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# Calculus II

Course Code: MATH 1220
Faculty: Science & Technology
Department: Mathematics
Credits: 3.0
Semester: 15 weeks
Learning Format: Lecture, Tutorial
Typically Offered: Fall, Summer, Winter
course overview

MATH 1220 is an introduction to integral calculus. It develops the concept of the integral and its applications. Other topics include techniques of integration, improper integrals, sequences and series of numbers, Taylor series, polar coordinates, parametric equations, and separable differential equations.

### Course Content

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

### Methods of Instruction

Lectures, problem sessions and assignments

### Means of Assessment

Evaluation will be carried out in accordance with Douglas College 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.

### Learning Outcomes

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

course prerequisites

MATH 1120 or MATH 1123

curriculum guidelines

Course Guidelines for previous years are viewable by selecting the version desired. If you took this course and do not see a listing for the starting semester/year of the course, consider the previous version as the applicable version.

course schedule and availability
course transferability

Below shows how this course and its credits transfer within the BC transfer system.

A course is considered university-transferable (UT) if it transfers to at least one of the five research universities in British Columbia: University of British Columbia; University of British Columbia-Okanagan; Simon Fraser University; University of Victoria; and the University of Northern British Columbia.

For more information on transfer visit the BC Transfer Guide and BCCAT websites.

assessments