Calculus 2
Techniques of integration, differential equations, series, and parametric and polar curves. 2816 problems, in the order most textbooks teach them.
Start here
Six core problems from across the course. Pick one, try it on paper, then open the solution.
Interactives
Taylor Polynomial Explorer
A scroll-through tutorial: build the polynomial one derivative at a time, measure its error, move its center, and find where more terms stop helping.
Taylor and Maclaurin polynomials Assumptions statedPumping Work Explorer
A scroll-through tutorial: cut the water in a tank into slabs, add up the work to lift each one, and watch the sum become an integral for a cylinder, a cone and a sphere.
Work: springs, pumping and cables Assumptions statedTechniques of integration
Integration by parts
Undo the product rule: ∫u dv = uv − ∫v du.
90 problemsTrigonometric integrals
Powers and products of sines and cosines, with identities.
49 problemsPartial fractions
Split a rational function into simple pieces, then integrate each.
59 problemsImproper integrals
Infinite intervals and infinite integrands: replace the bad bound with a limit.
28 problemsTrapezoidal, midpoint and Simpson's rules
The trapezoidal, midpoint and Simpson's rules, and how close they come.
56 problemsPartial fraction decomposition
Split a rational function into simpler fractions, and integrate it term by term.
60 problemsIntegrals of powers of sine and cosine
∫ sinᵐx cosⁿx dx: save one factor when a power is odd, use half-angle identities when both are even.
57 problemsTrigonometric substitution
Integrals with √(a² − x²), √(a² + x²): substitute x = a sin θ or a tan θ, integrate, and draw the triangle to get back to x.
35 problemsComparison test for improper integrals
Decide whether an improper integral converges by comparing it with a simpler one.
13 problemsLaplace transforms from the definition
The Laplace transform from its definition, as an improper integral.
22 problemsDifferential equations
Separable differential equations
Separate the variables, integrate both sides, and use the initial condition.
70 problemsFirst-order linear equations
The integrating factor method for y′ + p(x)y = q(x).
70 problemsEuler's method
Step along the slope field: yₙ₊₁ = yₙ + h·f(tₙ, yₙ).
60 problemsThe logistic equation
Logistic growth: equilibria, the solution P(t) = K / (1 + Ae^(−rt)), and when the population reaches a level.
60 problemsNewton's law of cooling
Newton's law of cooling: find k from one reading, then predict a temperature or a time.
60 problemsOrder and linearity of differential equations
The order of a differential equation, whether it is linear (and homogeneous), and checking a solution.
60 problemsDifferential equations by direct integration
Solve y′ = f(x) or y″ = f(x) by integrating, with or without an initial condition.
60 problemsMixing tank problems
Salt in a tank with brine flowing in and out: set up and solve the differential equation.
60 problemsInitial-value problems needing partial fractions
Initial-value problems where the integral needs partial fractions or an arctangent.
60 problemsSequences and series
Geometric series
Σ arⁿ converges to a/(1 − r) exactly when |r| < 1.
70 problemsSeries convergence tests
Divergence, p-series, integral, comparison and ratio tests: which one, and why.
66 problemsRadius of convergence
Where a power series converges, found with the ratio test.
70 problemsTaylor and Maclaurin polynomials
Polynomial approximations built from derivatives at a point.
32 problemsLimits of sequences
Does the sequence converge? The first few terms, then the limit as n → ∞.
61 problemsTelescoping series
Telescoping series: split the term, watch the middle cancel, and take the limit of the partial sums.
60 problemsAbsolute and conditional convergence
Absolutely convergent, conditionally convergent, or divergent? The alternating series test and a comparison for |aₙ|.
14 problemsError bounds for series
How many terms are enough? Error bounds from the alternating series test and the integral test.
20 problemsComparison and limit comparison tests
Compare with a p-series: the limit comparison test decides most rational and algebraic terms.
60 problemsPower series from the geometric series
Power series from 1/(1 − u) = Σ uⁿ: substitute, multiply, and read off the interval of convergence.
60 problemsTaylor series about a point
Taylor polynomials and series about a point other than 0.
14 problemsMaclaurin series by substitution
Maclaurin series built from known ones by substitution and multiplication.
21 problemsThe Lagrange error bound
Bound the error of a Taylor polynomial with M|x − a|ⁿ⁺¹/(n + 1)!, or find the degree that makes it small enough.
48 problemsPartial sums of a series
The first partial sums of a series, and what they say about convergence.
12 problemsThe root test
Decide convergence with the root test: the limit of the n-th root of |aₙ|.
60 problemsThe binomial series
The first terms of the binomial series (1 + u)ʳ for a fractional or negative r.
55 problemsLimits using Taylor series
Evaluate a 0/0 limit by replacing the numerator with its Taylor polynomial.
42 problemsSumming a series by differentiating or integrating
Sum Σ n xⁿ or Σ xⁿ/n by differentiating or integrating the geometric series.
20 problemsParametric and polar
Tangent lines to parametric curves
dy/dx = (dy/dt)/(dx/dt), and the tangent line at a parameter value.
70 problemsArea in polar coordinates
½∫ r² dθ: petals, cardioids and circles.
20 problemsSecond derivatives of parametric curves
dy/dx and d²y/dx² for a parametric curve, without eliminating the parameter.
29 problemsArc length of parametric curves
Arc length of a parametric curve: ∫ √((dx/dt)² + (dy/dt)²) dt.
53 problemsConverting between polar and rectangular coordinates
Between polar and rectangular: points both ways, and simple curves.
60 problemsEliminating the parameter
Eliminate the parameter to find the curve's equation in x and y, and name it.
60 problemsHorizontal and vertical tangents of parametric curves
Points where a parametric curve has a horizontal or a vertical tangent.
62 problemsArea and surface area with parametric curves
The area under a parametric curve, and the surface area when it is rotated about the x-axis.
29 problemsTangent lines to polar curves
The slope of the tangent line to a polar curve r = f(θ) at a given angle.
60 problemsArc length in polar coordinates
Arc length of a polar curve, L = ∫ √(r² + (dr/dθ)²) dθ.
28 problemsEquations of conic sections
Equations of parabolas, ellipses and hyperbolas from their foci, vertices and directrices.
51 problemsConics in polar form
A conic r = ed/(1 ± e cos θ): its eccentricity, its type and its directrix.
60 problemsApplications of integration
Surface area of revolution
Surface area of revolution: 2π ∫ (radius) √(1 + (y′)²) dx.
60 problemsWork: springs, pumping and cables
Work as force times distance, added up: springs, pumping water out of a tank, winding up a cable.
60 problemsMass from a density
Mass from a linear density along a rod, or a radial density across a disk.
60 problemsCenters of mass and centroids
Centers of mass of point masses and rods, and centroids of plane regions.
60 problemsHydrostatic force
Force of water on a submerged vertical plate: pressure grows with depth, so integrate over strips.
60 problemsThe theorem of Pappus
Volumes of revolution by the theorem of Pappus: area times the distance the centroid travels.
60 problemsProbability density functions
Normalize a probability density, then find a probability and the mean.
60 problems