Course Description and Outline

This course is a continuation of Calculus I. After a review of limits, derivatives, and integrals, we will develop various integration techniques (integration by parts, trig substitution, etc.) and then go on to apply them in various situations, such as finding areas, volumes, and lengths of curves. We will also study some applications of integration to physics, engineering, economics, etc. The second half of the course will be devoted to the study of infinite sequences and series. The main question for this part is whether an infinite sum adds up to something, and the ultimate goal will be to come up with an infinite series representation for a given function. What time is left will be devoted to the study of parametric equations and polar coordinates. A section-by-section outline of the course is given below. 

The more general objective of this course is to continue providing you with a deeper understanding and working knowledge of calculus, while in the process strengthening your analytical skills, increasing your ability to communicate mathematics symbolically and orally, making you comfortable with reading and understanding mathematics on your own, and developing an appreciation for calculus as one of the greatest intellectual developments in history.

Here is a more detailed outline of the course.  All sections refer to Calculus: Concepts and Contexts, 4th edition, by James Stewart.

Review of differentiation and integration
  • Derivatives, rules of differentiation, Riemann sums, Fundamental Theorem of Calculus. Handouts and practice sheets will be given.
Chapter 5: Integrals
  • 5.5 The substitution rule 
  • 5.6 Integration by parts 
  • 5.7 Additional techniques of integration (including review of trigonometry) 
  • 5.10 Improper integrals (including review of limits and L’Hôpital’s Rule) 
Chapter 6: Applications of Integration
  • 6.1 More about areas 
  • 6.2 Volumes 
  • 6.3 Volumes by cylindrical shells 
  • 6.4 Arc length 
  • 6.5 Average value of a function 
  • 6.6 Applications to physics and engineering 
  • 6.7 Applications to economics and biology 
Chapter 8: Infinite Sequences and Series
  • 8.1 Sequences (and e-N proofs from Appendix D) 
  • 8.2 Series 
  • 8.3 The integral and comparison tests; estimating sums 
  • 8.4 Other convergence tests 
  • 8.5 Power series 
  • 8.6 Representations of functions as power series 
  • 8.7 Taylor and Maclaurin series 
The following material will be covered if we have time. It cannot be found in our textbook so I will provide copies form another source:

Polar coordinates
  • Graphing and finding areas in polar coordinates 
Parametric equations
  • Parametrizing curves; Limits, derivatives, velocity, acceleration, and arc length for parametric curves