Mathematics 101 study topics, part 3:

• 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). Skip Pappus's Theorem.
  • Exercises: 1-18.
• Section 7.8:
  • Probability density, expectation, variance (you should remember the formulas).
  • Exercises: 1-13.
• Section 7.9:
  • Separable equations, first order linear equations: general solutions, initial value problems.
  • As an applied example, we will discuss the logistic equation, which is also mentioned in Section 7.7. You may want to look it up there.
  • Exercises: 1-16.