Definite integral registers using infinitesimals

I propose a theoretical model of representation registers for definite integral notation. The two registers, adding up pieces (AUP) and multiplicatively-based summation (MBS), are developed from modes of interpreting integral notation identified by S. Jones (2015). In this model, the AUP register affords modeling with definite integral notation, while the MBS register affords sense-making with and evaluation of integrals. These registers are illustrated in the context of a Calculus I class that used an informal infinitesimals approach; in this class differentials such as dx directly represented infinitesimal quantities instead of serving as a reminder of a quantity that existed before a limit was taken. Theoretical implications of extending Duval's register theory (2006) in this way are also explored.