Lectures, problem sessions and assignments
Introduction to the Integral
- sigma notation
- Riemann sums
- the definite integral
- the Fundamental Theorem of Calculus
- antiderivatives; elementary substitutions
- applications to area under and between curves, volume and work
Techniques of Integration
- parts
- trigonometric substitution
- trigonometric integrals (products and powers)
- partial fractions (linear factors and distinct quadratic factors)
- rationalizing substitutions
- improper integrals
Applications of Integration
- areas between curves
- volumes by cross sections and cylindrical shells
- work
- separable differential equations
- arc length
Infinite Series
- sequences
- sum of a geometric series
- absolute and conditional convergence
- comparison tests
- alternating series
- ratio and root test
- integral test
- power series
- differentiation and integration of power series
- Taylor and Maclaurin series
- polynomial approximations; Taylor polynomials
Parametric Equations and Polar Coordinates
- areas and arc lengths of curves in polar coordinates
- areas and arc lengths of functions in parametric form
Optional Topics (included at the discretion of the instructor)
- tables of integrals
- approximation of integrals by numerical techniques
- Newton's law of cooling, Newton's law when force is proportional to velocity, and logistics curves
- a heuristic "proof" of the Fundamental Theorem of Calculus
- the notion of the logarithm defined as an integral
- further applications of Riemann sums and integration
- binomial series
At the conclusion of this course, the student should be able to:
- compute finite Riemann sums and use to estimate area
- form limits of Riemann sums and write the corresponding definite integral
- recognize and apply the Fundamental Theorem of Calculus
- evaluate integrals involving exponential functions to any base
- evaluate integrals involving basic trigonometric functions and integrals whose solutions require inverse trigonometric functions
- choose an appropriate method and apply the following techniques to find antiderivatives and evaluate definite integrals:
- integration by parts
- trigonometric and rationalizing substitution
- completing the square for integrals involving quadratic expressions
- partial fractions
- integrals of products of trigonometric functions
- apply integration to problems involving areas, volumes, arc length, work, velocity and acceleration
- be able to determine the convergence or divergence of improper integrals either directly, or by using the comparison test
- determine if a given sequence converges or diverges
- determine if a sequence is bounded and/or monotonic
- determine the sum of a geometric series
- be able to choose an appropriate test and determine series convergence/divergence using:
- integral test
- simple comparison test
- limit comparison test
- ratio test
- root test (optional)
- alternating series test
- distinguish and apply concepts of absolute and conditional convergence of a series
- determine the radius and interval of convergence of a power series
- approximate a differentiable function by a Taylor polynomial, determine the remainder term, and compute the error in using the approximation
- find a Taylor or Maclaurin series representing specified functions by:
- "direct" computation
- means of substitution, differentiation or integration of related power series
- find the area of a region bounded by the graph of a polar equation or parametric equations
- find the lengths of curves in polar coordinates or in parametric form
- solve first order differential equations by the method of separation of variables; apply to growth and decay problems
Evaluation will be carried out in accordance with ÁñÁ«ÊÓƵ policy. The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester. Evaluation will be based on the following criteria:
| Weekly quizzes | 0-40% |
| Tests | 20-70% |
| Assignments | 0-15% |
| Attendance | 0-5% |
| Class participation | 0-5% |
| Tutorials | 0-10% |
| Final examination | 30-40% |
Note: All sections of a course with a common final examination will have the same weight given to that examination.
Consult the ÁñÁ«ÊÓƵ bookstore for the current textbook. Examples of textbooks under consideration include:
Stewart, Calculus: Early Transcendentals, Cengage Learning, current edition
Anton, Bivens, and Davis, Calculus: Early Transcendentals, Wiley, current edition
Briggs, Cochran, and Gillet, Calculus: Early Transcendentals, Pearson, current edition
Edwards and Penney, Calculus: Early Transcendentals, Pearson, current edition
A graphing calculator may also be required.