Math 50B --- Integral Calculus
Catalog Description
The second in the series of three calculus courses. Integral Calculus develops a set of advanced symbolic and numerical integration techniques, building on skills developed in the first course in the series, Differential Calculus. The course includes applications of integration, sequences and series, and the use of the Taylor polynomial to approximate functions. Students are introduced to parametric and polar equations and to the solution of differential equations.
Special notes or advisories: A graphing calculator is required.
Prerequisites
MATH-50A (or equivalent) with "C" or better
Describe representative skills without which the student would be highly unlikely to succeed: Students must be well-grounded in the art and theory of differentiation in order to be successful in this course. Students must be able to differentiate a variety of mathematical functions by hand, including functions involving trigonometric, logarithmic, exponential, and power functions.
Course Learning Outcomes
What should the student be able to do as a result of taking this course? State some of the objectives in terms of specific, measurable student accomplishments.
- Read, write, and speak accurately about mathematical ideas and use correct mathematical notation.
- Use graphing technology to visualize functions, explore mathematical concepts, and verify their work.
- Apply differentiation and integration techniques and theory to functions of one variable.
- Apply the concepts of the derivative and integral to solve real-world problems and applications.
- Demonstrate the fundamental relationship between the derivative and the integral.
- Apply the theory of infinite series and Taylor polynomials to approximate functions of one variable.
- Use numerical, graphical, symbolic, and verbal representations to solve problems and communicate with others.
Course Content
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Themes: What themes, if any, are threaded throughout the learning experiences in this course?
- Problem solving
- Writing
- Technology
- Communication
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Concepts: What concepts do students need to understand to demonstrate course outcomes?
- The use of the graphing calculator and/or mathematical software as a fundamental problem-solving tool.
- The presentation of mathematical solutions in a logical coherent structure, including the use of fundamental writing skills, grammar, and punctuation.
- Limits and their role in the major theorems in calculus.
- Continuity and its role in the major theorems in calculus.
- Differentiation and its role in the major theorems in calculus.
- Integration and its role in the major theorems in calculus.
- Sequences and series and their role in the major theorems in calculus.
- Differential equations and their role in the major theorems in calculus.
- The Fundamental Theorem of Calculus and its role in the major theorems in calculus.
- Mathematical modeling and its role in the major theorems in calculus.
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Issues: What primary issues or problems, if any, must students understand to achieve course outcomes.
- The appropriate use of technology in the problem-solving process.
- The connection between mathematics and the real world.
- The role of the student in becoming a successful learner.
- The recognition that the problem-solving skills learned in this class are applicable in future mathematics classes and classes in related fields, such as physics, engineering, etc.
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Skills: What skills must students master to demonstrate course outcomes?
- Integration
- Use techniques in numerical integration, including the trapezoidal method and Simpson's rule, to estimate a definite integral.
- Calculate integrals using techniques including:
- integration by parts,
- trigonometric integrals,
- trigonometric substitution,
- partial fraction decomposition,
- improper integrals.
- Use integrals to find the area between curves, volume, arc length, and surface area of a solid of revolution.
- Sequences and series
- Determine convergence of sequences.
- Find the limit of a convergent sequence.
- Compute the limit of a sequence, using limit laws and theorems.
- Find the sum of a series as a limiting value of a sequence of partial sums.
- Determine convergence of infinite series, by using tests, including:
- integral test,
- comparison test,
- limit comparison test,
- alternating series test,
- ratio test,
- root test.
- Determine whether a series converges absolutely or conditionally.
- Find the radius of convergence of a power series.
- Represent functions as power series.
- Create new power series, by using:
- multiplication and division,
- differentiation,
- integration.
- Use Taylor Polynomials to approximate functions, and calculate the remainder term.
- Parametric and polar equations
- Sketch the graph of a system of parametric equations.
- Use the graphing calculator and/or mathematical software to graph parametric equations in the plane.
- Convert a variety of equations in Cartesian form into polar form, and vice-versa.
- Construct a set of parametric equations that produce a given geometric locus or path.
- Sketch the graph of a given equation in polar form:
- by hand,
- by using the graphing calculator and/or mathematical software.
- Calculate arc length of a path determined by a set of parametric or polar equations.
- Differential Equations
- Use differential equations in mathematical modeling.
- Analyze differential equations, using numerical and graphical approaches, including:
- direction fields,
- Euler's method.
- Solve separable differential equations.
- Model population growth.
- Integration
Representative Learning Activities
What will the students be doing (i.e., Listening to lectures, participating in discussions and/or group activities, attending a field trip, etc.)? Relate the activities directly to the Course Learning Outcomes.
- Listening to lectures
- Participating in group activities and/or assignments
- Participating in class assignments and/or discussions
- Completing homework assignments
- Using the graphing calculator and/or mathematical software to complete activities designed to foster a deeper level of understanding of the concepts and skills developed in this class
Assessment Tasks
How will the student show evidence of achieving the Course Learning Outcomes? Indicate which assessments (if any) are required for all sections.
- Representative assessment tasks:
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- In-class examinations and/or quizzes
- Homework assignments
- Take-home examinations and/or quizzes allow the instructor to include questions and/or exercises that require the use of the graphing calculator and/or mathematical software to supplement the analysis. Extra time allows the students to develop their writing and presentation skills
- Writing assignments designed to develop communication of mathematical concepts
- Group and/or individual projects and presentations
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Required assessments for all sections – to include but not limited to:
- Homework assignments
- At least two proctored, closed-book examinations, plus a comprehensive final examination
Examples of Appropriate Texts or Other Readings
- Author: Stewart; Title: Calculus, Early Transcendentals, Sixth Edition; Date: 2007