In this presentation, we explore the accumulation function in three dimensions; specifically characterizing an accumulation approach to line integrals. The goal of an accumulation approach is to go beyond strictly conceptualizing integration or accumulation as simply "area under the curve." We build up the notion of the accumulation of line integrals via a characterization of an accumulation approach to arc length. I extend existing curriculum on the accumulation of arc length, in the context of Calculus II, to that of scalar line integrals in Calculus IV. In this presentation, I will discuss the characterizations of the accumulation of arc length and line integrals, as well as present the accompanying curricular unit and activities, and results from enacting these activities in a Calculus IV classroom.

Our goal is to numerically approximate the deformation of the Earth's surface caused by an ellipsoidal magma cavity over a long period of time. We begin by using the cavity geometry to make an assumption of axisymmetry. From this assumption we can derive the constitutive equations that arise from a generalized Hooke's law. Along with the system's equilibrium equations and an imposed symmetry we then derive the variational form of the problem to be used in a finite element method.