Definite Integrals (390 words)

On clean paper, legibly print or write, in clear and concise English, a description of the development of the definite integral over a segment [a, b]. You may use a specific function or a general function, f. Your description should be logically organized, didactic, and sufficient that any individual with pre-calculus training could follow it (you do not have to explain what a derivative is). It should also include the following:

* a basic statement of the problem (curve defined by y = f(x) on an interval [a, b];
* a list of conditions necessary for the Riemann integral to exist;
* an explanation of the role that partitions, minimums or maximums and limits play in the integral;
* the role that the fundamental theorem of calculus plays in evaluating definite integrals.

Note that you can write on the Riemann integral or the Darboux integral (whichever one you are more familiar with).

The limit of the sum of the area of the rectangles can be represented as the integral of the function of F(x), from the limits of a and b. This sum is also the Reimann Sum, representing the area bounded by the curve and the x-axis. The limit of the Reimann sum is the Reimann limit, as the partition gets finer. The mesh of the partitions must become smaller and smaller, so that in the limit, it is zero. For a function f to be considered Reimann integrable, two conditions must be satisfied: first, the function must be bounded, second, the function must be continuous almost everywhere in the interval [a,b], i.e. the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure....

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