book

Make: Calculus

by Joan Horvath, Rich Cameron

August 2022

Intermediate to advanced

328 pages

8h 26m

English

Make: Community

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Who this Book Is ForWhat We Assume you Know AlreadyTeaching and Learning With This BookDeveloping a Hands-on Calculus Course3D Printable ModelsChapter LayoutAcknowledgmentsAbout the Authors
Building CalculusThe Steadily-Increasing WallThe Curved WallNegative ChangesExamples to TryMeasuring Real-World ChangeInstantaneous SlopeLooking AheadSecond Fundamental TheoremChapter Key PointsTerminology and SymbolsSolutions
What is Calculus?FunctionsWhen Our Brick Models FailLimitsDerivatives and CurvesFundamental Theorem ModelDimensional AnalysisEqual, but Not the SameChapter Key PointsTerminology and SymbolsReferences
OpenscadOpenSCAD WorkflowIdiosyncrasies of OpenSCADNavigating on the ScreenCommentsThe ModelsExample 1: Changing a ParameterExample 2: Changing a Model With the CustomizerSome Models Have Small PartsDownloading the Models: Github3D Printing3D Printing WorkflowMaterialsPrinting TipsIf you do not Have a 3D PrinterChapter Key PointsTerminology and SymbolsLearning More
The Derivative-Integral ModelModel ParametersUsing Other LEGO BricksTesting Your DerivativesCustomizer WorkaroundsPlotting Curves and Derivatives Not in the CustomizerPaper ModelsInstantaneous SlopeTangent LinesThe Mean Value TheoremExamplesDerivatives of Other Powers of xSines and CosinesDegrees, Radians, and PiExponential GrowthOffset CalculationEuler’s Number, eLogarithmsExponential Curve OffsetExperiments to TryChapter Key PointsTerminology and SymbolsReferences

Maxima, Minima, Inflection PointsSecond DerivativesInflection PointsOther Inflection Point SituationsSketching a Curve From Its DerivativesCalculating DerivativesThe Chain RuleDerivatives of Products and QuotientsDerivative of a ProductDerivative of a QuotientL’Hôpital’s RuleOther Ways of Writing DerivativesPartial DerivativesModeling the SurfaceModeling the Partial DerivativesHigher-Order Partial DerivativesChapter Key PointsTerminology and SymbolsExercise AnswersReferences
What is an Integral?Assembling an IntegralThe Second Part of the Fundamental Theorem of CalculusComputing IntegralsIndefinite Integrals (Antiderivatives)Area Under a CurveArea of a RegionComputing an AverageThe Mean Value Theorem, Reprised3D Printing IntegralsIntegrals of Powers of XIntegrals of Sine and CosineIntegrals of ExponentialsApplication: PID ControllersExperiments to TryChapter Key PointsTerminology and SymbolsReferences
3D CoordinatesVolumes of RevolutionVolume of a ConeMethod of DisksCavalieri’s PrincipleCalculating With Method of DisksVolumes of Other Solids of RevolutionRevolution ModelsSurfaces of RevolutionComputing Volume of More General SolidsCalculating VolumeChecking Our ResultsPrinting This ModelIntegral of a Product or QuotientIntegral of a QuotientDoing the AlgebraPrinting and Experimenting With the ModelVolume Under a SurfaceChapter Key PointsTerminology and SymbolsReferences
Ordinary Differential EquationsExponential Growth or Decay EquationRadioactive DecayOther ExponentialsThe Logistic EquationMath of EpidemicsDifference EquationsBrick Model RepriseNumerical Models of DerivativesNumerical Models of Higher DerivativesError in Numerical SolutionsError, Exponential EquationError, Logistic EquationNumerical Models of IntegralsWorking with Real DataChapter Key PointsTerminology and SymbolsReferences
Going Around in CirclesPhase ShiftsSine and Cosine Derivative RelationshipsApproximating Sine and CosineSimple Harmonic MotionSecond Order Ordinary Differential EquationsSpring ExperimentPendulum ExperimentSystems of Differential EquationsReprising the Logistic EquationThe Lotka-Volterra EquationsPopulation Behavior Over TimeExploring the Lotka-Volterra EquationsCreating the ModelsPhase SpacePhase-Space ModelSlope FieldsStable PointChanging Population RatiosAttofox ProblemSeparation of VariablesChapter Key PointsTerminology and SymbolsReferences
Calculus Models of CircuitsSimulating CircuitsDefinitions and Units of Electrical ComponentsResistor, Capacitor, and Inductor CircuitsRC CircuitsCapacitive Touch SensingLC CircuitsRL and RLC CircuitsFiltersAccelerometers and GyroscopesAccelerometer MouseSetting up a Circuit Playground Classic or ExpressArduino Sketch StructureAlgorithm for the Accelerometer MouseCircuit Playground Sketch for Accelerometer MouseSetting up the MouseTesting Out the MouseLight-Up PendulumMaking the LED PendulumLED Pendulum SketchOther Circuit Playground Accelerometer Project IdeasPID ControllersTemperature ControlBall and BeamInverted PendulumsChapter Key PointsTerminology and SymbolsReferences
Cartesian, Polar, Cylindrical, and Spherical CoordinatesCreating the ModelsIntegrals and Derivatives in Polar CoordinatesVector BasicsVector AdditionMethod of ShellsMultiplying a Vector by a ScalarComplex NumbersThe Complex PlaneRaising Complex Numbers to a PowerVectors Meet CalculusVector Multiplication: Dot ProductApplying the Dot Product: WorkVector Multiplication: Cross ProductApplying the Cross Product: TorqueVector FieldsGrad, Div, and CurlChapter Key PointsTerminology and SymbolsReferences
Sequences vs. SeriesSeriesInfinite SeriesSeries Expansions of FunctionsPower SeriesTaylor and Maclaurin SeriesMaclaurin Series of Sine, Cosine, and ExponentialModeling ConvergenceSinusoid ModelsExponential ModelPrinting the ModelsBroader ApplicationsLimits and SeriesEuler’s Equationde Moivre’s TheoremProving Euler’s EquationChapter Key PointsTerminology and SymbolsReferences
Calculating Integrals and DerivativesIntegration by PartsTrigonometric IdentitiesCofunctionsDouble Angles and Sums of AnglesSquared FunctionsTrigonometric SubstitutionMath Modeling in Real LifeChapter Key PointsTerminology and SymbolsResources for Further StudyUseful Websites and Search SuggestionsCalculation ResourcesBooks

Content preview from Make: Calculus

7 INTEGRALS AND VOLUME

In the last chapter, we introduced the basics of computing integrals, along with simple examples that mostly looked at applications like area, which are computed in two dimensions. In this chapter, we move up to looking at integrals to compute the volume of a 3D object. We have already seen that you can use an integral to calculate the area under a curve.

If we would like to check our calculations by measuring the area, there are not very good physical ways to do that. There are a lot of physical ways to measure volume, though. Since an integral can be used to compute the area under a curve, it follows that we can also use them to figure out the volume of a solid. There are several different ways to do that, and we will ...