Integration, the second major theme of calculus, deals with areas, volumes, masses, and averages such as centers of mass and gyration. We will learn analytical techniques for computing these quantities. Unfortunately, these analytical techniques do not always work and numerical schemes are required. In this lab we will concentrate on rectangular approximations to areas. These turn out to not always be very efficient numerical schemes, but they will be very useful in the theoretical development of the integral.
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 and then approximating the area above each subinterval. For example, dividing the interval [0,4] into four uniform pieces produces the subintervals [0,1], [1,2], [2,3], and [3,4].