Calculus 2 Topics and Recommended Exercises

Dr. Brenton LeMesurier, Spring 2007

We will start with a few items from Chapters 3 and 4 that were not covered in Introductory Calculus (Math 120), then move to the main topics of integration (Ch. 6,7) and infinite sequences and their sums (Ch. 11), ending with a look at curves (Sect. 8.1, Ch. 10) and an introduction to differential equations (Ch. 9).

Calculus of some inverse functions and more on L'Hôpital's rule (from Ch. 3 and 4)

• A review of the inverses of the trigonometric functions (Appendix D, pages A31-34), and then the finding of their derivatives.
  Study Section 3.6, pages 229 and 230, including Exercises 41-50 and 53.

  This also gives two useful integrals, seen in the last line of the table on page 469 at the start of Chapter 7. You should learn the basic properties of at least the inverses of sine, cosine and tangent, including their derivatives, and these two indefinite integrals.

• Next, the last of the mainstream functions are met (ones common and important enough to be on your calculators!); the hyperbolic functions and their inverses.
  Study Section 3.9 including Exercises 7, 8, 20, 21, 23(a,b,d), 30, 31, 32, 36, 42, 43, 44, 47.
• L'Hôpital's rule for computing limits where simple substitution gives meaningless indeterminate forms like "0/0" and "∞/∞" was seen in Introductory Calculus (Math 120), but now we see how to use it for some other cases, such as limits where substitution fails because it gives indeterminate forms like "0.∞", "0⁰" and "1^∞".
  Study Section 4.4 including Exercises 39, 40, 43, 55, 61, 65.

Basic Applications of Integration (Chapter 6)

Next we start into the first major topic of the semester; Chapter 6, on applications of integration to questions like computing areas and volumes. We will cover Sections 1, 2, 3 and 5. (The omitted section 4 is the physics application of work.)

• Computing the area between two curves (Section 6.1) mostly relies on one formula, A = ∫_a^b [f(x)-g(x)]dx. The main idea is to understand the geometry; sketching the graph of the region and thus working out which integral, or which combination of several such integrals, gives the total area.
  Study Section 6.1 (skipping Example 4), including Exercises 1, 2, 5, 6, 9, 10, 29 and 38.
• In some cases, we cannot evaluate the definite integral. Thus it is worth looking at numerical approximations, by revisiting the Midpoint Rule (see page 384), which is the natural best version of Riemann sum approximations.
• Next up will be computing the volume of a solid (Section 6.2), which is even more a matter of using the geometry to work out the functions and integrals needed, before using one of several basic definite integral formulas.
  Study Section 6.2 (skipping Examples 7 and 9), including Exercises 1, 2, 11, 12, 31, 32, 39 and 42.
• One final case of volumes is solids whose surface is produced by rotating the graphs of functions about a vertical axis; then slicing into cylindrical shells is the best approach.
  Study Section 6.3, including Exercises 1, 2, 3, 4, 8, 21 and 22.
• Computing the average value of a function on an interval [a,b], like average velocity between two times, leads to simply the integral divided by the length of the interval:
  Study Section 6.5, including Exercises 3 and 4 for the mathematical basics, and 15 and 17 for two physical applications.

Techniques of Integration (Chapter 7)

Chapter Seven will cover the core methods for computing indefinite and definite integrals. One recurring theme is eliminating or simplifying products and quotients of functions, since we have no general product or quotient rule for integrals.

• Integration by Parts is the closest thing we have to the non-existent "General Power Rule for Integrals".
  Study Section 7.1 including Exercises 1 to 10, 176, 18, 21, 24, and 51 and 52 (which involve computing volumes)
• Products of trig. functions can usually be integrated using either simple substitutions (like u=sin x or u=cos x), or trig identities for writing sin² x and cos² x in forsm with no products of functions. It is also often useful to write all other trig. functions first in terms of just sines and cosines.
  Study Section 7.2, mainly to Example 4 and the Strategy on page 478, ands Exercises 1, 2, 5, 6, 9, 17 and 27. Note also the integrals of tangent and secant.
• The method of Section 7.2 are surprisingly useful for integrating functions involving the square root of a quadratic, using inverse trigonometric substitutions like x = sin u, x = tan u or x = sec u.
  Study Section 7.3 and Exercises 1-10, 24 and 29.
• The last method of integration is really a method of algebraic simplification: writing a rational function (a quotients of two polynomials) as a sum of simple terms, eliminating the product of factors in the denominator of the rational function.
  For now, study Section 7.4 to "Case I", Examples 1, 2 and 3, and try some exercises such as 1, 2, 3, 13, 14, 15 and 21.
• Section 7.4 finishes the techniques for finding indefinite integrals and Section 7.5 is a summary of strategies for choosing which of these methods to try. In conjunction, we have
• Section 7.6, Integration Using Tables, but only briefly and omitting the part about computer algebra systems. Reduction formulas are the main emphasis.
  Exercises 7, 13, 23 and 24.
• Section 7.7, Approximate Integration. We have seen the Midpoint Rule already, so the man new ideas are Simpson's rule, which gives far more accuracy for just a little more effort, and result about quickly the accuracy improves as you use more intervals in the calculations. The Trapezoid is rather less useful, so you can omit the parts involving it from the following exercises.
  Exercises 5, 11, 15, and 21.
• Section 7.8, Improper Integrals: for example, computing the areas of regions that are infinitely wide, or infinitely high at some points.
  Exercises 1, 3, 5, 9, 11, 17, 21, 27, and 49, 53 on "comparison".

Infinite Sequences and Series (Chapter 11)

We will then skip to the second man topic of the course, Infinite Sequences and Series, before ending the semester with further applications of calculus (mostly integration) mostly from Chapter 10.

• Section 11.1 is about Infinite Sequences, meaning simply infinite lists of numbers {a₁, a₂, a₃, ... } = {a_n}, which is really a function whose argument is a natural number, or some other range of integers: a_n = f(n). An important use of sequences is that the numbers are successively better approximations to the solution of a problem, so that the exact answer is the limit as n goes to infinity, like a horizontal asymptote.
  Exercises 1, 2, 3, 5, 9, 11, 15, 19, 39, 43, 52, 53, 55, 59.
• Section 11.2 introduces Infinite Series, meaning the sum of all term in a sequence, a₁ + a₂ + a₃ + ... = Σ a_n. A very familiar example is an infinite decimal like 0.33... = 3/10 + 3/100 + ... = Σ 3/(10)ⁿ. A series is equal to the improper integral ∫₁^∞ f(x) dx with f(n) = a_n and f(x) = a_n for x between n and n+1. Like improper integrals, some series do not converge to any number though, like 1 - 1 + 1 - 1 ... Σ (-1)ⁿ and 1/1 + 1/2 + 1/3 + ... = Σ 1/n.
  Study Section 11.2, omitting Examples 6 and 9, and do Exercises 1, 2, 3, 5, 9, 11, 15, 19, 35, 41, 52.
• Section 11.3: Series (infinite sums) are often best evaluated approximately by simply summing a large number of terms, a₁ + a₂ + a₃ + ... + a_N, the N-th partial sum. So often the first question to answer is whether a series converges to a number at all, and for this one of various useful methods is comparing to an improper integral like ∫₁^∞ f(x) dx Section 11.3 introduces this idea of checking for convergence by the Integral Comparison Test; we mostly study the important example of p-series 1/1 + 1/2^p + 1/3^p + ... = Σ 1/n^p, p a constant.
  Study Section 11.3 to Example 4, and do Exercises 1, 2, 3, 5, 9, 12, 19.
• Section 11.4 continues the ides of working out whether a series converges by comparison; this time to another series for which we already know whether it converges or not. The simplest test comes from idea that for two sequences of positive numbers {a_n} and {b_n}, if 0 ≤ a_n ≤ b_n then 0 ≤ &Sigma a_n ≤ &Sigma b_n, including the case where one or both of the sums is infinite.
  Study Section 11.4, to Example 4 for now, and do Exercises 1, 2, 3, 5, 7, 13, 15.
• Section 11.5, Exercises 1, 2, 5, 8, 13, 14, 21, 23, 29.
• Section 11.6, Exercises 1, 2, 3, 5, 12, 31, 33.
• Section 11.7, do selected exercises for review on convergence testing.
• Section 11.8, Exercises 1, 2, 3, 5, 7, 9, 29, 35.
• Section 11.9, Exercises 1, 2, 4, 5, 13, 16, 24, 27, 29.
• Section 11.10, Exercises 1, 2, 3, 8, 13, 23, 27, 33, 37, 41, 49, 53, 56, 58.

Curves, Parametric Equations and Polar Coordinates (Chapter 10 and Section 8.1)

• Section 8.1 shows how to compute the length of a curve, revisited in Section 10.3.
  Exercises 2, 7, 12.

Chapter 10 combines calculus with geometry, looking at curves in the plane and the regions inside them using both cartesian and polar coordinates.

• Section 10.1 introduces parametric curves, where both the x and y position coordinates are given as functions of some other variable like t for time.
  Exercises 1, 2, 9 and 11.
• Section 10.2, Exercises 1, 5, 19, 25.
• Section 10.3 Read only the part about arc length, omitting surface area.
  Exercises 1, 5, 9.
• Section 10.4, Exercises 16, 17; 35, 42; 57, 63.

Differential Equations (Chapter 9)

A few topics only will be covered, as preview to seeing differential equations in later courses in mathematics, physics, etc.

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Brenton leMesurier, Department of Mathematics, College of Charleston
Last revised Tuesday January 9, 2007