The Definite Integral
In the previous chapter of our book, we became acquainted with the concept of the indefinite integral: the collection of primitive functions of f was called the indefinite integral of f. Now we introduce a very different kind of concept that we also call
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The Definite Integral
In the previous chapter of our book, we became acquainted with the concept of the indefinite integral: the collection of primitive functions of f was called the indefinite integral of f . Now we introduce a very different kind of concept that we also call integrals—definite integrals, to be precise. This concept, in contrast to that of the indefinite integral, assigns numbers to functions (and not a family of functions). In the next chapter, we will see that as the name integral that they share indicates, there is a strong connection between the two concepts of integrals.
14.1 Problems Leading to the Definition of the Definite Integral The concept of the definite integral—much like that of differential quotients —arose as a generalization of ideas in mathematics, physics, and other branches of science. We give three examples of this. Calculating the Area Under the Graph of a Function. Let f be a nonnegative bounded function on the interval [a, b]. We would like to find the area A of the region S f = {(x, y) : x ∈ [a, b], 0 ≤ y ≤ f (x)} under the graph of f . As we saw in Example 13.23, the area can be easily computed assuming that f is nonnegative, monotone increasing, continuous, and that we know a primitive function of f . If, however, those conditions do not hold, then we need to resort to a different method. Let us return to the argument Archimedes used (page 3) when he computed the area underneath the graph of x2 over the interval [0, 1] by partitioning the interval with base points xi = i/n and then bounding the area from above and below with approximate areas (see Figure 14.1). Similar processes are often successful. The computation is sometimes easier if we do not use a uniform partitioning. Example 14.1. Let 0 < a < b and f (x) = 1/x. In this case, a uniform partitioning will not help us as much as using the base points xi = a · qi (i = 0, . . . , n), where q = n b/a. © Springer New York 2015 M. Laczkovich, V.T. S´os, Real Analysis, Undergraduate Texts in Mathematics, DOI 10.1007/978-1-4939-2766-1 14
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14 The Definite Integral
Fig. 14.1
The function is monotone decreasing, so the area over the interval [xi−1 , xi ] is at least 1 xi−1 · (xi − xi−1 ) = 1 − = 1 − q−1 (14.1) xi xi and at most 1 xi · (xi − xi−1 ) = − 1 = q − 1. xi−1 xi−1 Thus we have the bounds n b −1 n a n 1 − b = n(1 − q ) ≤ A ≤ n(q − 1) = n a −1 for the area A. Since
x b (b/a)1/n − 1 n b lim n − 1 = lim = = log (b/a) n→∞ n→∞ a 1/n a x=0 and similarly
a (a/b)1/n − 1 = lim n 1 − n = − lim n→∞ n→∞ b 1/n a x =− = − log (a/b) = log (b/a) , b x=0
we have A = log(b/a).
(14.2)
14.1 Problems Leading to the Definition of the Definite Integral
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We could have recovered the same result using the method in Example 13.23. Even though the function 1/x is monotone decreasing and not increasing, from the point of view of this method, that is of no significance. Moreover, since log x is a primitive function of 1/x, the area that we seek is A = log b − log a = log(b/a). Let us not
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