Integration adds up infinitely many infinitely thin pieces, and series add up infinitely many numbers — 26 interactive demonstrations of both. Fill the area under a curve with rectangles, watch area grow as a function of its right edge, and see why its slope is the curve you started with: the Fundamental Theorem, where slope and area undo each other. Stretch an axis to see why substitution works, and fill a rectangle with two areas to see integration by parts. Race the trapezoid rule against Simpson’s parabolas, and fill Gabriel’s horn with paint it could never cover. Spin regions into solids and slice them into disks and shells. Stack blocks into a tower that leans as far as you like, rearrange a series to add up to any number you choose, and watch Taylor polynomials wrap themselves around a function.
See also: Differential Calculus for limits and derivatives, Vector Calculus for integrals in several variables and polar curves, Fourier Analysis for series made of sines and cosines, and Complex Analysis for why power series converge in discs.
Rectangles that fill the area under a curve, area as a function that grows, and the theorem that says slope and area undo each other
Antiderivatives as families of curves, substitution as stretching the axis, integration by parts as two areas that fill a rectangle, and partial fractions
When there is no formula: trapezoids, Simpson’s parabolas, the race between rules, and throwing darts at random
Infinite regions with finite area. The p-test, Gabriel’s horn that holds paint but cannot be painted, and the area under the bell curve
Spin a region about an axis and slice the solid into disks, washers or shells; stack cross-sections of any shape
Integrals add up more than area: the length of a curve, the balance point of a shape, and the work to pump out a tank
Adding infinitely many numbers. Geometric series, a tower of blocks that leans arbitrarily far, convergence tests, and a series you can rearrange to any sum
Polynomials that hug a function more and more tightly, the hidden reason a series stops converging, and Euler’s formula built term by term