Mathematics 101 topics for study

• Section 5.2:
  • Interpreting area under graph as a limit of Riemann sums
  • Evaluating such sums and their limits in easy cases
  • Exercises: 1-11; these are nice exercises in computing sums and limits, but have limited practical value for the purpose of integration
• Section 5.3:
  • Partitions, Riemann sums, definite integral, integrability
  • Setting up and evaluating Riemann sums
  • Terminology: limits of integration, integrand, etc.
  • Exercises: 1-10
• Section 5.4:
  • Properties of definite integrals: Theorem 3
  • Using Theorem 4.3 and interpretation of integral as area to simplify and compute integrals
  • Mean Value Theorem for integrals, average value of function
  • Exercises: 1-38 (but do not spend too much time on this)
• Section 5.5:
  • Fundamental Theorem of Calculus
  • Applications: integrating elementary functions, differentiating integrals, finding areas of planar regions
  • Exercises: 1-46, 49-50. Note that in problems 21-32 it really is important to draw a picture.
• Section 5.6:
  • Elementary integrals, page 5.6. You should memorize the formulas 1-8 (note that 1-6 are special cases of 7), 9-12, 15-17, 19-20.
  • Substitution in indefinite and definite integrals
  • Trigonometric integrals. You should either memorize the formulas for integrals of tan x and cot x, or make sure that you can find them when needed (the derivation is very short and it is useful to remember it). You should also remember the tricks used to integrate powers of sine and cosine: substitution, converting even powers of sines to cosines and vice versa, and using the cos(2x) formulas.
  • Exercises: 1-48.
• Section 5.7:
  • Finding areas of regions under graph of a function, or between two graphs
  • Exercises: 1-25. (Some of these exercises may involve integration formulas from the textbook, including those that you do not need to memorize.)
• Section 6.1:
  • Integration for parts: indefinite and definite integrals. (You do not need to remember the "reduction formulas" at the end of the section, but it is useful to read this paragraph and try one or more of the exercises 31-34.)
  • Exercises: 1-30; 36-37 are also recommended but will not be needed as part of your preparation for midterm 1.
• Section 6.2:
  • Inverse trigonometric substitutions: sine, tangent, secant.
  • Completing the square (this should already be familiar to you).
  • Skip "other inverse substitutions", page 361, and "the tan(t/2) substitution", page 362. These methods can be useful sometimes, but they often lead to complicated calculations and we do not have the time to cover them in depth.
  • Exercises: 1-28, 40-46.
• Section 6.3:
  • Integrating rational functions: reduction to the case P(x)/Q(x) with the degree of P less than the degree of Q.
  • Partial fractions: setting up a partial fraction decomposition (remember that the cases of quadratic and repeated factors are a little bit different from distinct linear factors!); two methods of finding the coefficients.
  • Exercises: 1-26 are fairly straightforward; 27-34 require combining partial fractions with other techniques (such as substitution).
• Section 6.5:
  • Improper integrals: infinite intervals, unbounded functions
  • Evaluating improper integrals as limits of proper integrals
  • Comparison test
  • Exercises: in 1-29 you should calculate the integral as a limit of proper integrals; in 30-41 use the comparison test.
• Section 6.6:
  • Basic approximate integration formulas: Trapezoid Rule, Midpoint Rule
  • Error estimates: given n, how large can the error be? How large does n need to be in order to achieve a prescribed accuracy?
  • Exercises: 1-11. Some of the computational exercises can get a bit tedious, even if you use a calculator. You probably do not need to do too many of them.
  • For the trapezoid, midpoint, and Simpson's methods, you should memorize the formulas for T_n, M_n, S_n, but you do not have to memorize the error estimates.
• Section 6.7:
  • Simpson's Rule: computational formula and error estimate.
  • Exercises: 1-10.
• Section 6.8:
  • Other aspects of numerical integration: improper integrals (skip Taylor's formula and Romberg integration).
  • Exercises: 1-9.
• Section 7.1:
  • Volumes of solids of revolutions: slices, cylindrical shells (you should memorize these two formulas).
  • Exercises: 1-23.
• Section 7.2:
  • Volumes by slices
  • Exercises: 1-19.
• Section 7.3:
  • Arclength of a graph of function (memorize the formula)
  • Area of a surface of revolution (memorize the first two formulas on page 427 - the next two are identical except that x and y are interchanged).
  • Exercises: 1-31. Note that the arclength and area integrals are often difficult or impossible to evaluate without using numerical methods.
• Section 7.4:
  • Mass as the integral of density.
  • Center of mass.
  • The formulas on page 436 (you do not need to memorize them, but make sure that you understand the general principle - dividing a region into smaller parts and computing the center of mass of each one).
  • Note: in this and the next section, we do the best that can be done using single variable calculus, but to develop a systematic approach (one universal formula, as opposed to several different-looking formulas for various special cases) we would really need calculus of many variables.
  • Exercises: 1-14.
• Section 7.5:
  • Centroid of a planar region (the formulas on page 437 are the same as those on page 436, but with the density equal to 1).
  • Pappus's Theorem.
  • Exercises: 1-18 look fairly straightforward; in 19-24 you will likely want to use Pappus's theorem.
• We will skip Section 7.6. Selected topics from Section 7.7 will be covered in connection with the differential equations in Section 7.9.