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).

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This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

To evaluate these integrals, the fundamental theorem in calculus must be used. The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral. It expresses that the integral of a function F is equal to the function itself plus an arbitrary constant. This constant can be eliminated by substituting the upper and lower limit given in the definite integral. Thus, the limits of a given integral play an important role, for it determines the value of the integral of the function, which represents the area under the curve....
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