Integration, the second major theme of calculus, deals with areas, volumes, masses, and averages such as centers of mass and gyration. In lecture you have learned that the area under a curve between two points a and b can be found as a limit of a sum of areas of rectangles which are in some sense ``under'' the curve of interest. As these sums, and their limits, are often tedious to calculate, there is clear motivation for the analytical techniques which will be introduced shortly in class. However, not all ``area finding'' problems can be solved using analytical techniques, and the Riemann sum definition of area under a curve gives rise to several numerical methods which can often approximate the area of interest with great accuracy.
Suppose f(x) is a non-negative, continuous function defined on some interval [a,b]. Then by the area under the curve y=f(x) between x=a and x=b we mean the area of the region bounded above by the graph of f(x), below by the x axis, on the left by the vertical line x=a, and on the right by the vertical line x=b.
All of the numerical methods in this lab depend on subdividing the interval [a,b] into subintervals of uniform length. For example, dividing the interval [0,4] into four uniform pieces produces the subintervals [0,1], [1,2], [2,3], and [3,4].