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Integral Calculus

A brief introduction to integral calculus

How do you find the area under a curve? What about the length of any curve? Is there a way to make sense out of the idea of adding infinitely many infinitely small things? Integral calculus gives us the tools to answer these questions and many more. Surprisingly, these questions are related to the derivative, and in some sense, the answer to each one is the opposite of the derivative.

Unit 1: Integrals

Accumulations of change introduction: IntegralsApproximation with Riemann sums: IntegralsSummation notation review: IntegralsRiemann sums in summation notation: IntegralsDefining integrals with Riemann sums: IntegralsFundamental theorem of calculus and accumulation functions: IntegralsInterpreting the behavior of accumulation functions: IntegralsProperties of definite integrals: IntegralsFundamental theorem of calculus and definite integrals: IntegralsReverse power rule: Integrals
Indefinite integrals of common functions: IntegralsDefinite integrals of common functions: IntegralsIntegrating with u-substitution: IntegralsIntegrating using long division and completing the square: IntegralsIntegrating using trigonometric identities: IntegralsTrigonometric substitution: IntegralsIntegration by parts: IntegralsIntegrating using linear partial fractions: IntegralsImproper integrals: IntegralsProof videos: Integrals

Unit 2: Differential equations

Differential equations introduction: Differential equationsVerifying solutions for differential equations: Differential equationsSketching slope fields: Differential equationsReasoning using slope fields: Differential equations
Approximation with Euler’s method: Differential equationsSeparation of variables: Differential equationsParticular solutions to differential equations: Differential equationsExponential models: Differential equationsLogistic models: Differential equations

Unit 3: Applications of integrals

Average value of a function: Applications of integralsStraight-line motion: Applications of integralsNon-motion applications of integrals: Applications of integralsArea: vertical area between curves: Applications of integralsArea: horizontal area between curves: Applications of integralsArea: curves that intersect at more than two points: Applications of integralsVolume: squares and rectangles cross sections: Applications of integrals
Volume: triangles and semicircles cross sections: Applications of integralsVolume: disc method (revolving around x- and y-axes): Applications of integralsVolume: disc method (revolving around other axes): Applications of integralsVolume: washer method (revolving around x- and y-axes): Applications of integralsVolume: washer method (revolving around other axes): Applications of integralsArc length: Applications of integralsCalculator-active practice: Applications of integrals

Unit 4: Parametric equations, polar coordinates, and vector-valued functions

Arc length: parametric curves: Parametric equations, polar coordinates, and vector-valued functionsPlanar motion: Parametric equations, polar coordinates, and vector-valued functionsArea: polar regions (single curve): Parametric equations, polar coordinates, and vector-valued functions
Area: polar regions (two curves): Parametric equations, polar coordinates, and vector-valued functionsArc length: polar curves: Parametric equations, polar coordinates, and vector-valued functionsCalculator-active practice: Parametric equations, polar coordinates, and vector-valued functions

Unit 5: Series

Convergent and divergent infinite series: SeriesInfinite geometric series: Seriesnth-term test: SeriesIntegral test: SeriesHarmonic series and p-series: SeriesComparison tests: SeriesAlternating series test: SeriesRatio test: SeriesAbsolute and conditional convergence: Series
Alternating series error bound: SeriesTaylor and Maclaurin polynomials intro: SeriesLagrange error bound: SeriesPower series intro: SeriesFunction as a geometric series: SeriesMaclaurin series of eˣ, sin(x), and cos(x): SeriesRepresenting functions as power series: SeriesTelescoping series: SeriesProof videos: Series

Course challenge

Test your knowledge of the skills in this course. Have a test coming up? The Course challenge can help you understand what you need to review.
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