121616949-math.220 - know that one interpretation of Z D 1...

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206 Chapter 9 Applications of Integration Such an integral, with a limit of infinity, is called an improper integral . This is a bit unfortunate, since it’s not really “improper” to do this, nor is it really “an integral”—it is an abbreviation for the limit of a particular sort of integral. Nevertheless, we’re stuck with the term, and the operation itself is perfectly legitimate. It may at first seem odd that a finite amount of work is sufficient to lift an object to “infinity”, but sometimes surprising things are nevertheless true, and this is such a case. If the value of an improper integral is a finite number, as in this example, we say that the integral converges , and if not we say that the integral diverges . Here’s another way, perhaps even more surprising, to interpret this calculation. We
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Unformatted text preview: know that one interpretation of Z D 1 1 x 2 dx is the area under y = 1 /x 2 from x = 1 to x = D . Of course, as D increases this area increases. But since Z D 1 1 x 2 dx =-1 D + 1 1 , while the area increases, it never exceeds 1, that is Z ∞ 1 1 x 2 dx = 1 . The area of the infinite region under y = 1 /x 2 from x = 1 to infinity is finite. Consider a slightly different sort of improper integral: Z ∞-∞ xe-x 2 dx . There are two ways we might try to compute this. First, we could break it up into two more familiar integrals: Z ∞-∞ xe-x 2 dx = Z-∞ xe-x 2 dx + Z ∞ xe-x 2 dx. Now we do these as before: Z-∞ xe-x 2 dx = lim D →∞-e-x 2 2 fl fl fl fl fl D =-1 2 , and Z ∞ xe-x 2 dx = lim D →∞-e-x 2 2 fl fl fl fl fl D = 1 2 ,...
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