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We will develop the definite integral as a means to calculate the area of certain regions in the plane. Given two real numbers a < b and a function f (x) defined on the interval [a, b], define the region R(f, a, b) to be the set of points (x, y) in the plane with a≤x≤b and with y between 0 and f (x). Note that this region may lie above the x-axis, or below, or both, depending on whether f (x) is positive or negative. In computing the area of R(f, a, b), it will be convenient to count the regions above the x-axis as having "positive area", and those below as having "negative area".
We can split up the interval [a, b] into n smaller intervals (for some integer n) of width Δx = (b - a)/n. Let
|si = a + i(Δx)|
for i = 0, 1, , n, so that the n intervals are given by [s0, s1], ,[sn-1, sn].
Let Mi be the maximum value of f (x) on the interval [si-1, si]. Similarly, let mi be the minimum value of f (x) on the interval [si-1, si]. Consider the region made up of n rectangles, where the i-th rectangle is bounded horizontally by si-1 and si and vertically by 0 and Mi. As shown below, this region contains R(f, a, b).
Moreover, we know how to compute the area of this region. It is simply
|(M1) + (M2) + ... + (Mn) = Mi|
We denote this nth upper Riemann sum by Un(f, a, b). Replacing Mi in the above with mi, we obtain a region contained in R(f, a, b).
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