As the comments have said, the Zariski topology is not hausdorff, and hence is not metrizable, which means you can't talk about convergence. For example, the Zariski topology of any algebraic curve is just the cofinite topology!
Though, the real thing is that the Zariski topology doesn't exist for doing analysis. Generally, classic algebro-geometric spaces come in two types - the characteristic 0 type (varieties over a field of characteristic 0), and the positive characteristic type (varieties over a characteristic p field). You can consider the analytic properties of any characteristic 0 space by looking at the corresponding analytification, as described by Serre's GAGA theorems (e.g. http://en.wikipedia.org/wiki/GAGA#GAGA). These are "analytic spaces", which are the equivalent of complex manifolds with singularities. Positive characteristic spaces (for example, varieties over a finite field), generally only have finitely many points, so it doesn't make much sense to do analysis on it.
However, I think you can still talk about integration by looking at functionals on the space of differential forms, which can be defined in any characteristic! (after all, integration is just a way of assigning a number to a differential form). I know this is one way of defining the Jacobian variety in the characteristic 0 case, and I suspect this also makes sense in characteristic $p$.
If you're interested in integration in characteristic $p$, perhaps you should read about coleman integrals, which aren't really characteristic $p$, but at least it's $p$-adic! See for example:
$p$-adic integrals and Cauchy's theorem
or just google "coleman integral" or "p-adic integral".
