Richard Wong

Assistant Teaching Professor in Mathematics at Rice University.

You can find the course syllabus here.

What tools do we have to calculate an integral like $\int_0^1 xe^{-x^2} \ dx$ or $\int_0^1 e^{-x^2} \ dx$?

When your calculator tells you that the value of $\int_0^1 e^{-x^2} \ dx \approx 0.746824$, how do you know it’s correct? How accurate is your calculator? Is it possible for an infinite region to have finite area? Can we calculate the area of a fractal?

You’ve previously seen in Math 101 that calculus is an important and powerful tool that allows us to describe the physical world around us. In Math 102, we will push the notion of integration to its utmost limits. In this class, we will build our toolbox for integration; we will study the behavior of the infinite, and we will learn how to quantify how accurate our approximations are.

In this course, you will develop the critical thinking and questioning skills needed to answer these complex questions. Moreover, you will become fluent in precisely communicating your ideas through the mathematical language of calculus.

Schedule

  Learning Outcome Textbook Section Lectures
CII1 Techniques of integration. Describe and interpret real-world scenarios as integration problems. Apply various techniques to compute integrals, including $u$-substitution, partial fraction decomposition, and integration by parts. Compare and contrast different methods of integration. 5.5, 7.1, 7.4 1-5
CII2 Polar coordinates. Visualize and interpret polar coordinates and polar functions. Convert between Euclidean and polar coordinates to compute areas. Calculate trigonometric integrals. 7.2, 10.3-10.4 6-9
CII3 Parametric equations. Describe curves that are not graphs of functions. Visualize and interpret parametric equations. Convert between functions and parametric equations to compute arclengths and areas. Calculate integrals using trig substitution. 7.3, 10.1-10.2 10-14
CII4 Improper integrals. Compare the behavior of functions at infinity. Determine if an integral over an infinite region exists. Evaluate integrals over infinite regions. Determine if a function can be integrated over a discontinuity. Given an improper integral, find an appropriate comparison. 7.8 15-18
CII5 Sequences and series. Describe the end term behavior of a sequence. Describe what it means for a sequence and/or series to converge. Evaluate and geometrically interpret common series. Describe examples of sequences and series that do not converge. Describe the relationship between series and integration. 11.1-11.3 19-22
CII6 Series tests. Identify and apply the appropriate test to determine convergence or divergence of a given series, including the comparison test, alternating series test, and ratio test. Compare the difference between convergence, absolute convergence, and conditional convergence. 11.4-11.7 23-29
CII7 Taylor series. Approximate functions (and/or their derivatives/integrals) using power series. Determine precise error bounds when using Taylor approximation. Use Taylor series to solve differential equations. 9.1, 11.8-11.11, 17.4 30-38
CII8 Complex numbers. Describe and interpret real world scenarios using complex numbers. Describe a geometric interpretation of the complex numbers. Explain why Euler’s formula $e^{i\pi} + 1 = 0$ holds. Appx. H 39-41

Learning Objectives:

The goals of the course are that you: 

  • Build your toolbox for integration; study the behavior of the infinite; and learn how to quantify the accuracy of your approximations.
  • Develop the reasoning and questioning skills needed to explore these (mathematical) topics and apply them to real-life situations.
  • Develop the collaboration and communication skills needed to convey your (mathematical) ideas.

Below you will find the explicit learning objectives associated to each of these goals.

Calculus II Objectives (CII)

  1. Techniques of integration. Describe and interpret real-world scenarios as integration problems. Apply various techniques to compute integrals, including $u$-substitution, partial fraction decomposition, and integration by parts. Compare and contrast different methods of integration.
  2. Polar coordinates. Visualize and interpret polar coordinates and polar functions. Convert between Euclidean and polar coordinates to compute areas. Calculate trigonometric integrals.
  3. Parametric equations. Describe curves that are not graphs of functions. Visualize and interpret parametric equations. Convert between functions and parametric equations to compute arclengths and areas. Calculate integrals using trig substitution.
  4. Improper integrals. Compare the behavior of functions at infinity. Determine if an integral over an infinite region exists. Evaluate integrals over infinite regions. Determine if a function can be integrated over a discontinuity. Given an improper integral, find an appropriate comparison.
  5. Sequences and series. Describe the end term behavior of a sequence. Describe what it means for a sequence and/or series to converge. Evaluate and geometrically interpret common series. Describe examples of sequences and series that do not converge. Describe the relationship between series and integration.
  6. Series tests. Identify and apply the appropriate test to determine convergence or divergence of a given series, including the comparison test, alternating series test, and ratio test. Compare the difference between convergence, absolute convergence, and conditional convergence.
  7. Taylor series. Approximate functions (and/or their derivatives/integrals) using power series. Determine precise error bounds when using Taylor approximation. Use Taylor series to solve differential equations.
  8. Complex numbers. Describe and interpret real world scenarios using complex numbers. Describe a geometric interpretation of the complex numbers. Explain why Euler's formula $e^{i\pi} + 1 = 0$ holds.

Mathematical Reasoning Objectives (MR)

  1. Reason abstractly and quantitatively.  Use mathematics to model real-world situations and to interpret and solve problems.  Make appropriate assumptions and approximations to simplify a complicated situation. Draw pictures or study examples to provide insight.  Attend to the meaning of quantities instead of just computing them. Consider the units involved.
  2. Make sense of problems and persevere in solving them.  Understand what a problem is asking for. Analyze the givens, constraints, and goals of a problem. Break large/complex problems into smaller/simpler problems. Check answers using alternate methods.
  3. Build intuition.  Investigate and create specific examples and counterexamples. Look for and make use of structure or repeated patterns. Look for both general methods and shortcuts.  Seek to understand unexpected results.
  4. Use appropriate tools strategically.  Consider the available tools when solving a mathematical problem. Understand and use stated assumptions, definitions, and previously established results.
  5. Construct viable arguments.  Make conjectures, and use assumptions, definitions, and/or previously established results to prove or disprove them. Use strategies such as direct proofs, contrapositive statements, proof by cases, or proof by contradiction.
  6. Be creative and explore mathematics. Do more than find a solution. Look for novel or elegant solutions. See what changes when you add, change, or weaken hypotheses. Draw connections between topics and ideas.

Mathematical Communication Objectives (MC)

  1. Ask questions.  Notice, identify, and clarify sources of confusion. Look for connections and relationships between ideas. Explore topics and ideas deeply.
  2. Use and develop mathematical fluency.  Use standard mathematical notations and terms (as discussed in class or demonstrated in course materials).  Clearly indicate and explain any use of non-standard shorthand, notation, or tools.
  3. Analyze and constructively critique the reasoning of others.  Actively listen and summarize key ideas to check comprehension. Test conjectures against examples and potential counterexamples. Assess and reconcile various approaches to problems. Work together to find errors and fix flaws.
  4. Explain and justify your reasoning.  Communicate and justify your conclusions to others. Indicate the general strategy or argument, and identify the key step or idea(s). Listen and reflect to the critiques of others.  Work together to find errors and fix flaws.
  5. Attend to precision.  Communicate precisely to others. Use clear definitions, and carefully state any assumptions or results used.  Be able to explain heuristics/arguments in depth.
  6. Be clear and concise.  Use the appropriate amount of generality or specificity in arguments.  Avoid use of any extraneous assumptions, hypotheses, or statements. Indicate any figures or examples that you have in mind.
  7. Review, reflect, and revise. Review previous work and assess the positives and negatives. Reflect on your strategies and look for ways to improve. Use feedback to grow and develop your mathematical reasoning or communication skills.