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Definition of the Definite Integral

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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 axb 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".

Figure %: Computing the Area of R(f, a, b)

We can split up the interval [a, b] into n smaller intervals (for some integer n ) of width Δx = (b - a)/n . Let

s i = a + i(Δx)    

for i = 0, 1,…, n , so that the n intervals are given by [s 0, s 1],…,[s n-1, s n] .

Figure %: Splitting up the interval [a, b] into n smaller intervals

Let M i be the maximum value of f (x) on the interval [s i-1, s i] . Similarly, let m i be the minimum value of f (x) on the interval [s i-1, s i] . Consider the region made up of n rectangles, where the i -th rectangle is bounded horizontally by s i-1 and s i and vertically by 0 and M i . As shown below, this region contains R(f, a, b) .

Figure %: Containing R(f, a, b) with rectangles

Moreover, we know how to compute the area of this region. It is simply

(M 1) + (M 2) + ... + (M n) = M i    

We denote this n th upper Riemann sum by U n(f, a, b) . Replacing M i in the above with m i , we obtain a region contained in R(f, a, b) .

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