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Calculus II

Advanced integration techniques, sequences and series, parametric and polar curves. Builds the full integration toolkit and introduces the paradigm of infinite processes — can an infinite sum converge to a finite number?

5 modules·82 concepts·48 practice problems·~45 hours
Practice Calc IIBrowse Modules

Prerequisites

Calculus I

Exam Relevance

AP Exams1 exam
AP Calculus BC40% of exam
University Exams1 exam
University Calculus II100% of exam

Module Breakdown

1.Advanced Integration Techniques

17 concepts·10 problems

The workhorse module. Integration by parts, trigonometric integrals, trig substitution, partial fractions, and improper integrals — the full toolkit for any integral you encounter in engineering.

17 concepts covered
U Substitution And Basic Integration ReviewIntegration By Parts FormulaChoosing U And Dv The Liate StrategyRepeated Integration By PartsTabular Integration By PartsIntegration By Parts For Definite IntegralsIntegrals Of Sinm CosnIntegrals Of Tanm SecnTrig Substitution Sqrt A2 Minus X2Trig Substitution Sqrt A2 Plus X2Trig Substitution Sqrt X2 Minus A2Partial Fraction Decomposition Distinct Linear FactorsPartial Fraction Decomposition Repeated And Quadratic FactorsIntegration Using Partial FractionsImproper Integrals Type I Infinite LimitsImproper Integrals Type Ii Discontinuous IntegrandComparison Test For Improper Integrals

2.Applications of Integration

14 concepts·10 problems

The geometric and physical payoff: volumes by disk/washer and shell, arc length, surface area, hydrostatic force, centroids, and probability density functions.

14 concepts covered
Disk And Washer MethodShell MethodChoosing Disk Washer Vs ShellVolume By Known Cross SectionsArc Length FormulaSurface Area Of RevolutionWork By Variable Force ExtensionsHydrostatic Pressure And ForceCenter Of Mass In One DimensionCentroids Of Planar RegionsMoments And Centroids Via IntegrationPappuss TheoremChoosing An Integration Strategy For ApplicationsProbability Density Functions And Expected Value

3.Differential Equations Introduction

10 concepts·8 problems

A preview of the full Diff Eq course. Separable ODEs, exponential growth/decay, Newton's law of cooling, and logistic growth — translating word problems into differential equations.

10 concepts covered
Antiderivatives And Initial Value Problems ReviewDirection Fields And Solution CurvesSeparable Odes IdentificationSolving By Separation Of VariablesInitial Value Problems For OdesExponential Growth ModelExponential Decay And Half LifeNewtons Law Of CoolingLogistic Growth ModelModeling With Differential Equations

4.Sequences & Series

28 concepts·10 problems

The conceptual pivot of the course. Sequences, series, convergence tests, power series, Taylor and Maclaurin series — one of the most powerful ideas in mathematics and physics.

28 concepts covered
Sequences And Summation Notation ReviewLimits At Infinity ReviewSequence Definition And NotationLimit Of A SequenceMonotone Convergence TheoremBounded SequencesSqueeze Theorem For SequencesPartial Sums And Series DefinitionGeometric Series Convergence And SumDivergence Test Nth Term TestIntegral Test For SeriesP SeriesComparison Test For SeriesLimit Comparison TestAlternating Series TestAlternating Series Error BoundRatio TestRoot TestAbsolute Vs Conditional ConvergencePower Series DefinitionRadius Of ConvergenceInterval Of ConvergenceDifferentiation Of Power SeriesIntegration Of Power SeriesTaylor Polynomial ApproximationMaclaurin Series For Standard FunctionsTaylor Series With Remainder LagrangeUsing Series For Approximation

5.Parametric Equations & Polar Coordinates

13 concepts·8 problems

Alternative ways to describe curves. Parametric equations describe paths traced by a moving point; polar coordinates describe curves via distance and angle. Both essential in physics and engineering.

13 concepts covered
Trigonometric Functions And Identities ReviewParametric Curves Definition And GraphingEliminating The ParameterDerivatives Of Parametric Curves Dy DxSecond Derivative For Parametric Curves D2y Dx2Arc Length For Parametric CurvesArea Under A Parametric CurvePolar Coordinate System And PlottingConversion Between Polar And CartesianGraphs Of Polar CurvesSymmetry In Polar GraphsArea In Polar CoordinatesArc Length In Polar Coordinates

Reference Textbooks

  • Stewart — Calculus: Early Transcendentals
  • Thomas — Calculus

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