The following is a chapter by chapter guide intended to help you organize the material we have covered in class as you study for your exam. It is only intended to serve as a guidline, and may not explicitly mention everything that you need to study.
It is a good idea to review the basics of differentiation and integration from Calculus I, especially the product rule, quotient rule and chain rule for derivatives, and the substitution method for integrals, and the Fundamental Theorem of Calculus. You should know how to differentiate and integrate elementary functions (polynomials, trig functions, exponentials and logarithms).
Please review all homework, quiz and workshop problems for the chapters given below, and make sure you can solve all of them. You can also look at these problems for additional practice for each chapter.
6.2: Make sure you can find volume as the integral of cross-sectional area. For this, it is important to be able to find a formula for the cross-sectional area, and the limits of integration. You should also know how to compute masses and populations given density functions, and how to find the average value of a continuous function. You should know the statement of the Mean Value Theorem for Integrals.
6.3 & 6.4: Learn all the methods of finding a volume of revolution. Depending on the problem, certain methods are much easier to use than others! It is really important to be able to tell which to use. Practice problems from the text book.
7.1: Learn the integration by parts formula. Also learn how to pick u and v' appropriately. Take a look at all the example problems.
7.2: You should know how to obtain the results summarized in the table at the end of this chapter. Also make sure you know which cases can be solved using a simple u-substitution instead of integration by parts and reduction formulas, since this is a much faster method, and will save you a lot of time on the exam. You do not need to memorize the integral of sec(x) or csc(x). These will be given to you if you reequire them.
7.3: Learn to identify which trig substitution is appropriate for which integrals, and how to make the substitution. The chapter summary is quite helpful here.
7.4: Know the definitions of the hyperbolic trig functions, and how to use this to differentiate and integrate such functions. You should also know how to find the integrals of products of powers of hyperbolic functions, using methods similar to those covered in Section 7.2. You do not need to know the material at the end of the section, on how to express ordinary trig functions in terms of exponentials using complex numbers.
7.5: Learn to recognize proper and improper rational functions, and how to convert improper rational functions to the sum of a polynomial and a proper rational function using long division. Then make sure you know how to indentify which terms appear in the partial fractions decomposition by looking at the denominator, and how to solve for all the constants. Finally, make sure you know how to integrate each term in the decomposition.
7.6: Know the limit definition for improper integrals, and review L'Hôpital's rule. The methods of integration covered earlier in the semester are often needed to evaluate these improper integrals, especially integration by parts.
7.8: You should be able to interpret the trapezoidal rule and the midpoint rule geometrically, and interpret Simpson's rule as their weighted average. You do not need to memorize the error bound formulas in this section, but make sure you know how to apply them if they are are given to you.
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