RedAssign6[1].2 - In order to get the correct answer you...

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Unit: Improper Integrals Module: Convergence and Divergence of Improper Integrals The Second Type of Improper Integral www.thinkwell.com [email protected] Copyright 2001, Thinkwell Corp. All Rights Reserved. 1468 –rev 06/14/2001 If a function is not continuous on the integration interval, then the standard procedure will not work. Use the discontinuity as an endpoint for the integral. This is the second type of improper integral . Remember: A definite integral is considered an improper integral if it has one of these properties: the integration is over an infinite interval, or its integrand is undefined at a point within the interval of integration. A second type of improper integral occurs when the function is not continuous over the interval of integration. In this example, the function is undefined and the curve is discontinuous at x = 0. This makes the integral improper.
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Unformatted text preview: In order to get the correct answer, you have to set up the integral so that the point of discontinuity is an endpoint. If you didn’t realize that the function is discontinuous at x = 0, you might set up the problem with limits of integration between negative one and one. This would result in a solution with the area under the curve equal to negative two. This answer is incorrect. Notice that the function is squared so it is positive everywhere; the correct answer must also be positive. To set up the integral correctly, use the point of discontinuity, x = 0, as a limit of integration. Because the function 2 1 x is symmetric about the y-axis, find the area under the curve between zero and one and then double the result. Evaluating the antiderivative, − 2 x , as → x results in −∞ . This improper integral diverges....
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