In extending the concept of integrals, we have so far considered three possible domains: intervals, regions, and solids. We further generalize this and consider integrals defined over curves or surfaces. We will also allow vector fields to be our integrands.
As we shall see, some of these new integrals can be computed using theorems analogous to the Fundamental Theorem of Calculus. Although seemingly diverse, these theorems are essentially variants of the same idea: that the integral of a function inside a set can be retrieved by inspecting the function's behavior on the set's boundary.