This question may be too low-level even for this site.
For simplicity let us assume the elliptic curve to be $\mathbb{C}/\{1, \tau\}$ where $\tau=x+iy$. Let the metric on the Riemann surface to be given by $$ |dz|=C $$
My question is, what is the area of the elliptic curve? Formally I can integrate the volume form to be $$ \int_{z\in E}\frac{i}{2}dz\wedge \overline{dz}\rightarrow \frac{y C^{2}}{2} $$ But it this rigorous? What is the very definition of finding the volume associated to the norm of a cotangent vector in a complex manifold? Somehow I could not find this in literature.