The accumulator A(x) = ∫ax f(t) dt, and why A′(x) = f(x)

Year 12 Mathematics Advanced · FTC Part 1 · drag x and watch the shaded area become the height of A · the slope of A at x is the height of f at x · moving a shifts A by a constant and leaves its slope alone
f(t)

Top: f(t), shaded from a to x · Bottom: A(x)

f area above (+) below (−) A(x)
1.5
0

A(x) is the signed area from a to x. Slide a and watch every A-value move by the same amount.

At the current x
A(x) = ∫ax f(t) dt –
closed form –
f(x), the height of f –
A′(x), the slope of A –
centred difference, δ = 0.001 –

Slope of A against height of f

Turn the tangent on, then drag x. The tangent tilts exactly as the height of f changes: steep where f is large, flat where f = 0, downhill where f is negative.

What the lower limit does

A(x) = F(x) − F(a). Changing a changes F(a), a constant, so the whole A-curve slides up or down and its slope at every x is untouched. That constant is the +C of the indefinite integral.