Calculus II Notes 8.8 - Calculus II-Stewart Dr. Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 8.8 8.8 Improper Integrals The fundamental theorem of calculus applies to functions that are continuous on a closed bounded interval. An integral of this type is called proper. We now consider two other cases. A type I improper integral is continuous on an unbounded interval, and a type II is an integral of an unbounded function. Type I: Infinite Intervals Definition We define 1) f ( x ) a dx = lim b →∞ f ( x ) a b dx if it converges, 2) f ( x ) −∞ b dx = lim a →−∞ f ( x ) a b dx if it converges, and 3) f ( x ) −∞ dx = f ( x ) c dx + f ( x ) −∞ c dx if both of these converge. Example A Evaluate, if possible, e 2 x 0 dx . Solution : e 2 x 0 dx = lim b →∞ e 2 x 0 b dx = lim b →∞ e 2 b 2 − − e 2 0 ( ) 2 = 1 2 . Example B Evaluate, if possible, 1 x 1 dx . Solution : 1 x 1 dx = lim b →∞ 1 x 1 b dx = lim b →∞ ln b ( ) ln1 ( ) [ ] = . Example C Evaluate, if possible, 1 x 2 1 dx .
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Calculus II Notes 8.8 - Calculus II-Stewart Dr. Berg Spring...

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