PP 8.8_1 - Improper Integrals Know: To evaluate the...

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Improper Integrals
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Know: To evaluate the definite integral of a function without an infinite discontinuity over the interval [a, b], we use the Fundamental Theorem of Calculus. To do: extend the concept to functions with infinite discontinuities and allow the interval to be unbounded. - - 0 2 3 2 and 1 : Examples dx e dx x x
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Definition = t a t a t a dx x f f(x)dx a t dx x f ) ( lim then , number every for exists ) ( If (a) - 0 2 : Example dx e x 2 1 2 2 1 0 2 2 1 0 2 + - = - = - - - t t x t x e e dx e The integral exists for every t.
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( 29 2 1 2 2 1 0 2 0 2 lim lim + - = = - - - t t t x t x e dx e dx e 2 1 2 1 0 = + = 2 1 0 2 = - dx e x When the limit exists, like in this example, we say that the improper integral is convergent. Otherwise, the improper integral is divergent.
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Definition -∞ - = b t t b b t dx x f f(x)dx b t dx x f ) ( lim then , number every for exists ) ( If (b) - - 1 : Example dx e x t t x t x e e e dx e - - - - + - = - = 2 1 1 1 1 The integral exists for every t.
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( 29 t t t x t x e e dx e dx
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This note was uploaded on 01/21/2011 for the course PHYS 4A 60865 taught by Professor L. oldewurtel during the Fall '09 term at Irvine Valley College.

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PP 8.8_1 - Improper Integrals Know: To evaluate the...

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