Recent research on integration has shown the importance of quantities-based meanings for integrals. However, this research body is still in need of detailed empirical accounts of "how" students develop such understandings across an entire unit on integration. This paper contributes by providing one such account, based on a quantities-based orientation called "adding up pieces" (AUP). Our study examines how three separate pairs of students learned definite integrals and integral functions within interview settings over four consecutive interview-lessons, meant to correspond to four consecutive in-class sessions. The first interview developed the partition, target quantity, and sum structure for definite integral, which were solidified in the second interview. The third and fourth interviews extended this AUP structure to a variable upper bound, output, and function structure for integral functions in preparation for the Fundamental Theorem of Calculus (FTC).