Rates and Totals

Calculus splits into two halves. Differentiation finds rates — how fast a quantity changes at a single instant. Integration accumulates tiny pieces into a whole. A speedometer differentiates your position over time; an odometer integrates your speed into total distance.

These two operations mirror each other perfectly. The Fundamental Theorem of Calculus proves that integration and differentiation undo one another, the way addition undoes subtraction. This single insight connects slopes of curves to areas under them.