BackCalculus II Final Review: Integration, Series, Parametric and Polar Equations
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Integration Techniques and Applications
Antiderivatives and Basic Integration
Antiderivatives, or indefinite integrals, are functions whose derivatives yield the original function. Integration is the reverse process of differentiation.
Power Rule for Integration: For ,
Integration of Rational Functions: Use substitution or partial fractions for rational expressions.
Integration of Exponential and Trigonometric Functions: Standard formulas include , , .
Examples:
Integration by Substitution
Substitution is used when an integral contains a function and its derivative. Set , then .
Example: (let )
Integration by Parts
Integration by parts is based on the product rule for differentiation. The formula is:
Example: (let , )
Trigonometric Integrals
Integrals involving trigonometric functions often use identities or substitution.
Partial Fractions
Partial fraction decomposition is used to integrate rational functions where the degree of the numerator is less than the denominator.
Example:
Improper Integrals
Improper integrals involve infinite limits or discontinuous integrands. Evaluate as limits:
Example:
Applications: Arc Length and Surface Area
Arc Length Formula: For from to :
Surface Area of Revolution (about y-axis):
Example: Find the arc length of from to .
Sequences and Series
Convergence of Sequences
A sequence converges to if .
Example: converges to 0.
Example: converges to .
Convergence Tests for Series
Several tests determine whether an infinite series converges or diverges.
Test | Process | Conditions |
|---|---|---|
Geometric Series Test | Check if | Converges if |
Nth Term Test | Compute | If limit , series diverges |
Integral Test | Test | positive, continuous, decreasing |
p-Series Test | Series | Converges if |
Direct Comparison Test | Compare to | Both positive terms |
Limit Comparison Test | Compute | Both positive terms, limit finite and |
Alternating Series Test | Check decreasing, | Series |
Ratio Test | Compute | Converges if limit |
Root Test | Compute | Converges if limit |
Power Series and Taylor Series
Power Series:
Interval of Convergence: Values of for which the series converges.
Taylor Series:
Examples:
Taylor series for about :
Taylor series for about :
Parametric Equations and Polar Coordinates
Parametric Equations
Parametric equations express and as functions of a parameter .
Example: ,
Eliminating the Parameter: Solve for in one equation and substitute into the other to get a Cartesian equation.
Calculus with Parametric Curves
Arc Length:
Surface Area (about x-axis):
Area Enclosed:
Polar Coordinates
Polar coordinates represent points as , where is the distance from the origin and is the angle from the positive x-axis.
Conversion: ,
Area in Polar Coordinates:
Arc Length in Polar Coordinates:
Examples:
Convert to rectangular: ,
Convert to polar: ,
Summary Table: Key Integration Formulas
Integral | Result |
|---|---|
Additional info:
This review covers topics from Calculus II, including integration techniques, applications of integration, sequences and series, parametric equations, and polar coordinates.
Students should be familiar with all standard integration techniques, convergence tests for series, and calculus with parametric and polar forms.
