An integral over a discrete set is simply a sum! In the general case, the integral of a function f with respect to a measure \mu can be computed by:
<br />
\int_A f d\mu = \sum_{a \in A} f(a) \mu(a)<br />
There is a discrete analog of a derivative called a difference:
<br />
\Delta_hf(x) = f(x + h) - f(x)<br />
(when h is omitted, assume it's 1)
And difference equations have many similarities with differential equations. For example, one can "solve" for the Fibonacci sequence which is defined by a linear second-order homogenous difference equation:
<br />
\Delta^2 F + \Delta F - F = 0 | F(0) = 0, F(1) = 1<br />
whose solution technique is directly analogous to that of similar differential equations: (use F(r) = a^r as a putative solution, get two linearly independent solutions, and take a linear combination that satisfies the initial conditions)
There's a more general concept here called a
skew derivation (or \sigma-derivation) of which both the ordinary derivative and this finite difference are examples.
And, of course, there's the antidifference operator, also called the summation operator, which bears a similar to indefinite integrals. For instance, you can even do summation by parts.
