Fibo Math Lab — FTC: Accumulation & Rate of Change

Fibo Math Lab

The Fundamental Theorem of Calculus

Accumulation & rate of change · Calculus

Skill goal: See that accumulation and rate of change undo each other (Fundamental Theorem of Calculus).

1 · Observe

Let A(x) = ∫0x f(t) dt — the accumulated area under f from 0 up to x. The shaded region grows as x moves right:

Drag the teal dot (or use the slider) to move the upper limit x. Watch the shaded area accumulate.

2 · Manipulate

Choose f(t):

2.00

The accumulation function A(x)

This second plot graphs A(x) itself. The gold tangent line shows how fast A is growing at the current x — compare its slope with f(x) in the readouts below.

3 · Predict

With f(t) = 2t selected, answer before dragging:

Q1. What function is A(x) = ∫0x 2t dt ?

Q2. Then what is A′(x)? How does it compare with f(x)?

4 · See

Live readouts — the area is computed numerically (Simpson’s rule), the slope numerically too:

2.00
x (upper limit)
4.00
f(x)
4.00
shaded area A(x)
4.00
tangent slope A′(x)
✓ Tangent slope A′(x) equals f(x) — FTC Part 1, live!

5 · Explain

FTC Part 1 — differentiating an accumulation function returns the original function:

d/dx ∫ax f(t) dt = f(x)

Why it makes sense: when x creeps right by a tiny dx, the new sliver of area is ≈ f(x)·dx, so area grows at rate f(x).

FTC Part 2 — a definite integral is just the antiderivative’s net change. Example:

∫13 2t dt = [t²]13 = 9 − 1 = 8

  • Part 1: accumulate, then differentiate → back to f.
  • Part 2: antidifferentiate, then evaluate → the accumulated area.
  • Together: integration and differentiation undo each other.

6 · Challenge

Three rounds — each is checked automatically (tolerance 0.01).

Score: 0 / 3

Keep learning