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 |
## Area and Volume RevisitedNow that we understand what double integrals are, we can use them to compute areas of regions and volumes of solids of revolution. The following video shows how. ## AreasPics from other LMsThe area of a region $R$ is just $\iint_R 1 \,dA$. Rewriting that as an iterated integral gives us the formulas for area that we derived a long time ago. If $R$ is a Type I region, bounded by $y=g(x)$, $y=h(x)$, $x=a$ and $x=b$, then the area of $R$ is \begin{eqnarray*} \iint_R 1 \, dA &=& \int_a^b \int_{g(x)}^{h(x)} 1\, dy\,dx\\ &=& \int_a^b \Bigl(h(x)-g(x)\Bigr) dx. \end{eqnarray*} This is our familiar formula for the area between two curves. If $R$ is a Type II region, bounded by $x=g(y)$, $x=h(y)$, $y=c$ and $y=d$, then the area of $R$ is \begin{eqnarray*} \iint_R 1 \, dA &=& \int_c^d \int_{g(y)}^{h(y)} 1\, dx\,dy\\ &=& \int_c^d \Bigl(h(y) - g(y)\Bigr)\, dx. \end{eqnarray*} ## VolumesNow suppose that $R$ lies above the
$x$-axis. If we rotate the region $R$ around the $x$ axis, then the
volume of the solid of revolution is $\displaystyle{\iint_R 2\pi y
\,dA}$. If $R$ is a Type I region and we integrate first over $y$ and
then over $x$, we get
\begin{eqnarray*}
\iint_R 2 \pi y dA & = & \int_a^b \int_{g(x)}^{h(x)} 2 \pi y \,dy\, dx \\
& = & \int_a^b \left . \pi y^2 \right |_{g(x)}^{h(x)} \,dx \\
& = & \int_a^b \pi \Bigl(h(x)^2 - g(x)^2\Bigr) dx.
\end{eqnarray*}
This is our familiar formula for volumes by Likewise, if $R$ lies to the right of the $y$ axis and we rotate around the $y$ axis, then the volume the solid of revolution is $\displaystyle{\iint_R 2\pi x\, dA}$. Integrating $dx \, dy$ gives volume by washers and integrating $dy\,dx$ gives volume by cylindrical shells. |