N\}$ of that curve.
Now we use linear interpolation (or a higher order method) to get the curve
$$l:[a,b] \rightarrow \mathbb{R}^n.$$
My question is the following: How can we estimate the error of the following quantity (I defined the quantity here for a 2 dimensional curve $(\gamma_1,\gamma_2)$ and it is fine if the answer also only treats the 2 dimensional case and leaves the generalisation to me)
$$\int_a^b \int_0^t \gamma_1(s)^{i_1}\gamma_2(s)^{i_2} dsdt \quad i_1,i_2 \in \mathbb{N}$$
in dependence of the number $N$ of sampled points? I.e. we approximate $(\gamma_1,\gamma_2)$ by $(l_1,l_2)$. Equivalently, my question is how to approximate
$$\int_a^b \int_0^t| \gamma_1(s)^{i_1}\gamma_2(s)^{i_2}-l_1(s)^{i_1}l_2(s)^{i_2}| dsdt \quad i_1,i_2 \in \mathbb{N}$$
In general the error should decrease with $N$ getting bigger. I know that for a function $f: [a,b] \rightarrow \mathbb{R}$ the interpolation error can be bound by $$\left|f(x)-l(x)\right| \leq \frac{\left(b-a\right)^{2}}{8} \max _{a \leq x \leq b}\left|f^{\prime \prime}(x)\right|.$$ So we could try to bound
$$\|\gamma_1(t)-l_1(t)\| \leq \frac{\left(b-a\right)^{2}}{8} \max _{a \leq t \leq b}\left|\gamma_1^{\prime \prime}(t)\right|$$ but I don't think this really helpes.