In everyday language, integration is the process of adding together a continuously accumulating effect: one variable keeps affecting another, and we want the total over a chosen interval. Speed and distance is the clean case — speed sets how fast distance piles up.

Concept Two ways to reach 600 km

A car starts at position 0 and travels at a constant 120 km/h for 5 hours. The direct way to find the distance is arithmetic:

The same 600 is the area of the rectangle beneath the speed-time graph.

Height 120 (km/h), width 5 (hours).

Arithmetic and integration agree because the rate never changes.

Concept From a number to a function

What if the car travels for 6 hours, or 7, or any other time? Since the speed stays constant, we describe it with the function . Once we know the complete function, integration finds the total accumulated effect over any interval.

Starting at position 0 makes the constant , so the distance is simply — a single formula that answers the question for 5 hours, 6 hours, or any .

Example Evaluating the definite integral

To turn the general formula back into a single number for the first 5 hours, evaluate the definite integral.

⟹ 600 km — the same answer as the direct multiplication

Note Indefinite versus definite

	Indefinite integral	Definite integral
Result	A general formula, plus a constant	A single number
Here		600 (over 0 to 5)
Answers	"What is the distance function?"	"What is the distance for this interval?"

Summary

Integration totals a continuously accumulating effect over an interval — here, speed accumulating into distance.
Direct arithmetic and integration agree: 120 × 5 and the area under the speed graph both give 600 km.
The indefinite integral turns a rate into a general formula: .
The definite integral evaluates that formula over specific limits to a single number: 600 km from 0 to 5 hours.
Start with the everyday idea, then express it in equations — the same content, two languages.

Speed to Distance — the Simplest Integral