To integrate elliptic integral, we often cut riemann surface and glue them together, and gain a torus. We do this in order to avoid indeterminacy of integral, in other word, to avoid the condition that integral is not path independent.

However, on the torus, integral is not stil path independent. They differ up to two circles which generated first homology group of torus (in the picture,$r0$ and $r1$).

My question: What is the merit (necessity) of gluing to make torus and integrate on it? Indeterminancy cannot be solved by glueing process.