ABSTRACT
This chapter explores the definite integral from multiple perspectives and in a general context. One can use a partition to construct an estimate for the area by approximating the area of each sub-region bounded by the graph. In the context of developing the basic theory of the definite integral, the most commonly used types of approximating sums fall into two main categories. The chapter introduces and explores the utility of upper sums and lower sums. At first glance, any attempt at “partitioning first, estimating later” may seem like kicking the can down the road, since one is still faced with approximating the area of a region bounded above by the graph and below by a (sub)interval. However, one can gain traction by investigating the behavior of these approximating sums as one refines a given partition.
