# Integration of a (0,1)-form on the boundary of a Riemann surface

In Simon Donaldson's book, he says that for any (0,1)-form $$\theta$$ on a compact connected Riemann surface $$X$$, the integral of $$\partial\theta$$ over $$X$$ is zero by Stokes' theorem - but that seems like he is using that $$\int_{\partial X}\theta=0,$$ and since $$X$$ might have boundary, how do we know that this is true? Is it even true? He makes a ton of small mistakes, so he might have meant to say a closed Riemann surface.