Lesson 11 of 12
Interactive lesson · prerequisites

Accumulation Is a Running Total

Accumulation means keeping a total while new pieces arrive. The pieces can be distances, quantities, areas or anything else that adds. The important idea is that the total itself changes as the endpoint moves forward.

Learning goalInterpret accumulation as a total that changes as new contributions are added.
totalₙ = contribution1 + ··· + contributionₙ
Step 1

Move the idea

Drag the slider

Try it: change one value

Move the slider

Change one parameter and watch what changes with it.

Step 2 · The aha

An accumulated quantity is not one frozen answer; it is a running total indexed by how far you have gone.

01

Keep every earlier piece

A running total remembers what came before. Adding a new contribution changes the total from its previous value rather than replacing it.

02

The endpoint matters

Stop after three pieces and you have one total; stop after six and you have another. So the accumulated total can itself be viewed as a function of the endpoint.

03

Bridge to integrals

An integral is continuous accumulation: instead of adding a few discrete pieces, calculus adds contributions across a continuously varying interval.

Step 3 · One-question check

Did it click?

If the first four contributions are 1, 2, 3 and 4, what is the accumulated total after the fourth contribution?

Continue the path
Next: turn a running total into continuous accumulated area