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Unformatted text preview: 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION In defining a definite integral , we dealt with a function f defined on a finite interval [ a , b ] and we assumed that f does not have an infinite discontinuity (Section 5.2). ( ) b a f x dx 7.8 Improper Integrals In this section, we will learn: How to solve definite integrals where the interval is infinite and where the function has an infinite discontinuity. TECHNIQUES OF INTEGRATION IMPROPER INTEGRALS In this section, we extend the concept of a definite integral to the cases where: The interval is infinite f has an infinite discontinuity in [ a , b ] IMPROPER INTEGRALS In either case, the integral is called an improper integral. One of the most important applications of this idea, probability distributions, will be studied in Section 8.5 TYPE 1INFINITE INTERVALS Consider the infinite region S that lies: Under the curve y = 1/ x 2 Above the xaxis To the right of the line x = 1 INFINITE INTERVALS You might think that, since S is infinite in extent, its area must be infinite. However, lets take a closer look. INFINITE INTERVALS The area of the part of S that lies to the left of the line x = t (shaded) is: Notice that A ( t ) < 1 no matter how large t is chosen. 2 1 1 1 1 1 ( ) 1 t t A t dx x x t = =  =  INFINITE INTERVALS We also observe that: 1 lim ( ) lim 1 1 t t A t t = = INFINITE INTERVALS The area of the shaded region approaches 1 as t . INFINITE INTERVALS So, we say that the area of the infinite region S is equal to 1 and we write: 2 2 1 1 1 1 lim 1 t t dx dx x x = = INFINITE INTERVALS Using this example as a guide, we define the integral of f (not necessarily a positive function) over an infinite interval as the limit of integrals over finite intervals. IMPROPER INTEGRAL OF TYPE 1 If exists for every number t a , then provided this limit exists (as a finite number). ( ) t a f x dx ( ) lim ( ) t a a t f x dx f x dx = Definition 1 a IMPROPER INTEGRAL OF TYPE 1 If exists for every number t a , then provided this limit exists (as a finite number). Definition 1 b ( ) b t f x dx ( ) lim ( ) b b t t f x dx f x dx = CONVERGENT AND DIVERGENT The improper integrals and are called: Convergent if the corresponding limit exists. Divergent if the limit does not exist. ( ) a f x dx ( ) b f x dx Definition 1 b IMPROPER INTEGRAL OF TYPE 1 If both and are convergent, then we define: Here, any real number a can be used. ( ) a f x dx ( ) a f x dx ( ) ( ) ( ) a a f x dx f x dx f x dx  = + Definition 1 c IMPROPER INTEGRALS OF TYPE 1 Any of the improper integrals in Definition 1 can be interpreted as an area provided f is a positive function. IMPROPER INTEGRALS OF TYPE 1 For instance, in case (a), suppose f ( x ) 0 and the integral is convergent.is convergent....
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This note was uploaded on 01/06/2012 for the course MATH 2414.S01 taught by Professor Alans.grave during the Fall '11 term at Collins.
 Fall '11
 AlanS.Grave
 Continuity

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