https://teddyrookbook.com/math/calculus-2/Recent content in Calculus 2: Integrals and Series on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/calculus-2/area-between-curves/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/area-between-curves/Integrate top minus bottom, or right minus left. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/volume-slicing/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/volume-slicing/Add up the areas of thin cross-sections. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/disks-washers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/disks-washers/Revolve a region: V = π∫(R² − r²) dx. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/shells/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/shells/V = 2π∫(radius)(height) dx. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/arc-length/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/arc-length/L = ∫√(1 + (f′(x))²) dx. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/surface-of-revolution/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/surface-of-revolution/S = 2π∫ f(x)√(1 + (f′(x))²) dx. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/work/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/work/Work is force times distance, added up over thin pieces. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/fluid-force/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/fluid-force/Pressure grows with depth, so integrate strips across the surface. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/center-of-mass/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/center-of-mass/Balance points of thin plates. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/hyperbolic-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/hyperbolic-functions/cosh and sinh, their identities and derivatives. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/integration-by-parts/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/integration-by-parts/∫u dv = uv − ∫v du reverses the product rule. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/trig-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/trig-integrals/Use identities for powers of sine, cosine, tangent and secant. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/trig-substitution/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/trig-substitution/Replace √(a² − x²) and similar roots with trig functions. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/partial-fractions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/partial-fractions/Split a rational function into pieces you can integrate. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/integration-strategy/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/integration-strategy/Recognize which method fits an integral. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/numerical-integration/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/numerical-integration/Approximate integrals with trapezoids and parabolas. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/improper-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/improper-integrals/Integrals over infinite intervals or across infinite values are limits. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/improper-comparison/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/improper-comparison/Compare with a simpler integral to decide convergence. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/de-intro/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/de-intro/An equation with a derivative; a solution is a function that satisfies it. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/slope-fields-euler/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/slope-fields-euler/Sketch solutions from slopes, and step along them numerically. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/separable-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/separable-equations/Move each variable to its own side and integrate. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/logistic-equation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/logistic-equation/dP/dt = rP(1 − P/K) models growth with a limit. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/parametric-calculus/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/parametric-calculus/dy/dx = (dy/dt)/(dx/dt). Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/polar-area/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/polar-area/A = ½∫r² dθ. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/polar-tangents-length/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/polar-tangents-length/Treat a polar curve as parametric in θ. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/sequences/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/sequences/A sequence converges when its terms approach one number. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/monotone-bounded/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/monotone-bounded/An increasing sequence with an upper bound converges. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/series-partial-sums/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/series-partial-sums/A series converges when its partial sums approach a number. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/geometric-telescoping/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/geometric-telescoping/Sum geometric series and series whose terms cancel. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/divergence-test/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/divergence-test/If the terms do not go to 0, the series diverges. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/integral-test/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/integral-test/Compare a series with an improper integral; Σ1/nᵖ converges when p > 1. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/comparison-tests/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/comparison-tests/Compare with a known series. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/alternating-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/alternating-series/Terms that shrink to 0 and alternate in sign give a convergent series. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/absolute-conditional/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/absolute-conditional/Test Σ|aₙ| to learn how a series converges. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/ratio-root-tests/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/ratio-root-tests/Look at the limit of |aₙ₊₁/aₙ| or |aₙ|^(1/n). Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/series-strategy/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/series-strategy/Match the series to the test that fits. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/series-estimates/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/series-estimates/Bound the error of a partial sum. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/power-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/power-series/Find where Σcₙ(x − a)ⁿ converges. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/functions-as-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/functions-as-series/Build series from 1/(1 − x). Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/differentiate-integrate-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/differentiate-integrate-series/Work term by term inside the interval of convergence. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/taylor-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/taylor-series/cₙ = f⁽ⁿ⁾(a)/n!. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/taylor-error/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/taylor-error/Bound the error of a Taylor polynomial. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/known-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/known-series/Build new series from those for eˣ, sin x, cos x and ln(1 + x). Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/binomial-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/binomial-series/(1 + x)ᵏ as a series for any k. Calculus 2: Integrals and Series · Math Ladderhttps://teddyrookbook.com/math/calculus-2/series-applications/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/calculus-2/series-applications/Use series for limits, integrals and approximations. Calculus 2: Integrals and Series · Math Ladder