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General Information

Course ID (CB01A and CB01B)
MATHD001B
Course Title (CB02)
Calculus II
Course Credit Status
Credit - Degree Applicable
Effective Term
Fall 2024
Course Description
This course examines the fundamentals of integral calculus.
Faculty Requirements
Course Family
Not Applicable

Course Justification

This course satisfies the CSU GE/Breadth requirement, Area B4 Mathematics, and Quantitative Reasoning, and also satisfies the IGETC requirement, Area 2 Mathematical Concepts, and Quantitative Reasoning. This course is CSU and UC transferable. This course is a required core course for the A.S.-T degree in Mathematics. This is the second course in a sequence of four courses in calculus. This course emphasizes single variable integral calculus.

Foothill Equivalency

Does the course have a Foothill equivalent?
No
Foothill Course ID

Course Philosophy

Course Philosophy
This course introduces the problem of finding the area under the graph of a function over an interval-first by approximating the areas of rectangles fitted in the desired region, and then by invoking limits to find the exact area. The two seemingly unrelated problems - one, of finding the slope of the tangent line and two, of finding the area under a curve - are in fact very intimately related, as the student learns, through the fundamental theorem of calculus. In fact, differentiation (MATH 1A) and integration (MATH 1B) are inverse processes.



The solution to the area problem is useful in solving other problems. For example, it can be used to find arc length of curves, as well as volume and surface area of certain solids. It has applications in many fields of mathematics, such as, probability and differential equations, as well as other disciplines, such as physics and economics.



This course contains historical notes relevant to the material covered in the sections they appear. They are not part of the actual course outline. These historical notes are to be used at the discretion of the instructor.

Formerly Statement

Course Development Options

Basic Skill Status (CB08)
Course is not a basic skills course.
Grade Options
  • Letter Grade
  • Pass/No Pass
Repeat Limit
0

Transferability & Gen. Ed. Options

Transferability
Transferable to both UC and CSU
CSU GEArea(s)StatusDetails
CGB4CSU GE Area B4 - Mathematics/Quantitative ReasoningApproved
IGETCArea(s)StatusDetails
IG2XIGETC Area 2 - Mathematical Concepts and Quantitative ReasoningApproved
C-IDArea(s)StatusDetails
MATHMathematicsApprovedMATH D001A & MATH D001B required for C-ID MATH 210 MATH D001A & MATH D001B & MATH D001C required for C-ID MATH 900 S MATH D001B & MATH D001C required for C-ID MATH 220

Units and Hours

Summary

Minimum Credit Units
5.0
Maximum Credit Units
5.0

Weekly Student Hours

TypeIn ClassOut of Class
Lecture Hours5.010.0
Laboratory Hours0.00.0

Course Student Hours

Course Duration (Weeks)
12.0
Hours per unit divisor
36.0
Course In-Class (Contact) Hours
Lecture
60.0
Laboratory
0.0
Total
60.0
Course Out-of-Class Hours
Lecture
120.0
Laboratory
0.0
NA
0.0
Total
120.0

Prerequisite(s)

MATH D001A or MATH D01AH

Corequisite(s)

Advisory(ies)

ESL D272. and ESL D273., or ESL D472. and ESL D473., or eligibility for EWRT D001A or EWRT D01AH or ESL D005.

Limitation(s) on Enrollment

(Not open to students with credit in the Honors Program related course.)

Entrance Skill(s)

General Course Statement(s)

(See general education pages for the requirements this course meets.)

Methods of Instruction

Lecture and visual aids

Discussion of assigned reading

Discussion and problem-solving performed in class

In-class exploration of internet sites

Quiz and examination review performed in class

Homework and extended projects

Guest speakers

Collaborative learning and small group exercises

Collaborative projects

Problem solving and exploration activities using applications software

Problem solving and exploration activities using courseware

Assignments

  1. Required readings from text
  2. Problem solving exercises
  3. A selection of homework, quizzes, group projects and exploratory worksheets
  4. Optional project synthesizing various concepts and skills from course content

Methods of Evaluation

  1. Periodic quizzes and/or assignments from sources related to the topics listed in the curriculum will be evaluated for completion and accuracy to assess studentsʼ comprehension and ability to communicate the course content as well as to provide timely feedback to students on their progress.
  2. At least three one-hour exams without projects or at least two one-hour exams with projects are required. Students are expected to provide complete and accurate solutions to problems that include both theory and applications by integrating methods and techniques studied in the course. Students shall receive timely feedback on their progress.
  3. One two-hour, comprehensive, final examination is required, in which students are expected to display comprehension of course content and be able to choose methods and techniques appropriate to the various types of problems that cover course content.
  4. Projects (optional)

    Projects may be used to enhance students' understanding of topics studied in the course in group or individual formats, whether communicating their understanding orally through classroom presentation or in writing. Evaluation is to be based on completion and comprehension of course content, and students shall receive timely feedback on their progress.

Essential Student Materials/Essential College Facilities

Essential Student Materials: 
  • Students will use technology (computers and graphing calculators) to examine mathematical concepts graphically and numerically
Essential College Facilities:
  • None.

Examples of Primary Texts and References

AuthorTitlePublisherDate/EditionISBN
James Stewart, Daniel Clegg & Saleem Watson "Calculus: Early Transcendentals", 9th Ed. Cengage 2021.

Examples of Supporting Texts and References

AuthorTitlePublisher
Hass, Weir, and Thomas, "University Calculus, Early Transcendentals", 4th Ed. Pearson 2020.
Anton, Bivens, and Davis, "Calculus, Early Transcendentals Combined", 10th Ed. Wiley 2011.
Hughes-Hallett, et al., "Calculus, Single and Multivariable", 7th Ed. Wiley 2017.
Mathematics Multicultural Bibliography, available on the De Anza Math department resources website.

Learning Outcomes and Objectives

Course Objectives

  • Analyze and explore aspects of the integral calculus.
  • Analyze and evaluate the definite integral as a limit of a Riemann sum and examine its properties
  • Examine the Fundamental Theorem of Calculus
  • Find definite, indefinite, and improper integrals using various techniques
  • Apply the definite integral to applications
  • Examine differential equations

CSLOs

  • Analyze the definite integral from a graphical, numerical, analytical, and verbal approach, using correct notation and mathematical precision.

  • Formulate and use the Fundamental Theorem of Calculus.

  • Apply the definite integral in solving problems in analytical geometry and the sciences.

Outline

  1. Analyze and explore aspects of the integral calculus.
    1. Using a variety of perspectives: verbally, graphically, numerically, and symbolically.
    2. The following functions are incorporated in this development: linear, polynomial, rational, exponential, logarithmic, power, trigonometric, inverse trigonometric, piece-wise, and parametric functions.
  2. Analyze and evaluate the definite integral as a limit of a Riemann sum and examine its properties
    1. Apply Riemann Sums to approximate area under the graph of a function
    2. Define and evaluate the definite integral as a limit of the Riemann sum
    3. Apply the definition of the definite integral to calculate the area under the graph of a function exactly
    4. Examine the properties of the definite integral
    5. Historical Note:
      1. Georg F. B. Riemann (1826-1866, Germany)
      2. Gottfried Leibniz (1646-1716, Germany): The indefinite integral symbol
      3. Jean Baptiste Joseph Fourier (1768 - 1830, France): The definite integral symbol
  3. Examine the Fundamental Theorem of Calculus
    1. Use the Fundamental Theorem of Calculus to calculate definite integrals using antiderivatives
    2. Use the Fundamental Theorem of Calculus to obtain an antiderivative function
  4. Find definite, indefinite, and improper integrals using various techniques
    1. Integrate by using substitution
    2. Integrate by parts
    3. Integrate by using trigonometric substitution
    4. Integrate after performing partial fraction decomposition
    5. Integrate curves that are defined parametrically
    6. Use the table of integrals to evaluate integrals
    7. Develop techniques for approximating definite integrals and associated errors
      1. Approximate definite integrals by left sums, right sums, midpoint rule, trapezoidal rule and Simpson's rule
      2. Evaluate errors to numerical integration
      3. Estimate the value of definite integrals within given error bounds
    8. Define and evaluate improper integrals
    9. Historical Note:
      1. Thomas Simpson (1710-1761, England): he didn't develop Simpson's Rule; it was developed before he was even born. He wrote texts and worked on probability!
      2. Joseph Louis Lagrange (1736-1813, France): Developed integration by parts
      3. Maria Agnesi (1718-1799, Italy): story of "Curve of Agnesi"
  5. Apply the definite integral to applications
    1. Find the area bounded by two or more curves
    2. Find the volume of solids with known cross section
    3. Find the volume of solids of revolution
    4. Find the arc length of
      1. Plane curves
      2. Parametric curves
    5. Examine some applications of the definite integral to other areas in Mathematics such as but not limited to
      1. Find the surface area of a solid of revolution
      2. Find the average values of a function.
      3. Present a logical development of the natural log as a definite integral and the corresponding development of exponential function as the inverse of the natural log
    6. Examine some applications of the definite integral to Physics such as but not limited to
      1. Work
      2. Center of Mass
      3. Hydrostatic Force
    7. Examine some applications of the definite integral to other subjects such as but not limited to
      1. Probability - Use the definite integral to define and calculate probabilities and expected value, from a probability distribution function.
      2. Applications in Economics such as consumer and producer surplus
      3. Applications in Biology such as cardiac output and the flux of blood flow
    8. Historical Note:
      1. Earlier attempts all over the world at solving area, volume problems with a hint of calculus-like ideas:
        1. Areas and Volumes: Eudoxus' (408-355 B.C., Asia Minor, modern day Turkey) method of exhaustion; Archimedes (287 - 212 B.C., Sicily) found sums of "infinite series" (e.g. area of circle) using double reductio ad absurdum that actually incorporated some of the technical details of limits
        2. Area of circle: Liu Hui (220 - 280, China): Principle of exhaustion and Calvalieri's principle; Seki Kowa (1642-1708, Japan): inscribed rectangles - early "Riemann sum"
        3. Ancient Egyptians: Volume of the frustum of pyramid
        4. Ibn al-Haytham (965-1039, Persia), AKA Alhazen: Method of compression to find the volume of the solid formed by rotating the parabola around a line perpendicular to the axis of the curve.
      2. Evangelista Toricelli (1608-1647, Italy) and Gabriel's Horn
      3. Pappus (290-350 Egypt): Centroid and its relation to volume and surface area.
  6. Examine differential equations
    1. Establish the concept of a differential equation and its slope field
    2. Use Euler's method to approximate the solution to a differential equation
    3. Use separation of variables to solve a differential equation
    4. Establish exponential model of growth and decay
    5. Establish logistic model of growth (optional)
    6. Historical Note:
      1. Leonhard Euler (1707-1783, Switzerland): Concept of differential equations
      2. Jakob Bernoulli (1654-1705, Switzerland): Development of separation of variables
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