Archimedes measured curves two thousand years before calculus.
Humanity only knew how to measure rectangles. How did we ever measure the exact area under a wiggly curve?
Slice it thin, stack it up, and you get the whole — integration.
Experiment
Hands-on experiment
Predict first — filling under the curve with 4 rectangles leaves an error. If you slice the rectangles infinitely thin, what happens to the error?
Fill under y = x² (interval 0–4)
Rectangle sum
14.000
True value
21.333…
Error
7.333
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Area formulas start from the rectangle. Triangles and trapezoids all reduce to rectangles in the end. But the area under a curve? There was no formula at all.
The idea is almost shamelessly simple — fill the space under the curve with skinny rectangles and add them up. There's an error? Then slice thinner. Endlessly.
In the limit of 'endlessly thinner', the error vanishes to 0 and the exact area remains. And this area is no mere geometry puzzle — stack speed and you get distance, stack power and you get energy, stack probability density and you get probability. Wherever a 'accumulated total' is needed, there's an integral.
Insight
Insights from the video
“The spirit of integration is the audacity that 'slice infinitely and the error disappears'.”
Finitely many rectangles always leave an error. But if the error can be made arbitrarily small by adding more, then in the limit the error is exactly 0 — the logic you trained in the Limit lab builds an area here.
“The area under a speed graph is distance — that's the integral's true face.”
Speed at each instant × a short time = a short distance. Stack them all and you get total distance. This structure — 'accumulate the rate of change and you get the total change' — is the prototype of every application of integration.
Misconception
Common misconceptions
The integral is defined as the reverse of differentiation.
The definition of the integral is 'slice and stack' (the limit of Riemann sums). Being the reverse of differentiation is not the definition — it's an astonishing theorem discovered later (the Fundamental Theorem of Calculus). It's a classic case of a concept known in the wrong order.
An integral is always the area of a shape.
Parts below the x-axis count as negative. An integral is a 'signed accumulation' — like distance accumulating negatively when you drive in reverse. Area is just one of the integral's many faces.
Formula
Writing it as math
Write the rectangle-stacking experiment as formulas and you get the definition of the definite integral.
Riemann sum — what you did in the experiment
The total area of n rectangles of width Δx. The slider in the experiment was exactly n.
Definition of the definite integral
The limit as the rectangles get endlessly thinner. The ∫ symbol is an elongated S for Sum — it literally means 'stack infinitely'.
The faces of accumulation
Stack speed and you get distance; stack probability density and you get probability. 'Area under the graph' can always be read as 'accumulated rate of change'.
In Real Life
Where you meet it in real life
Your electricity bill
Power (kW) is the instantaneous usage rate; energy (kWh) is its accumulation — the kWh on your bill is the integral of your power graph.
Probability and statistics
The area under the normal-distribution curve is probability. Every statistic that computes 'the top 2.5%' stands on integration.
The odometer
A car computes distance traveled by accumulating speed over time, no GPS needed — a mechanical integrator.
Drug concentration and dosage
The area under the blood-concentration curve (AUC) is the total amount absorbed — pharmacology's key metric is an integral.
Connection
Concepts connect
Previous concept
Limits
For 'slice endlessly thinner' to make sense, you need limits. Integration is a house built on limits.
← Limits labLeads to next
Fundamental Theorem of Calculus
The shocking discovery that stacking (integration) and instantaneous change (differentiation) are mirror images of each other — mathematics' greatest shortcut is the next station.
Go to the Fundamental Theorem of Calculus lab →