When can the order of limit and integral be exchanged?

The most useful results are the Lebesgue dominated convergence and monotone convergence theorems.

@Tim: You wrote in a comment: "For both theorems you mentioned, they apply to a discrete sequence of functions. In my questions, the index is continuous. How would that be coped with?"

If $\lim_{y\to a} f(x,y)$ exists, then $\lim_{n\to\infty} f(x,y_n)$ exists, for every sequence $\{y_n\}_{n=1}^\infty$ that approaches $y$, and conversely. You can use that to show that the dominated convergence theorem and the monotone convergence theorem still work in the "continuous" setting.