MATH 54 - Calculus II
Course Description
In this course, you will learn about integration, which you can think of as the inverse process
for differentiation. You will also learn about the significance of the integral, what it
represents, its applications and the different techniques you can use to calculate it. After
this, you will investigate sequences and series, the different convergence tests and the
Taylor and Maclaurin series, all of which will allow us to approximate definite integrals and
create new functions. You will then learn about expressing curves using parametric
equations or polar coordinates, and how to differentiate and integrate these functions.
Finally, you will learn about vectors and three-dimensional space as a preparation for Math
55.
Course Learning Outcomes
After completion of the course, the student should be able to:
- Find the integrals of given functions using suitable integration techniques.
- Solve different types of problems by applying the concept of integration.
- Determine the convergence or divergence of a given sequence or series.
- Discuss calculus of parametric and polar equations.
- Extend the concepts and operations in plane analytic geometry to three dimensions.
- Describe what vectors are and perform vector operations.
- Determine the equations of lines, planes, cylinders, and quadric surfaces in three dimensions.
Course Outline
UNIT 1. Integrals
- Areas and Distances
- The Definite Integral
- The Fundamental Theorem of Calculus
- Indefinite Integrals
- The Substitution Rule
- Average Value of a Function
- Areas of Plane Regions between Curves
- Volumes of Solids by Slicing, Disks, Washers, and Cylindrical Shells
UNIT 2. Techniques of Integration
- Integration by Parts
- Trigonometric Integrals
- Trigonometric Substitution
- Integration of Rational Functions by Partial Fractions
- Strategy for Integration
- Approximate Integration
- Improper Integrals
UNIT 3. Further Applications of Integration
- Arc Length of a Curve
- Area of a Surface of Revolution
- Applications to Physics and Engineering
- Separable Differential Equations
- Laws of Growth and Decay
- Models for Population Growth
UNIT 4. Infinite Sequences and Series
- Sequences
- Series
- The Integral Test and Estimates of Sums
- The Comparison Tests
- Alternating Series
- Absolute Convergence and the Ratio and Root Tests
- Strategy for Testing Series
- Power Series
- Representations of Functions as Power Series>
- Taylor and Maclaurin Series
- Applications of Taylor Polynomials
UNIT 5. Polar and Parametric Curves
- Tangent Lines and Arc
- Length of Parametric Curves
- Areas and Lengths in Polar Coordinates
UNIT 6. Vectors and The Geometry of Space
- The Three-Dimensional Coordinate System
- Vectors in Three Dimensions
- The Dot Product
- The Cross Product
- Equations of Lines and Planes
- Cylinders and Quadric Surfaces