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].

Tue Nov 5 14:24:02 EST 1996