Lecture11 - Lecture 11 MATH 138-W12-003 I Improper...

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Lecture 11: MATH 138-W12-003 January 27, 2012 I) Improper Integrals, Section 7.8 Previously, when we discussed integration we had something like R b a f ( x ) dx we assumed two things implicitly: 1) the bounds a, b were both finite and 2) the function f ( x ) does not have an infinite discontinuity, for example like 1 /x at x = 0 . In this lecture we extend our notation of integration to include these two cases as well. We deal with each of these types separately. Infinite Intervals: To make things concrete consider the example f ( x ) = 1 /x 3 about the x -axis and on the inter- val x [1 , ) . The lower bound of 1 is somewhat arbitrary in that we could have chosen any finite value of x above 0 , since when x = 0 we have another problem in that the integrand blows up (becomes infinite in magnitude). What is perhaps surprising is that even though the interval is infinite, there are some integrands that have finite area. Indeed, let’s compute the area for an arbitrary upper bound, say t , using our knowledge of calculus. A ( t ) = Z t 1 1 x 3 dx = - 1 2 x 2 t 1 = 1 2 1 - 1 t 2 . One property to appreciate is that for any finite value of t the second term reduces the magnitude and therefore the area is necessarily less than 1 / 2 .
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