Suppose someone cuts a strange shape out of paper and challenges us to find the area of that paper. If we knew the shape of the boundary in terms of some equation, we might be able to set up and solve a double integral. In most cases, however, we do not have a formula for the boundary of regions, so we are forced to apporoximate the area.
The basic idea is to think of a region drawn on a sheet of graph paper. If we count the number of squares that are completely enclose within the figure, then we can compute a lower bound on the are of the region. Similarly, if we want an upper bound on the area of a region, then we can add up the areas of all boxes for which some point in the box is also in the region.
For the following exercises, you will need to load some special functions (written by your instructor) into Maple with the command: