In previous chapters we have explored a relationship between vectors and integration. Our most tangible result: if is the vector-valued velocity function of a moving object, then integrating from to gives the displacement of that object over that time interval.
This chapter explores completely different relationships between vectors and integration. These relationships will enable us to compute the work done by a magnetic field in moving an object along a path and find how much air moves through an oddly-shaped screen in space, among other things.
Our upcoming work with integration will benefit from a review. We are not concerned here with techniques of integration, but rather what an integral “does” and how that relates to the notation we use to describe it.