Integration Techniques, L’Hôpital’s Rule, and Improper Integrals 8 Copyright © Cengage Learning. All rights reserved.

Improper Integrals 8.8 Copyright © Cengage Learning. All rights reserved.

3 Evaluate an improper integral that has an infinite limit of integration. Evaluate an improper integral that has an infinite discontinuity. Objectives

4 Improper Integrals with Infinite Limits of Integration

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,

6 Improper Integrals with Infinite Limits of Integration

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.

8 Generalization:

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

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.

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 You can evaluate this integral as shown below. Figure 8.24

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:

18 My generalization:

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