lec35 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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± ± ± ± ± ± ± ± Lecture 35 18.01 Fall 2006 Lecture 35: Improper Integrals Definition. An improper integral , defined by M f ( x ) dx = lim f ( x ) dx a M →∞ a is said to converge if the limit exists (diverges if the limit does not exist). e kx dx = 1 /k ( k > 0) Example 1. 0 M M e kx dx = ( 1 /k ) e kx = (1 /k )(1 e kM ) 0 0 Taking the limit as M → ∞ , we find e kM 0 and e kx dx = 1 /k 0 We rewrite this calculation more informally as follows, 0 e kx dx = ( 1 /k ) e kx 0 = (1 /k )(1 e k ) = 1 /k (since k > 0 ) e kx dx = 1 /k has an easier formula than the Note that the integral over the infinite interval M 0 correspondingfiniteintegral e kx dx = (1 /k )(1 e kM ) . As a practical matter, for large M ,the 0 term e kM isnegligible,soeventhesimplerformula 1 /k servesasagoodapproximationtothefinite integral. Infinite integrals are often easier than finite ones, just as infinitesimals and derivatives are easier than difference quotients. Application: Replace x by t = time in seconds in Example 1. R = rate of decay = number of atoms that decay per second at time 0 . At later times t > 0 the decay rate is Re kt (smaller by an exponential factor e kt ) Eventually (over time 0 t < ) every atom decays. So the total number of atoms N is calculated using the formula we found in Example 1, Re kt dt = R/k N = 0 The half life H of a radioactive element is the time H at which the decay rate is half what it was at the start. Thus e kH = 1 / 2 = kH = ln(1 / 2)
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This note was uploaded on 01/18/2012 for the course MATH 18.01 taught by Professor Brubaker during the Fall '08 term at MIT.

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lec35 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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