Technology Tips: Using Maple to Enhance Students' Understanding of Numerical Integration

Typically, calculus students are introduced to the simplest numerical approximations of the definite integral through the process of finding the areas of rectangles. Students are initially shown how to use the endpoints of each subinterval to find lower and upper sums, a process that gives them a bound on the actual area. They are then shown, sometimes through a series of labor-intensive computations or through visualization with graphs, that as the number of rectangles, or partitions, increases, the approximations become more and more accurate. Somewhere in this process students are probably also shown how to use midpoints to obtain slightly more accurate numerical approximations. At this point, most calculus courses lead students toward the fundamental theorem of calculus, at which time they learn that they can evaluate a definite integral by finding the antiderivative and evaluating between the limits of integration.