🧮Calculus II

A fast review of the definite integral and the Fundamental Theorem, sharpened for everything ahead.

Riemann Sums and the Definite Integral · Properties of Definite Integrals · The Fundamental Theorem, Part 1: The Accumulation Function · The Fundamental Theorem, Part 2: Evaluating with an Antiderivative · +2 more

The chain rule run backward — spotting the inner function whose derivative is already sitting in the integral.

The Idea Behind u-Substitution · Substitution in a Definite Integral · Integrals Involving Exponential Functions · Integrals That Produce a Logarithm · +1 more

The product rule run in reverse, trading one hard integral for an easier one plus a boundary term.

Integration by Parts as the Product Rule in Reverse · Choosing u and dv · Repeated Integration by Parts and the Tabular Method · Integration by Parts on a Definite Integral · +1 more

Powers and products of sine, cosine, tangent, and secant, cracked open by identity instead of raw substitution.

Powers of Sine and Cosine with an Odd Exponent · Powers of Sine and Cosine with Only Even Exponents · Integrals of Powers of Tangent and Secant · Products of Sines and Cosines with Different Periods · +1 more

A square root that won't simplify meets a right triangle, and the whole integral turns into trigonometry.

The Sine Substitution for sqrt(a^2 - x^2) · The Tangent Substitution for sqrt(a^2 + x^2) · The Secant Substitution for sqrt(x^2 - a^2) · Completing the Square Before a Trig Substitution

Splitting a tangled rational function into pieces simple enough to integrate one at a time.

Splitting a Rational Function into Simple Pieces · Distinct Linear Factors · Repeated Linear Factors · Irreducible Quadratic Factors · +1 more

No new formula here — just the pattern-reading that decides which technique an integral is actually asking for.

Reading an Integral to Find Its Technique · Integrals That Need Two Techniques in a Row · When Substitution Alone Isn't Enough · Integrals with No Elementary Antiderivative

When there's no antiderivative to fall back on, straight lines and parabolas estimate the area instead.

The Trapezoidal Rule · Simpson's Rule · The Error Bound for the Trapezoidal Rule · The Error Bound for Simpson's Rule

Stretching the interval to infinity, or letting the integrand blow up, and asking whether the area still adds up.

Improper Integrals with an Infinite Limit · Improper Integrals with an Infinite Discontinuity · p-Integrals on [1, infinity) · The Comparison Test for Improper Integrals · +2 more

Turning a slice, a shell, or a stretched-out curve into an integral you can actually solve.

Area Between Two Curves · Area Between Curves That Cross, and Integrating in y · Volumes by Disks and Washers · Volumes by Cylindrical Shells · +2 more

Work, pressure, mass, and probability, once you notice they're all just another sum of infinitesimal slices.

Work Done by a Variable Force · Pumping Liquid Out of a Tank · Hydrostatic Force on a Submerged Surface · Center of Mass and Moments · +1 more

Equations built from a rate of change, solved by finding the function hiding behind the rate.

Verifying a Solution to a Differential Equation · Separable Differential Equations · Direction Fields · Euler's Method · +2 more

A curve traced by a moving point, x and y each written as their own function of time.

Defining a Curve Parametrically · Eliminating the Parameter to Find a Rectangular Equation · The Slope of a Parametric Curve · Concavity of a Parametric Curve · +2 more

Locating a point by a distance and an angle instead of two perpendicular distances, and watching curves bloom.

Converting Between Polar and Rectangular Coordinates · Graphing a Polar Curve as Theta Sweeps · Cardioids, Roses, and Limacons · Tangent Lines to a Polar Curve · +2 more

A list of numbers marching toward infinity, and the handful of ways to prove whether it settles or runs away.

Defining a Sequence and Its Limit · Sequences Defined Recursively · Monotonic and Bounded Sequences · The Squeeze Theorem for Sequences · +1 more

Adding infinitely many numbers without getting nonsense — which sums settle down, and which run away forever.

Sigma Notation for a Series · Series and Partial Sums · The Geometric Series · Telescoping Series · +3 more

The full toolkit for deciding whether a series converges, derived rather than just asserted.

The Integral Test · The Direct Comparison Test · The Limit Comparison Test · The Alternating Series Test · +5 more

A polynomial with infinitely many terms, well-behaved near its center and dead on arrival past a fixed distance.

Defining a Power Series · Radius of Convergence · Interval of Convergence and the Endpoints · Representing Functions as a Power Series · +1 more

An infinite polynomial that reproduces a function exactly, built one matched derivative at a time.

Taylor Polynomials · Maclaurin Polynomials of Increasing Degree · The Lagrange Error Bound · Finding a Taylor or Maclaurin Series · +4 more