Question regarding integration over smooth $0$-manifolds

Question

Let $\mathcal M := [a, b]$, this is a smooth manifold with boundary $\partial\mathcal M = \{a,b\}$. Let $f \in C^\infty(\mathcal M)$, then for any smooth $0$-manifold $N$, $$\int_Nf :=\sum_{p\in N}\pm f(p)$$ where the positive sign is taken when the orientation is positive and the negative sign where it is negative. We can use Stokes' theorem to get the following result: $$\int_\mathcal Mdf =\int_{\partial\mathcal M}f= (\pm f(a)) + (\pm f(b))$$ which should give us the result $f(b) -f(a)$, in order to recover the fundamental theorem of calculus. My question is: how do we determine the orientation of $\partial \mathcal M$ in order to get the desired result? Can $\partial \mathcal M$ be consistently oriented?