Calculus II

Units

1. 1

   Areas and volumes

   6 lessons

   1. Area between curves expressed as functions of x and of ySketch a bounded region, choose a slice direction, and calculate its area with correct limits.
   2. Area between curves that intersect at more than two pointsFind crossings, calculate the area of every region, and justify a split-integral or absolute-value setup.
   3. Volumes with cross sections: squares, rectangles, triangles and semicirclesFind volume from a base region and the shape of its perpendicular cross sections.
   4. Volume with the disc method about axes and other linesFind a disc radius, set up its integral, and calculate volume about a horizontal or vertical line.
   5. Volume with the washer method about axes and other linesDraw both radii, set up washer integrals, and calculate volumes about horizontal or vertical lines.
   6. Volumes by cylindrical shellsBuild V = 2π ∫ (radius)(height) dx from one thin strip, then use it about the y-axis and about other vertical lines, checking against washers.
2. 2

   Techniques of integration

   10 lessons

   1. Integrating using integration by partsChoose and apply integration by parts to products, including repeated and cyclic cases.
   2. Trigonometric integralsIntegrate products of powers of sine and cosine, powers of tangent and secant, and products like sin 3x cos x by choosing the right identity.
   3. Trigonometric substitutionRemove a square root of a² − x², a² + x² or x² − a² with a trigonometric substitution, integrate, and return to x or convert the limits.
   4. Integrating using linear partial fractionsDecompose rational functions with distinct linear factors, then find indefinite and definite integrals.
   5. Partial fractions: repeated and quadratic factorsDecompose any rational function with repeated linear or irreducible quadratic factors, and integrate it into logarithms, powers and arctangents.
   6. Selecting techniques for antidifferentiation (BC)Choose a method, find an antiderivative, and check it by differentiating.
   7. Integration strategy: choosing a techniqueChoose a first move for any integral by asking a fixed order of questions, and recognise when no technique can give a formula.
   8. Numerical integration: trapezoid, midpoint and Simpson's rule with error boundsEstimate a definite integral from a table of values with three rules, predict whether each is high or low, and choose n to guarantee a tolerance.
   9. Evaluating improper integralsEvaluate an improper integral by writing its defining limit and deciding whether that limit is finite.
   10. Improper integrals: comparison and mixed typesDecide whether an improper integral converges, split it when it has two troubles, and use p-integrals and the comparison theorem when no antiderivative is handy.
3. 3

   Further applications of integration

   5 lessons

   1. The arc length of a smooth planar curve and distance travelledFind distance along a curve using an arc-length integral, and count every pass along a path.
   2. Surface area of a solid of revolutionDerive S = 2π ∫ (radius) ds from thin frustum bands and use it about the x-axis and the y-axis on curves where the integral simplifies.
   3. Work done by a variable forceCompute work as W = ∫ F dx for a hanging rope, a spring and a tank of water, slicing the load by distance.
   4. Hydrostatic pressure and forceFind the force of a still fluid on a flat plate by slicing it into horizontal strips with depth and width written as functions of y.
   5. Moments, centres of mass and centroidsFind balance points of point masses and centroids of uniform regions with x̄ = (1/A) ∫ x f dx and ȳ = (1/A) ∫ ½ f² dx, then use Pappus to check volumes.
4. 4

   Differential equations

   8 lessons

   1. Modelling situations with differential equations and verifying solutionsTranslate a rate description into a differential equation, verify a function, and use a condition to choose a solution.
   2. Sketching and reasoning with slope fieldsSketch slope segments, trace a solution through a given point, and recognize patterns and equilibria.
   3. Approximating solutions using Euler's methodEstimate a solution with Euler’s method and justify whether the estimate is above or below the actual value.
   4. General and particular solutions by separation of variablesFind a general solution, use an initial condition to select a particular solution, and state its largest domain.
   5. Exponential models with differential equationsSolve proportional-rate equations, use half-life or doubling time, and interpret a nonzero baseline.
   6. First-order linear equations and integrating factorsSolve y' + P(x) y = Q(x) by the integrating factor, with the constant in the right place, and fit an initial condition.
   7. Modelling with first-order equations: cooling, mixing and orthogonal trajectoriesTurn a cooling story, a mixing tank (fixed or changing volume) and a family of curves into first-order equations, solve them and check the results.
   8. Logistic models with differential equationsFind a logistic model’s limit, fastest-growth population, and solution from an initial value.
5. 5

   Parametric equations and polar coordinates

   10 lessons

   1. Parametric equations: graphing, direction and eliminating the parameterGraph a parametric path with arrows, eliminate t, and keep only the points reached.
   2. Polar coordinates and polar graphsPlot polar points and curves, convert coordinates, and recognize circles, cardioids, and roses.
   3. Defining and differentiating parametric equationsLocate points, find slopes and tangent lines, and identify horizontal and vertical tangencies.
   4. Second derivatives of parametric equationsCalculate d²y/dx² and use its sign to determine concavity at a given parameter value.
   5. Arc lengths of curves given by parametric equationsFind distance traveled along a parametric path, including when it retraces itself.
   6. Areas and surface areas for parametric curvesFind the area under a parametric curve and the surface area of a curve spun about an axis, with the right limits and sign.
   7. Polar coordinates and differentiating in polar formFind a polar curve’s tangent slope and tell whether its point moves toward or away from the origin.
   8. Area of a polar region and between two polar curvesCalculate polar area by choosing angles that trace a region once and, when needed, subtracting inner sectors.
   9. Arc length in polar coordinatesDerive and use L = ∫ √(r² + (dr/dθ)²) dθ, choosing one trace of the curve and keeping the absolute value.
   10. Conic sections in polar formRead e, d, the directrix, the vertices and the shape of a conic from r = ed/(1 ± e cos θ) or (1 ± e sin θ), and use it for an orbit.
6. 6

   Hyperbolic functions and complex exponentials

   3 lessons

   1. Hyperbolic functionsDefine cosh x and sinh x from the exponentials, graph them, show cosh²x − sinh²x = 1 as the hyperbola they trace, and use cosh to describe a hanging cable.
   2. Calculus of hyperbolic functionsDifferentiate and integrate sinh, cosh and tanh, write the inverse hyperbolic functions as logarithms, and evaluate ∫ dx/√(x² + 1) = sinh⁻¹ x + C.
   3. Complex numbers and Euler's formulaWrite a complex number as a point and as r e^{iθ}, prove e^{iθ} = cos θ + i sin θ from the Taylor series, and use it to multiply by rotating and scaling and to raise to powers.
7. 7

   Sequences and series

   20 lessons

   1. Sequences: limits and monotone convergenceFind a sequence's limit with the function trick and the landmark limits, and prove a recursive sequence converges before solving for its limit.
   2. Defining convergent and divergent infinite seriesDetermine convergence from partial sums and find a series’ sum when it exists.
   3. Working with geometric seriesDetermine whether a geometric series converges and find its sum when it does.
   4. Telescoping series and sums by partial fractionsFind the sum of a telescoping series by writing its partial sum, counting the leftover terms and taking the limit, and spot when it diverges.
   5. The nth term test for divergenceUse a term limit to prove divergence or recognize when the test gives no conclusion.
   6. The integral test, the harmonic series and p-seriesCheck integral-test conditions and use an improper integral or the p-series rule to classify a series.
   7. Comparison tests for convergenceChoose a known series and justify convergence or divergence by direct or limit comparison.
   8. Alternating series test for convergenceJustify convergence with both alternating series test conditions, and recognize when the nth-term test proves divergence.
   9. Alternating series error boundBound the error of an alternating-series approximation and determine whether it is high or low.
   10. Absolute or conditional convergenceClassify a series as absolutely convergent, conditionally convergent, or divergent, justifying both relevant tests.
   11. Ratio test for convergenceCompute consecutive-term ratio limits, apply the ratio test, and recognize when it is inconclusive.
   12. The root test and choosing a convergence testUse the root test on n-th powers and choose a first test for any series by asking seven questions in order.
   13. Radius and interval of convergence of power seriesFind a power series’ radius and justify its full interval by testing both endpoints.
   14. Power series: radius and interval of convergence in depthFind the radius and the exact interval of convergence of a power series, testing both endpoints, and say what termwise calculus does to each.
   15. Representing functions as power seriesBuild power series from a geometric series and determine where the resulting series converge.
   16. Finding Taylor polynomial approximations of functionsConstruct a Taylor polynomial from derivatives at its center and use it to approximate a nearby value.
   17. Lagrange error boundCalculate a Taylor polynomial error bound and use it to certify accuracy.
   18. Finding Taylor or Maclaurin series for a functionWrite first terms and a general term, then recover derivatives from coefficients.
   19. The binomial seriesWrite the series of (1 + x)ᵏ for any real k, find its first terms from derivatives, state where it is valid, and use it to approximate roots and build new series.
   20. Taylor series in use: limits, integrals and error controlUse known Maclaurin series to find limits, integrate functions with no elementary antiderivative, sum numerical series, and choose the right error bound for a stated tolerance.