Calculus II — Integral Calculus & Infinite Series

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• Net area and signed accumulation ~15 min
• Properties of the definite integral ~12 min
• The accumulation function ~15 min
• Fundamental Theorem, Part 1, revisited ~15 min
• Fundamental Theorem, Part 2, and evaluation ~15 min
• Substitution in definite integrals ~15 min
• Symmetry — even, odd, and periodic integrals ~15 min
• Average value and the Mean Value Theorem for integrals ~15 min
• Displacement, distance, and net change ~15 min
• Integrating to inverse-trig forms ~15 min

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• Area between two curves ~15 min
• Choosing dx or dy ~12 min
• Volumes by slicing ~15 min
• The disk method ~15 min
• The washer method ~15 min
• Cylindrical shells ~15 min
• Choosing a volume method ~12 min
• Arc length ~15 min
• Surface area of revolution ~15 min
• Work ~15 min
• Fluid force and hydrostatic pressure ~15 min
• Moments, centers of mass, and centroids ~15 min
• Probability density and expected value ~15 min

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• Integration by parts ~18 min
• Repeated parts and tabular integration ~15 min
• Trigonometric integrals ~20 min
• Trigonometric substitution ~20 min
• Completing the square before substituting ~15 min
• Partial fractions — distinct linear factors ~18 min
• Partial fractions — repeated and irreducible factors ~20 min
• Rationalizing and half-angle substitutions ~18 min
• Integral tables and computer algebra ~12 min
• A strategy for integration ~15 min
• Numerical integration — trapezoid and midpoint rules ~15 min
• Simpson's rule and error bounds ~18 min
• Improper integrals with infinite limits ~15 min
• Improper integrals with infinite discontinuities ~15 min
• Comparison tests for improper integrals ~15 min

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• What a differential equation is ~12 min
• Slope fields ~14 min
• Euler's method ~15 min
• Separable equations ~16 min
• Initial value problems and particular solutions ~13 min
• Exponential growth and decay ~15 min
• Newton's law of cooling and mixing problems ~17 min
• The logistic equation ~16 min
• Autonomous equations, phase lines, and stability ~15 min
• First-order linear equations and integrating factors ~17 min

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• Parametric Curves ~12 min
• Eliminating the Parameter ~12 min
• Derivatives of Parametric Curves ~12 min
• Tangent Lines and Concavity ~12 min
• Arc Length of a Parametric Curve ~12 min
• Area Under a Parametric Curve ~12 min
• Surface Area of a Parametric Revolution ~12 min
• Speed, Motion, and the Cycloid ~14 min

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• The Polar Coordinate System ~12 min
• Converting Between Polar and Rectangular ~14 min
• Graphing Polar Equations ~14 min
• Symmetry: Roses, Cardioids, and Limaçons ~14 min
• Slopes and Tangent Lines in Polar ~16 min
• Area Enclosed by a Polar Curve ~16 min
• Area Between Polar Curves and Finding Intersections ~18 min
• Arc Length in Polar Coordinates ~15 min
• Conic Sections in Polar Form ~15 min

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• Sequences and their notation ~12 min
• Limits of sequences ~14 min
• Limit tools for sequences ~16 min
• Monotone and bounded sequences ~14 min
• Recursive sequences and fixed points ~15 min
• Infinite series and partial sums ~14 min
• Geometric series ~15 min
• Telescoping series ~13 min
• The divergence test and the harmonic series ~15 min
• The algebra of series ~13 min

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• The Integral Test ~14 min
• p-Series ~10 min
• Remainder Estimates from the Integral Test ~12 min
• The Direct Comparison Test ~12 min
• The Limit Comparison Test ~12 min
• Alternating Series and the Alternating Series Test ~13 min
• The Alternating Series Error Bound ~10 min
• Absolute and Conditional Convergence ~12 min
• The Ratio Test ~13 min
• The Root Test ~12 min
• Choosing a Convergence Test ~12 min
• Rearrangement and the Fragility of Conditional Convergence ~11 min

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• Power series ~14 min
• Radius and interval of convergence ~16 min
• Testing the endpoints ~15 min
• Functions as geometric power series ~15 min
• Differentiating and integrating power series ~16 min
• Taylor polynomials ~16 min
• Taylor and Maclaurin series ~17 min
• Taylor's Theorem and the Lagrange remainder ~18 min
• Proving a Taylor series converges to its function ~17 min
• The binomial series ~16 min

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• The standard Maclaurin series library ~16 min
• Building new series from known ones ~18 min
• Multiplying and dividing series ~18 min
• Approximating values with error bounds ~17 min
• Evaluating limits with series ~15 min
• Integrating non-elementary functions ~16 min
• Euler's formula and complex exponentials ~18 min
• Series in physics and engineering ~16 min