Calculus II

Complete syllabus

Every chapter and lesson

1. 01

   Chapter 1 · 3 lessons

   Techniques of Integration

   1. 1.1

      Integration by parts

      9 min · Quick check

      Lesson goal: By the end you can apply the integration by parts formula to solve integrals involving products of functions.

      • Understand the formula for integration by parts: ∫u dv = uv - ∫v du
      • Identify appropriate choices for u and dv in given integrals
      • Apply the integration by parts method to evaluate integrals
      • Recognize when to use integration by parts multiple times
   2. 1.2

      Trig integrals & substitution

      9 min · Quick check

      Lesson goal: By the end you can evaluate integrals involving trigonometric functions using trigonometric identities and substitution techniques.

      • Define common trigonometric integrals and their properties
      • Use trigonometric identities to simplify integrals
      • Apply substitution methods to evaluate integrals involving trigonometric functions
      • Recognize patterns in trigonometric integrals for efficient solving
   3. 1.3

      Partial fractions

      9 min · Quick check

      Lesson goal: By the end you can decompose rational functions into partial fractions and integrate them.

      • Understand the concept of partial fraction decomposition
      • Identify proper and improper fractions and how to handle them
      • Apply the method of partial fractions to break down complex rational functions
      • Integrate the resulting simpler fractions
2. 02

   Chapter 2 · 3 lessons

   Further Integration

   1. 2.1

      Improper integrals

      9 min · Quick check

      Lesson goal: By the end you can understand and evaluate improper integrals.

      • Define improper integrals and their significance in calculus.
      • Identify the types of improper integrals: type I and type II.
      • Learn techniques for evaluating improper integrals using limits.
   2. 2.2

      Numerical integration

      8 min · Quick check

      Lesson goal: By the end you can apply numerical methods to approximate definite integrals.

      • Understand the need for numerical integration when analytical solutions are difficult.
      • Learn about various numerical integration techniques such as the Trapezoidal Rule and Simpson's Rule.
      • Apply numerical integration methods to compute approximate values of definite integrals.
   3. 2.3

      A strategy for integration

      8 min · Quick check

      Lesson goal: By the end you can develop strategies for solving complex integrals.

      • Identify different integration techniques such as substitution and integration by parts.
      • Learn how to choose the appropriate method based on the form of the integrand.
      • Practice combining multiple strategies to tackle challenging integrals.
3. 03

   Chapter 3 · 3 lessons

   Applications of Integration

   1. 3.1

      Volumes of revolution

      9 min · Quick check

      Lesson goal: By the end you can calculate the volume of solids of revolution using integration.

      • Define solids of revolution and the methods to generate them.
      • Explain the disk and washer methods for finding volumes.
      • Apply the shell method for calculating volumes of revolution.
      • Set up and evaluate integrals to find volumes of various solids.
   2. 3.2

      Arc length & surface area

      8 min · Quick check

      Lesson goal: By the end you can determine the arc length and surface area of curves using integration.

      • Define arc length and surface area in the context of curves.
      • Derive the formulas for arc length and surface area.
      • Apply integration techniques to compute arc lengths of given functions.
      • Calculate the surface area of solids formed by revolving curves.
   3. 3.3

      Physical applications

      8 min · Quick check

      Lesson goal: By the end you can apply integration to solve physical problems.

      • Identify physical applications of integration in real-world contexts.
      • Use integration to find quantities such as work, center of mass, and fluid pressure.
      • Set up integrals based on physical scenarios to solve problems.
      • Interpret the results of integration in terms of physical applications.
4. 04

   Chapter 4 · 3 lessons

   Differential Equations

   1. 4.1

      Separable equations

      9 min · Quick check

      Lesson goal: By the end you can identify and solve separable differential equations.

      • Define separable equations and their characteristics.
      • Demonstrate the process of separating variables.
      • Solve a basic separable equation step-by-step.
   2. 4.2

      Exponential models

      8 min · Quick check

      Lesson goal: By the end you can apply exponential models to real-world scenarios.

      • Define exponential growth and decay models.
      • Identify situations that can be modeled exponentially.
      • Solve problems using exponential equations.
   3. 4.3

      First-order linear equations

      8 min · Quick check

      Lesson goal: By the end you can solve first-order linear differential equations.

      • Define first-order linear equations and their standard form.
      • Explain the method of integrating factors.
      • Solve a first-order linear equation using the integrating factor method.
5. 05

   Chapter 5 · 3 lessons

   Sequences & Series

   1. 5.1

      Sequences

      8 min · Quick check

      Lesson goal: By the end you can define sequences and identify their types.

      • Understand the definition of a sequence.
      • Differentiate between finite and infinite sequences.
      • Identify arithmetic and geometric sequences.
      • Explore examples of sequences in real-world contexts.
   2. 5.2

      Series & convergence

      9 min · Quick check

      Lesson goal: By the end you can explain series and determine their convergence.

      • Define a series and its notation.
      • Differentiate between convergent and divergent series.
      • Understand the concept of partial sums.
      • Explore the significance of convergence in series.
   3. 5.3

      Convergence tests

      9 min · Quick check

      Lesson goal: By the end you can apply convergence tests to series.

      • Learn various convergence tests such as the Ratio Test and Root Test.
      • Apply the Comparison Test and Limit Comparison Test.
      • Understand the Integral Test for convergence.
      • Practice using these tests on different series.
6. 06

   Chapter 6 · 3 lessons

   Power Series

   1. 6.1

      Ratio & root tests

      8 min · Quick check

      Lesson goal: By the end you can apply the Ratio and Root Tests to determine the convergence or divergence of power series.

      • Understand the definitions of the Ratio Test and Root Test.
      • Learn how to apply the Ratio Test to a given power series.
      • Learn how to apply the Root Test to a given power series.
      • Interpret the results of the tests in terms of convergence and divergence.
   2. 6.2

      Power series

      9 min · Quick check

      Lesson goal: By the end you can define power series and identify their intervals of convergence.

      • Define what a power series is and its general form.
      • Identify the center and radius of convergence for a power series.
      • Determine the interval of convergence for a given power series.
      • Explore examples of power series and their applications.
   3. 6.3

      Taylor & Maclaurin series

      9 min · Quick check

      Lesson goal: By the end you can derive and use Taylor and Maclaurin series for functions.

      • Define Taylor and Maclaurin series and their significance.
      • Learn how to derive the Taylor series for a function at a given point.
      • Understand the special case of the Maclaurin series as a Taylor series at zero.
      • Apply Taylor and Maclaurin series to approximate functions.