121616949-math.221 - F in moving from x to x 1 In the case...

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9.7 Kinetic energy; improper integrals 207 so Z -∞ xe - x 2 dx = - 1 2 + 1 2 = 0 . Alternately, we might try Z -∞ xe - x 2 dx = lim D →∞ Z D - D xe - x 2 dx = lim D →∞ - e - x 2 2 fl fl fl fl fl D - D = lim D →∞ - e - D 2 2 + e - D 2 2 = 0 . So we get the same answer either way. This does not always happen; sometimes the second approach gives a finite number, while the first approach does not; the exercises provide examples. In general, we interpret the integral Z -∞ f ( x ) dx according to the first method: both integrals Z a -∞ f ( x ) dx and Z a f ( x ) dx must converge for the original integral to converge. The second approach does turn out to be useful; when lim D →∞ Z D - D f ( x ) dx = L , and L is finite, then L is called the Cauchy Principal Value of Z -∞ f ( x ) dx . Here’s a more concrete application of these ideas. We know that in general W = Z x 1 x 0 F dx is the work done against the force
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Unformatted text preview: F in moving from x to x 1 . In the case that F is the force of gravity exerted by the earth, it is customary to make F < 0 since the force is “downward.” This makes the work W negative when it should be positive, so typically the work in this case is defined as W =-Z x 1 x F dx. Also, by Newton’s Law, F = ma ( t ). This means that W =-Z x 1 x ma ( t ) dx. Unfortunately this integral is a bit problematic: a ( t ) is in terms of t , while the limits and the “ dx ” are in terms of x . But x and t are certainly related here: x = x ( t ) is the function that gives the position of the object at time t , so v = v ( t ) = dx/dt = x ± ( t ) is its velocity...
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