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 Anti-derivativesA Table of Common Anti-derivatives 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 |
## Three Different QuantitiesAs the name implies, the Fundamental Theorem of Calculus (FTC) is among the biggest ideas of Calculus, tying together derivatives and integrals. One half of the theorem gives the easiest way to compute definite integrals. The other half relates the rate at which an integral is growing to the function being integrated. It is the 5th of the Six Pillars of Calculus:
The Fundamental Theorem of Calculus relates three very different concepts: - The
**definite integral**$\int_a^b f(x)\, dx$ is the limit of a sum. $$\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x,$$ where $\Delta x = (b-a)/n$ and $x_i^*$ is an arbitrary point somewhere between $x_{i-1}=a + (i-1)\Delta x$ and $x_i = a + i \Delta x$. The name we give to the variable of integration doesn't matter: $$\int_a^b f(x) \,dx = \int_a^b f(s)\, ds = \int_a^b f(t)\, dt$$ - The
**indefinite integral**is the function $$I(x) = \int_a^x f(s)\, ds.$$ That is, it is a running total of the amount of stuff that $f$ represents, between $a$ and $x$. If $f$ is the height of a curve, then $I(x)$ is the area under the curve between $a$ and $x$. If $f$ is velocity, then $I(x)$ is the distance traveled between time $a$ and time $x$.
- An
**antiderivative**$F(x)$ of $f(x)$ is a function with $F'(x)=f(x)$. There are actually many different anti-derivatives of $f(x)$, but they differ by constants. For instance, $x^3$ and $x^3+7$ are both anti-derivatives of $3x^2$.
When studying the Fundamental Theorem of Calculus, it's very important to keep these straight! |