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 
is a non-negative, continuous function defined on some
interval 
. Then by the area under the curve 
between
x=a and x=b we mean the area of the region bounded above by the
graph of 
, 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 
into subintervals of uniform
length and then approximating the area above each subinterval. For
example, dividing the interval 
into four
uniform pieces produces the subintervals 
, 
,
, and 
.