mth.122.handout.09

mth.122.handout.09 - MTH 122 Calculus II Essex County...

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MTH 122 — Calculus II Essex County College — Division of Mathematics and Physics 1 Lecture Notes #9 — Sakai Web Project Material 1 Improper Integrals—Type I You may recall that on the last handout we did the following three integrations—where f is a standard normal curve—using Mathematica’s built-in numerical integration techniques. Z -∞ f ( x ) d x = 1 Z 0 -∞ f ( x ) d x = 1 2 Z 0 f ( x ) d x = 1 2 These values here are indeed exact, but that is not the main issue of this particular handout. The feature of these integrals that is remarkable, is the actual limits, because all your prior limits have been finite. Definition of an Improper Integral, Type I 1. If Z b a f ( x ) d x exists for every b a , then Z a f ( x ) d x = lim b →∞ Z b a f ( x ) d x provided the limit exists. 2 If the limit exists we say its is convergent , if the limit does not exists we say it is divergent . 2. Similarly, if Z b a f ( x ) d x exists for every b a , then Z b -∞ f ( x ) d x = lim a →-∞ Z b a f ( x ) d x provided the limit exists. 3
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This note was uploaded on 01/31/2011 for the course MTH 222 taught by Professor Ban during the Spring '10 term at Essex County College.

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mth.122.handout.09 - MTH 122 Calculus II Essex County...

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