This thesis analyses the metacognitive behaviour of mathematics undergraduates in order to understand more about their behaviour in solving single and multi-variable integration. The project offers a framework for other researchers to use in analysing metacognitive behaviour for the purposes of understanding more about student's thought processes in solving single and multi variable integration, as well as in other domains of mathematics. This is for the purpose of aiding educational practitioners rather than providing any pedagogical strategy, per se. Situated in the field of Advanced Mathematical Thinking (see Tall, 1991 for summary), the study was designed using Schoenfeld's (1985a) method of think-aloud verbal transcripts and protocol analysis, to investigate the mathematical thinking of 1st year undergraduates. Three groups of students were studied over a period of six months. The model was adapted through a 3-stage empirical process by allowing interventions by the researcher and analysing their impact on the student's thought processes. The study concentrated on self-regulatory metacognitive behaviour including Reflection, Organisation, Monitoring and Extraction, which developed a ROME model of analysis. The results of the study offered more mathematical interpretations of the students' self-regulatory behaviour solving single and multi variable integral problems. Through this mathematical efficacy in Calculus problem solving has been discussed in terms of ROME. The development of the ROME model of analysis has the potential to be used to analyse metacognitive behaviour of problem-solvers in other fields of mathematics.