# ch06_06 - 546 CHAPTER 6 Integration Techniques 6-38 0 x a...

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6-39 SECTION 6.6 . . Improper Integrals 547 There is something fundamentally wrong with this “calculation.” Note that f ( x ) = 1 / x 2 is not continuous over the interval of integration. (See Figure 6.2.) Since the Fundamental Theorem assumes a continuous integrand, our use of the theorem is invalid and our answer is incorrect . Further, note that an answer of 3 2 is especially suspicious given that the integrand 1 x 2 is always positive. y x 2 4 2 4 2 4 FIGURE 6.2 y = 1 x 2 Recall that in Chapter 4, we defined the definite integral by b a f ( x ) dx = lim n →∞ n i = 1 f ( c i ) x , where c i was taken to be any point in the subinterval [ x i 1 , x i ], for i = 1 , 2 ,..., n and where the limit had to be the same for any choice of these c i ’s. So, if f ( x ) → ∞ [or f ( x ) → −∞ ] at some point in [ a , b ], then the limit defining b a f ( x ) dx is meaningless.
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