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Integration Techniques, L’Hôpital’s Rule, and Improper Integrals 8 Copyright © Cengage Learning. All rights reserved.
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Improper Integrals 8.8 Copyright © Cengage Learning. All rights reserved.
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3 Evaluate an improper integral that has an infinite limit of integration. Evaluate an improper integral that has an infinite discontinuity. Objectives
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4 Improper Integrals with Infinite Limits of Integration
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5 The definition of a definite integral requires that the interval [a, b] be finite. And Fundamental Theorem of Calculus requires continuous f(x). Here we develop a procedure for evaluating integrals that do not satisfy these requirements— usually because either one or both of the limits of integration are infinite, or f has a finite number of infinite discontinuities in the interval [a, b]. Integrals that possess either property are improper integrals. A function f is said to have an infinite discontinuity at c if, from the right or left,
![5 The definition of a definite integral requires that the interval [a, b] be finite.](http://images.slideplayer.com./26/8735238/slides/slide_5.jpg "And Fundamental Theorem of Calculus requires continuous f(x). Here we develop a procedure for evaluating integrals that do not satisfy these requirements— usually because either one or both of the limits of integration are infinite, or f has a finite number of infinite discontinuities in the interval [a, b]. Integrals that possess either property are improper integrals. A function f is said to have an infinite discontinuity at c if, from the right or left,.")
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6 Improper Integrals with Infinite Limits of Integration
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7 Geometrically, this says that although the curves 1/x and 1/x^2 and look very similar for x>0, the region under to the right of 1/x^2 has finite area whereas the corresponding region under 1/x has infinite area. Note that both 1/x and 1/x^2 approach 0 as x-> infinity, but 1/x^2 approaches 0 faster than 1/x. The values of 1/x don’t decrease fast enough for its integral to have a finite value.
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8 Generalization:
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9 My example
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12 My examples Instead integrand approaches finite limit.
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14 Improper Integrals with Infinite Discontinuities
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15 Improper Integrals with Infinite Discontinuities The 2 nd type of improper integral is one that has an infinite discontinuity at or between the limits of integration.
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16 Example 6 – An Improper Integral with an Infinite Discontinuity Evaluate Solution: The integrand has an infinite discontinuity at x = 0, as shown in Figure 8.24. You can evaluate this integral as shown below. Figure 8.24
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17 My Examples Just showed that 2 nd integral diverges => diverges NOTE that failing to recognize that there is interior singular point leads to wrong answer:
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18 My generalization:
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