Introducing the Integral Calculus in Edwards & Penney

presentation of Edwards & Penney's work, by presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

[the following will not stray from Edwards & Penney's procedure, so it is largely quotation.]

Where differential calculus dealt with the derivative, integral calculus is based on the integral, which is used to define and calculate the area spanning from under function f 's graph to the x-axis and across the closed interval [a,b]. The region R's area, in the diagram below

is given by the integral of f from a to b, denoted by the symbol

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.272a.