In the previous chapter we defined the definite integral, based on its interpretation as the area of a region in the $xy$-plane. We also developed a bunch of theory to help us work with integrals. This abstract definition, and the associated theory, turns out to be extremely useful simply because "areas of regions in the $xy$-plane" appear in a huge number of different settings, many of which seem superficially not to involve "areas of regions in the $xy$-plane". Here are some examples.