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Unformatted text preview: 8.8 1 8.8 Improper Integrals Type I  Infinite Limits of Integration Consider the following example. Example 1. Infinite Areas? Suppose that b > 1 . Evaluate the following definite integral. (1) integraldisplay b 1 1 x 2 dx y = 1 /x 2 1 b 1 In other words, find the area under curve. It follows that integraldisplay b 1 1 x 2 dx = 1 x b 1 = 1 1 b (2) 8.8 2 Now, what happens in (2) as b → ∞ ? In this case we have the following. integraldisplay ∞ 1 1 x 2 dx = lim b →∞ integraldisplay b 1 1 x 2 dx = lim b →∞ 1 1 b = 1 Motivated by the previous example, we adopt the following definition. 8.8 3 Definition. Improper Integrals  Type I Definite integrals with infinite limits of integration are called improper integrals of type I . They are defined as follows. 1. If f ( x ) is continuous on [ a, ∞ ) , then integraldisplay ∞ a f ( x ) dx = lim b →∞ integraldisplay b a f ( x ) dx (3) 2. If f ( x ) is continuous on (∞ , b ] , then integraldisplay b∞ f ( x ) dx = lim a →∞ integraldisplay b a f ( x ) dx (4) 3. If f ( x ) is continuous on (∞ , ∞ ) , then integraldisplay ∞∞ f ( x ) dx = integraldisplay c∞ f ( x ) dx + integraldisplay ∞ c f ( x ) dx (5) for any real number c . Notice that this last case is handled by combining the first two. 8.8 4 In all three cases, we say the improper integral converges whenever the righthand side is finite. Otherwise, the improper integral diverges . Remark. So the improper integral in the first example converged and its value was 1 . Also, since f ( x ) = 1 /x 2 > we can say that the “area under the curve” is finite...in fact, the area is 1 . 8.8 5 The following result is very useful. Proposition 1. Suppose that f is continuous on [ a, ∞ ) and integraltext ∞ a f ( x ) dx converges. Then integraltext ∞ a cf ( x ) dx also converges for any c ∈ R . Proof. Let b > a . Then by the basic properties of the definite integral integraldisplay b a cf ( x ) dx = c integraldisplay b a f ( x ) dx It follows that integraldisplay ∞ a cf ( x ) dx = lim b →∞ integraldisplay b a cf ( x ) dx = c lim b →∞ integraldisplay b a f ( x ) dx = c integraldisplay ∞ a f ( x ) dx < ∞ 8.8 6 Example 2. Evaluate the improper integral integraldisplay ∞ 2 2 x 2 1 dx Notice that the integrand is continuous on [2 , ∞ ) so by (3) we have integraldisplay ∞ 2 2 x 2 1 dx = lim...
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 Spring '07
 Wei
 Calculus, Improper Integrals, Integrals, Limits, dx, LCT

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