Chapter 2: Calculus II — Integration, Series, and Parametric Forms

Extend integration into techniques, applications, sequences, series, power series, parametric equations, polar coordinates, and introductory differential equations.

What You Will Practice

Calculus II builds on integration and expands into more advanced ways of representing functions, curves, accumulation, and infinite processes.

Integration Techniques
Applications of Integration
Sequences
Series and Convergence
Power Series
Parametric and Polar Forms
Intro Differential Equations

Mini Lesson

1. Integration Techniques

Calculus II teaches methods for integrals that are not simple power-rule problems.

u-substitution: replace a complicated expression with u
integration by parts: ∫u dv = uv - ∫v du

Plain meaning: Choose the method that makes the integral easier.

2. Applications of Integration

Integrals can calculate area, volume, accumulated change, work, and average value.

Area under y = f(x) from a to b = ∫ f(x) dx

Example: If velocity is integrated over time, the result is displacement.

3. Sequences

A sequence is an ordered list of numbers.

aₙ = 1/n

Plain meaning: Ask what happens as n gets larger and larger.

4. Series and Convergence

A series is the sum of terms in a sequence. A series converges if the sum approaches a finite value.

1 + 1/2 + 1/4 + 1/8 + ... = 2

Plain meaning: Infinite adding can sometimes still lead to a finite total.

5. Power Series

Power series represent functions using infinite polynomial-like sums.

1 + x + x² + x³ + ...

Plain meaning: A function can sometimes be approximated by a long polynomial.

6. Parametric and Polar Forms

Parametric equations use a parameter, often time, to describe motion. Polar coordinates use radius and angle.

Parametric: x = x(t), y = y(t)
Polar: x = r cos(θ), y = r sin(θ)

7. Intro Differential Equations

Differential equations involve functions and their derivatives.

dy/dt = ky

Plain meaning: The rate of change depends on the current amount.

Interactive Calculus II Practice

Choose a topic and practice with instant feedback. Type simple answers such as x^3+C, converges, diverges, 2, or polar.

Typing tip: Use ^ for powers, +C for indefinite integrals, and type concept answers like converges, diverges, parametric, or polar.
Big idea: Calculus II is where integration becomes more flexible, infinite sums become useful, and curves can be represented in new coordinate systems.

Mastery Check

Before moving to Book 3 Chapter 3, students should be able to recognize these ideas.

Integration Techniques

I can recognize when substitution or integration by parts may be useful.

Applications

I know integrals can represent area, volume, accumulated change, and work.

Sequences and Series

I can explain convergence as approaching a finite value.

Power Series

I understand that functions can be represented by infinite polynomial-like sums.

Parametric and Polar

I can distinguish time-based curves from radius-angle coordinate systems.

Differential Equations

I can recognize that differential equations involve unknown functions and their derivatives.

Go to Chapter 3 Back to Book 3