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Chap7_Sec8

Chap7_Sec8 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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7 TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION

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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

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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

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TYPE 1—INFINITE INTERVALS Consider the infinite region S that lies: Under the curve y = 1/ x 2 Above the x -axis 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, let’s take a closer look.

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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 →∞ →∞ = - = ÷

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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 →∞ = =

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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

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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

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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.

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IMPROPER INTEGRALS OF TYPE 1 For instance, in case (a), suppose f ( x ) ≥ 0 and the integral is convergent.
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Chap7_Sec8 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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