So, in general, I can get this value:
$$\lim_{a \to \text{a constant}}{ \int{ \left( \sum_{x=x_1}^{x_2}{ f(a,x) }da \right)} } \tag{1}$$
What I'm after is this:
$$\sum_{x=x_1}^{x_2}{\left( \lim_{a \to \text{a constant}}{ \int{ f(a,x) }da } \right)} \tag{2}$$
So, essentially, I want to move the sum from inside an integral to outside an integral. The integral is very, very simple - it's continuous and Riemann integrable, etc. It can be done with elementary calculus. Then I plug in a value for the integrand.
The sum is also fairly simple; it is finite. The problem is that if I integrate first, I can't solve the resulting sum as it is. So I want to exchange the sum and the integral so that I can effectively get this value.
WHAT I'M AFTER
I'd like to know if I can always exchange these. Or, if not, when I can exchange these. I've already plugged in some values into my math software and both (1) and (2) gave the same result. I want to prove when I can do this, so I really also need sources or references that explain when I can do this.
SOME REFERENCES
Exchanging Sums and Integrals: Can we possibly exchange summation and integration with negative values?
Exchanging Sums and Limits: When can we exchange limits and summations?
I REALLY NEED A FAIRLY THOROUGH EXPLANATION BECAUSE I WANT TO PROVE THIS.