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Improper Integrals (9/24/08) There are two types of “improper integrals”: First Type: Definite integral taken over a ray or the whole real line, rather than over an interval of finite length. Second Type: Definite integral of a function which becomes unbounded (i.e., “blows up”) on the interval of integration.

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Example of First Type What is By this we mean, what is If this limit is a finite number, then we say the integral converges. If it is not a finite number, we say the integral diverges.

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Example of the second type What is As before, what this means is We make the same definition of converges and diverges

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Solutions of the examples

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Simpler approach In practice, we usually dispense with the “limit as b goes to” and simply use or 0. We understand that in the numerator will cause divergence, as will 1/0, whereas in the denominator gives a value of 0 to that term. For example, then

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Global Behavior With improper integrals of the first type, a function will behave like the ratio of its highest terms. Example: Does converge or diverge? Example: What about ?

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Basic Facts on Power and Exponential Functions converges for p > 1 and diverges for p 1. converges for p < 1 and diverges for p 1. converges for all a > 0.

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Assignment for Friday Hand-in #1 is due tomorrow (Thurs) at 4:45. Read Section 7.8 and do Exercises 1, 3, 5, 9-17 odd, 21, 27, 31, and 39.

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