Calculus
Derivative vs integral
Slope vs area, instantaneous vs accumulated — the two halves of calculus.
Two halves of one subject — and they undo each other.
Quick check
A car's speedometer reads its speed; its odometer reads total distance travelled. How are these related by calculus?
Two questions, one machine
Derivative asks 'how fast is this changing right now?' — slope of the tangent, instantaneous rate. Integral asks 'how much has accumulated in total?' — area under the curve. The Fundamental Theorem says these are inverse operations: do one, then the other, and you're back where you started.
Same idea, both sides
- Position ↔ velocity ↔ acceleration — differentiate to go down the chain, integrate to go back up.
- Rate of water flow ↔ total volume in the tank.
- Marginal cost (per extra unit) ↔ total cost.
- Probability density ↔ probability (area under the density curve).
Your turn
Water flows into a tank at rate r(t) = 3t litres/min. How much water enters between t = 0 and t = 2 min?
Compare
- Derivative — slope, instantaneous rate.
- Integral — area, total accumulation.
- FTC — they're inverses.