28 - Improper - Improper Integrals Up to this point we have...

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Unformatted text preview: Improper Integrals Up to this point we have integrated using closed intervals [a, b]. Now we will consider intervals such as x 2. ( 2 f (x) dx ) Integrals of this form are called improper integrals. a f (x) dx b - f (x) dx - f (x) dx 1 1 and Ex. Compare x 2 x for x > 0. 1 1 As x gets larger, x 2 is getting closer to zero faster than x . Let's evaluate 1 1 dx and 2 x 1 1 dx . We don't know how to find x an integral with an infinite limit, but we can evaluate the definite integral t 1 f (x) dx . t 1 1 dx 2 x t 1 1 dx x Let's let t increase. t 10 100 1000 10,000 100,000 1- 1 t 2 t -2 4.32456 18 61.2456 198 630.456 0.9 0.99 0.999 0.9999 0.99999 1 t , 1 - 1 and 2 t - 2 . When the limit exists, As t we say the integral converges to that limit or "is convergent"; if not, it diverges or "is divergent". Define a b f (x) dx = lim f (x) dx t a t - - f (x) dx = lim f (x) dx = t - t a b f (x) dx a - f (x) dx + f (x) dx Ex. 1 1 dx x Ex. 1 - 1 + x 2 dx Do: Convergent or Divergent? 1. x ( ln x ) 3 1 2 dx 2. 3 1 dx x ( ln x ) ...
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This note was uploaded on 10/16/2009 for the course MATH 1206 taught by Professor Llhanks during the Spring '08 term at Virginia Tech.

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