5.6 Again, practice is essential: you need to be able to recognize what is the "u" and what is the "dv". I recommend using "Formula 2" (the "u,v" formulation). Once you have chosen "u" and "dv", write down du and v -this requires a preliminary integration!-. Example 1-obvious case; Example 2-less obvious but you should know this "trick" (Exercises 4,6). Examples 3,4 each give an important wrinkle in applying "parts." Applying it several times if necessary (keep very careful track of your signs!! - Exercises 7,8,12); or applying it twice and then solving for the integral (Exercises 13,14). Be familiar with these maneuvers. Exercises 1-24, as many as you can do.
5.9 Remember the relation between left sum Ln, right sum Rn, trapezoidal rule Tn, midpoint rule Mn and Simpson's rule S2n, that is: Tn = (Ln+Rn)/2 and S2n = (2Mn+Tn)/3 (Example 4). This will help you derive the formula for Simpson's rule p.418 which you should know. Understand that Ln underestimates and Rn overestimates if f is increasing, and vice-versa if f is decreasing. Understand that Mn underestimates and Tn overestimates if f is concave-up, and vice-versa if f is concave-down. Exercises 5,7,8,21,23,24. The exercises require a calculator; understand the procedures well enough so that you can carry them out by hand for n = 2, 3 or 4
Use the Chapter Review for further reviewing.
October 25 2008