Improper Integrals If a function f is continuous on a closed and bounded interval [a,bl,we can define the definite integral (d Can we define an integral over an unbound interval [a,o)? For any b in [a,oo),we consider the integral r()d and study the limit m f(x)dx Main Menu cam5YTne
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Improper Integrals If a function f is continuous on a closed and bounded interval [a,b], we can define the definite integral ( ) b a f x dx Can we define an integral over an unbound interval [a, ∞) ? For any b in [a, ∞), we consider the integral ( ) b a f x dx and study the limit → b a b lim f (x)dx
If as b tends to o,this integral tends to a finite limit L, b lim f(x)dx=L b→0a then we write ()d-L and say that the improper integral 。f(x)d converges to L. Otherwise,we say that the improper integral f(x)dx diverges Main Menu ow5产
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. If, as b tends to ∞, this integral tends to a finite limit L, lim ( ) b b a f x dx L → = then we write and say that the improper integral Otherwise, we say that the improper integral converges to L. diverges. ( ) a f x dx L = ( ) a f x dx ( ) a f x dx
Improper Integrals Example 1 e-2 dx= Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Improper Integrals
(b) dx diverges Main Meny C 007e
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Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
(d) cosπdx diverges.. Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Improper Integrals (11.7.1) o dx converges if p 1 and diverges if 0<p<1. Main Menu 0
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Improper Integrals
Improper Integrals (A comparison test)Suppose that f and g are continuous and 0≤fx)≤g(x)for all x∈[a,oo). (Figure11.7.3) (11.7.2) f g(x)dx converges, then f(x)dx converges. (ii)If diverges, then g(x)dx diverges. g Figure 11.7.3 Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Improper Integrals
Example:Show that dx converges. 1V1+x3 [Main Menu☐ 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example : Show that converges. 1 + 3 1 x dx
Example:Show that dx diverges V1+x2 Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example : Show that diverges. 1 + 2 1 x dx