"Representational transformation diagrams" are used to compare and contrast standard textbook presentations of vector line integrals in undergraduate courses in both mathematics and physics. These presentations are taken as the lower anchor in a learning trajectory. Two principal approaches in the lower division are identified, roughly but not entirely correlated with these two disciplines. These textbook approaches are compared to existing characterizations for (single-variable) integration in the theory literature, notably "adding up pieces" (or "multiplicatively-based summation"), which is generalized here to chop, "multiply, add; accumulation from rate; quantitatively-based summation;" and a new characterization, parametric integrals. A review of upper-division textbooks establishes key features of an upper anchor in the learning trajectory. In conclusion, a hypothetical learning trajectory is presented, designed to scaffold student acquisition of rich concept images for vector line integrals.