Mws_gen_int_spe_impr - Chapter 07.07 Integrating Improper Functions After reading this chapter you should be able to 1 integrate improper functions

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07.07.1 Chapter 07.07 Integrating Improper Functions After reading this chapter, you should be able to: 1. integrate improper functions using methods such as the trapezoidal rule and Gaussian Quadrature schemes. What is integration? Integration is the process of measuring the area under a function plotted on a graph. Why would we want to integrate a function? Among the most common examples are finding the velocity of a body from an acceleration function, and displacement of a body from a velocity function. Throughout many engineering fields, there are (what sometimes seems like) countless applications for integral calculus. You can read about some of these applications in Chapters 07.00A-07.00G. Sometimes, the evaluation of expressions involving these integrals can become daunting, if not indeterminate. For this reason, a wide variety of numerical methods has been developed to simplify the integral. Here, we will discuss the incorporation of these numerical methods into improper integrals. Figure 1 Integration of a function
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07.07.2 Chapter 07.07 What is an improper integral? An integral is improper if a) the integrand becomes infinite in the interval of integration (including end points) or/and b) the interval of integration has an infinite bound. Example 1 Give some examples of improper integrals Solution The integral 2 0 2 4 dx x x I is improper because the integrand becomes infinite at 2 x . The integral 2 0 1 dx x x I is improper because the integrand becomes infinite at 1 x . The integral 0 tdt e I t is improper because the interval of integration has an infinite bound. The integral 0 1 dt t e I t is improper because the interval of integration has an infinite bound and the integrand is infinite at 1 t . If the integrand is undefined at a finite number of points, the value of the area under the curve does not change. Hence such integrals could theoretically be solved either by assuming any value of the integrand at such points. Also, methods such as Gauss quadrature rule do not use the value of the integrand at end points, and hence integrands that are undefined at end points can be integrated using such methods. For the case where there is an infinite interval of integration, one may make a change of variables that transforms the infinite range of integration to a finite one. Let us illustrate these two cases with examples.
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Integrating Improper Functions 07.07.3 Figure 2 A plate with a crack under a uniform axial load Example 2 In analyzing fracture of metals, one wants to know the opening displacement of cracks. In a large plate, if there is a crack length of a 2 meters, then the maximum crack opening displacement (MCOD) is given by a dx x a x E 0 2 2 2 MCOD where remote normal applied stress E Young’s modulus Assume m 02 . 0 a GPa 210 E and MPa 70 .
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida - Tampa.

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Mws_gen_int_spe_impr - Chapter 07.07 Integrating Improper Functions After reading this chapter you should be able to 1 integrate improper functions

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