Home The Fundamental Theorem of CalculusThree Different QuantitiesThe Whole as Sum of Partial Changes The Indefinite Integral as Antiderivative The FTC and the Chain Rule The Indefinite Integral and the Net ChangeIndefinite Integrals and AntiderivativesA Table of Common Antiderivatives The Net Change Theorem The NCT and Public Policy SubstitutionSubstitution for Indefinite IntegralsRevised Table of Integrals Substitution for Definite Integrals Area Between CurvesThe Slice and Dice PrincipleTo Compute a Bulk Quantity The Area Between Two Curves Horizontal Slicing Summary VolumesSlicing and Dicing SolidsSolids of Revolution 1: Disks Solids of Revolution 2: Washers Volumes Rotating About the $y$axis Integration by PartsBehind IBPExamples Going in Circles Tricks of the Trade Integrals of Trig FunctionsBasic Trig FunctionsProduct of Sines and Cosines (1) Product of Sines and Cosines (2) Product of Secants and Tangents Other Cases Trig SubstitutionsHow it worksExamples Completing the Square Partial FractionsIntroductionLinear Factors Quadratic Factors Improper Rational Functions and Long Division Summary Strategies of IntegrationSubstitutionIntegration by Parts Trig Integrals Trig Substitutions Partial Fractions Improper IntegralsType I IntegralsType II Integrals Comparison Tests for Convergence Differential EquationsIntroductionSeparable Equations Mixing and Dilution Models of GrowthExponential Growth and DecayLogistic Growth Infinite SequencesClose is Good Enough (revisited)Examples Limit Laws for Sequences Monotonic Convergence Infinite SeriesIntroductionGeometric Series Limit Laws for Series Telescoping Sums and the FTC Integral TestRoad MapThe Integral Test When the Integral Diverges When the Integral Converges Comparison TestsThe Basic Comparison TestThe Limit Comparison Test Convergence of Series with Negative TermsIntroductionAlternating Series and the AS Test Absolute Convergence Rearrangements The Ratio and Root TestsThe Ratio TestThe Root Test Examples Strategies for testing SeriesList of Major Convergence TestsExamples Power SeriesRadius and Interval of ConvergenceFinding the Interval of Convergence Other Power Series Representing Functions as Power SeriesFunctions as Power SeriesDerivatives and Integrals of Power Series Applications and Examples Taylor and Maclaurin SeriesThe Formula for Taylor SeriesTaylor Series for Common Functions Adding, Multiplying, and Dividing Power Series Miscellaneous Useful Facts Applications of Taylor PolynomialsWhat are Taylor Polynomials?How Accurate are Taylor Polynomials? What can go Wrong? Other Uses of Taylor Polynomials Partial DerivativesDefinitions and RulesThe Geometry of Partial Derivatives Higher Order Derivatives Differentials and Taylor Expansions Multiple IntegralsBackgroundWhat is a Double Integral? Volumes as Double Integrals Iterated Integrals over RectanglesOne Variable at the TimeFubini's Theorem Notation and Order Double Integrals over General RegionsType I and Type II regionsExamples Order of Integration Area and Volume Revisited 
BackgroundWhen talking about any mathematical quantity, it's important to keep track of three different questions:
This is especially true for integration, where the three questions have very different answers:
Of these applications, the easiest to understand is area under a curve, which is why we used area to introduce integrals. Likewise, we will use volume to introduce double integrals. But remember: ordinary integrals are about much more than area, and double integrals are about much more than volume. In the following video, we review the ideas and definitions of onedimensional integrals. Not how to compute an integral, but what it is and what it's good for. On the next page, we'll see how the exact same ideas work in two and three dimensions. We'll tackle 'how do you compute it?' in the next learning module. Remember the main idea of integration: The whole is the sum of the parts. Integration is a procedure for computing bulk quantities like area, volume, mass, distance, moment of inertia, and wealth. If $f(x)$ is the density of a quantity on interest, then
