Accumulation Function CalcLens

Pick a starting point a, then slide x to the right. The area swept out under f is recorded as the height of a second graph. That second graph is the accumulation function .

Type it the way you write it: 2x-2, sin(x), x^2, e^(-x^2), 1/x, sqrt(x).

Top: f, with the area from a to x shaded. Drag anywhere on this graph to move x.

Bottom: the accumulation function A. Its height is the shaded area above.

The Fundamental Theorem, on screen

Accumulation Function

The top graph is a function f. The bottom graph records the signed area collected under f, starting at a and sweeping right to x. That running total is a new function of x, written .

What to watch for

Try this

  1. With f(x) = 2x − 2 and a = 0, sweep from 0 to 3. A dips, bottoms out at x = 1, then climbs. Ask why the low point is exactly where f crosses the axis.
  2. Move a from 0 to 1. The whole A graph shifts vertically but its shape does not change — every antiderivative differs by a constant.
  3. Try f(x) = e−x². There is no formula for its antiderivative, yet A is a perfectly ordinary curve. The accumulation function exists whether or not anyone can write it down.

Keyboard

← →
Move x (focus the slider first)
Space
Start or stop the sweep
?
This dialog
Esc
Close