The tangent problem and the area problem looked like strangers — they were mirror images.
If you only record how far the tap was open (the flow), can you compute the tank's water level? And the reverse?
Integration and differentiation are mirror images that undo each other.
Experiment
Hands-on experiment
Predict first — if you only record how far the tap was open (the flow), can you know the tank's water level?
The tank experiment — the flow (top) creates the level (bottom)
Flow (the world of rates) — L/min
Level (the world of accumulation) — L
flow at t=15min
15.0 L/min
slope of the level graph at t=15min
14.8 L/min
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
Differentiation and integration were born from different questions. The derivative: 'the instantaneous rate of change'; the integral: 'the accumulation of pieces' — the tangent problem and the area problem. They looked like strangers.
Then Newton and Leibniz discovered: the instantaneous rate of change of an accumulated amount is the original function itself, and stacking the rate of change gives back the accumulation. The two problems were mirror images.
Why is this a revolution? Computing an integral by its definition (infinite slicing) is drudgery. But find a function whose derivative is f, and the integral is done — infinite labor became a single reverse lookup. The greatest shortcut in the history of mathematics.
Insight
Insights from the video
“One water tank contains all of calculus.”
The flow is a rate of change (the world of derivatives); the level is an accumulation (the world of integrals). Stack the flow and you get the level; the instantaneous slope of the level is the flow — this seemingly obvious observation is the whole theorem.
“F(b) − F(a): know the start and the end, and the entire middle is summarized.”
The total accumulated over an interval is the accumulation function's 'end value minus start value'. No need to add up every moment — just look at both ends. Like reading a bank balance difference instead of summing every transaction.
Misconception
Common misconceptions
Differentiation and integration are opposites because they're defined that way.
Their definitions are completely different (tangent vs. area). That they're opposites is a theorem requiring proof — hence the name 'Fundamental Theorem'. It isn't obvious; it's one of the most astonishing discoveries in mathematical history.
Computing an integral means slicing area into pieces and adding.
That's the definition, not the computation. Thanks to the theorem, the actual computation is 'find F whose derivative is f, then take F(b)−F(a)'. A classic case where the definition (slicing) and the method (reverse lookup) differ.
Formula
Writing it as math
The two directions of the relationship you observed in the water tank are the theorem's two parts.
Part 1: differentiate the accumulation and you're back
The instantaneous rate of change of the accumulation (level) is f at this very moment (flow). Stacking and watching change cancel each other.
Part 2: integrals are computed by antiderivatives
Just find F whose derivative is f, and the area comes out as end minus start — no infinite slicing needed.
In water-tank language
How much the level rose during the interval = the area under the flow graph over that time. It's exactly what you saw with your own eyes in the experiment.
In Real Life
Where you meet it in real life
Odometer and speedometer
The speedometer (rate) and odometer (accumulation) are a pair bound by the theorem. Record one and the other can be computed.
Battery level estimation
Your phone integrates current draw (rate) to estimate remaining battery (accumulation) — it's called coulomb counting.
Cumulative vs daily cases
The daily-cases graph (rate) and the cumulative-cases graph (accumulation) are in a derivative–integral relationship. A peak in one is an inflection point in the other.
Cash flow in finance
Stack monthly cash flows (rate) and you get the balance curve (accumulation). The entire accounting ledger runs on the Fundamental Theorem.
Connection
Concepts connect
Previous concept
Integrals
Only after suffering through stacking's laborious definition (Riemann sums) can you feel this shortcut's value in your bones.
← Integrals labLeads to next
Normal Distribution
Let's enter the world where 'area under a curve' becomes probability. The moment you measure the area of the bell curve, statistical inference begins.
Go to the Normal Distribution lab →