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"The student is completing Module 1, Applications of Integration, Chap 5, homework problems. They use Integration Techniques which are in a later chapter. Though we did a little of this last week, discovering, learning how to use, and using Integration Tables from an Appendix, we did much more of it today. Learning how to use the table requires recognizing that a standard table entry can be matched with some substitution in the problem at hand, e.g.. u=X**2 (X squared), and making the corresponding integration differential, du, equivalent to the original differential, dx, from the presented integration.
We did several problems during the two hours. In addition to using integration tables, we did a discrete approximation to the integral with Dx=.5, and Dx=.25, and then verified the convergence by actually evaluating the integral. As Dx became smaller, the approximation was closer to the actual integral value. We also evaluated the corresponding approximation errors to assure ourselves that the approximations were done accurately. The errors were 10% and 5%. We did three arc length integrations, and two surface area integrations. In addition to substitution to match tables, there is also integration by parts, L'Hopital's Rule, trig identities, and others. The student already recognizes that working as many problems as he can exposes him to a wider variety of detailed situations, which allow him to solve new problems more quickly because they are likely to be amenable to similar techniques. This coming week begins two weeks for Module 2, which is Integration Techniques. I recommended the student browse the chapter this weekend to become familiar with it."