Collection
From area to the integral: Riemann sums, antiderivatives, integration rules, units and improper integrals.
Apps
The definite integral as a limit of Riemann sums: a 3D column landscape under the curve, five example functions, choice of sample points (lower/upper/left/midpoint/right sum), a "refine" animation (columns grow ever thinner). Two singularity special cases show: for 1/√x the integral exists despite the pole, for 1/x² the sum grows beyond all bounds — the limit does not exist.
Analysis (one-dimensional)Try it
The integral as an oriented area balance and its four rules: additivity over intervals with a draggable intermediate point c (even outside [a, b]), swapping the limits with a negative strip width dx, the factor rule with a slider r and the sum rule with f, g and f + g. Below, every sub-area is an arrow: areas below the axis count negative.
Analysis (one-dimensional)Try it
A function f has infinitely many antiderivatives F + C — vertically shifted copies with the same slope f(x₀) at every point. Top: f with a ruler; bottom: the family with parallel tangents; a point P fixes C. Checking by differentiating exposes typical mistakes such as xⁿ → n·xⁿ⁻¹. Six functions from x² to eˣ.
Analysis (one-dimensional)Try it
A strip under a rate is height times width, with units on both edges: 5.94 m/s · 1 s = 5.94 m. The strips add up to distance, volume, energy or charge, no longer a rate. Below, the stock with its initial value: the integral is the change. Five contexts from velocity to solar power, plus the quiz “Which unit?”.
Analysis (one-dimensional)Try it
The limit z runs up to one million while the area A(z) is plotted live: under 1/x² it approaches 1, under 1/x it grows like ln z without bound. Free exponent p (convergent exactly for p > 1), side-by-side comparison, the pole case ∫₀¹ with ε → 0 and a three-step calculation instead of “plugging in ∞”.
Analysis (one-dimensional)Try itCollections
Instruments from the inside: the grand piano action, bowed strings on violin and cello, organ, trumpet, flute, saxophone, handpan, guitar and keyboard. Cut them open, play them yourself and hear how the sound is made.
11 appsOpen collection
Sentence structure, participles, the accusative with infinitive, verb forms, sequence of tenses, small words, rhetorical figures and the hexameter in real passages from Seneca and Ovid.
8 appsOpen collection
Galvanic cells, reaction mechanisms, equilibrium and the Haber-Bosch process, acids and bases, dyes, plastics, intermolecular forces and molecular structure to try out.
18 appsOpen collection
Photoelectric effect, potential well, Franck-Hertz, Millikan, diffraction gratings, standing waves, particles in fields, induction and oscillations to try out.
23 appsOpen collection
Gene regulation, PCR, genetic crosses, protein biosynthesis, evolution, cellular respiration, hormones and ecosystems to try out.
24 appsOpen collection
Planning and steering projects: the iron triangle, stakeholders, network diagrams, earned value, schedule risk, team size, Kanban and waterfall against Scrum.
8 appsOpen collection
From sequences to loans: arithmetic and geometric growth, savings plans, annuities, compounding within the year and perpetuities.
11 appsOpen collection
Reading the traces: burial mounds in section, radiocarbon dating, settlements in the landscape and the tomb of Tutankhamun.
5 appsOpen collection
Why robot arms are heavy, how six axes guide a hand and how control holds the load.
4 appsOpen collection
Tables, keys, joins, groups and subqueries you can touch: how a database really reads.
7 appsOpen collection
Generative AI in strong images: what it can do, how it writes, why it convinces, how to use it well and where to take care.
7 appsOpen collection
Engine, brakes, high voltage, CAN bus and chassis: technology and diagnostics of the modern vehicle.
9 appsOpen collection
Heat in the city centre: heat islands, shade, trees and unsealing, using the old town of Reutlingen as an example.
7 appsOpen collection
Reading the small print, spotting subscription traps, placing risk figures and making texts understandable.
5 appsOpen collection
From equation to curve: initial values, linear equations, substitution and the worlds in which they apply.
7 appsOpen collection
Rules and physics in sport: offside in football, spin and the Magnus effect, the walls in padel.
3 appsOpen collection
Mean, median, mode, box plot and levels of measurement: descriptive statistics to try out.
9 appsOpen collection
From coordinates to the inverse: vectors, dot and cross products, linear equations and matrices in pictures.
14 appsOpen collection
Production principles, waste, one-piece flow, value streams, bills of materials and routing: how a factory is organised.
9 appsOpen collection
What the CRA requires of manufacturers: checking whether it applies, meeting reporting deadlines, keeping software bills of materials.
5 appsOpen collection
Reducing, common denominators, multiplying, dividing and mixed numbers: fractions step by step.
13 appsOpen collection
Proofs of special beauty to touch: Zagier's windmills, the museum guards and Buffon's needle.
3 appsOpen collection
How oscillations break down: coefficients, symmetry, phasor chains, the Gibbs phenomenon and the plucked string.
7 appsOpen collection
What happens inside the machine: objects, methods, operators, slicing and sorting in Python, visible step by step.
12 appsOpen collection
From the Patent Motor Car to the Z3: famous machines and experiments cut open in 3D.
12 appsOpen collection
Keyboard, intervals, scales, chords, the circle of fifths and groove: music theory you hear and play.
9 appsOpen collection
How sound is made and shaped: synthesizers and samplers, envelopes and overtones, waveforms and timbre, voice and orchestra, room acoustics and sound scenography — all audible in your browser.
22 appsOpen collection
Staging space, sight, light, sound and paths: visitor flows, showcase lighting, room acoustics, sound islands, projection mapping and the pavilions of world expos to try out.
17 appsOpen collection
Mathematical questions open for decades, for which an AI model proposed solutions in 2026, as interactive apps: understand the problem, experience the result.
32 appsOpen collection
From neurons and gradients to transformers, RLHF and agents: machine learning you can touch.
45 appsOpen collection
DNA, replication, transcription, translation and cell division as explorable 3D models.
27 appsOpen collection
Perspective, ornament, structure and space: how art and architecture work with mathematics.
25 appsOpen collection
Hyperloop, autonomous vehicles, warehouses and conveyors: logistics as interactive simulation.
17 appsOpen collection
Heart, eye, voice, skeleton and nerves: the human body in 3D, layer by layer.
12 appsOpen collection
Monty Hall, Simpson, Berkson and base rates: where intuition fails us with numbers.
14 appsOpen collection
Planets, stars, black holes and space probes: the cosmos to explore.
12 appsOpen collection
Light switches for nerve cells, a telescope made of ice, mirror-image chemistry and interest rates: the 2026 Nobel Prizes to try out.
8 appsOpen collection
Planted participants, smoke in the room, one dollar for a lie and experimenters who are themselves the subjects: famous social psychology experiments to try yourself.
5 appsOpen collection
Relations, classes, orders, bijections, different infinities and topology: mathematics as the science of structure, hands-on.
11 appsOpen collection
Riemann, P vs NP, Poincaré and other great questions of mathematics, made tangible.
16 appsOpen collection
Counting, arithmetic and shapes: maths for beginners, playful and clear.
6 appsOpen collectionWant to use these apps in your courses? Let’s find out what works for you.
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