Owl: Define a running total: let A(x) be the signed area under a continuous rate curve f from a fixed start a up to a moving right edge x. As you slide x, A grows or shrinks.
Surprisingly, the derivative A′(x) brings f back — the original rate. Differentiating "area so far" recovers the integrand.
That is half of the Fundamental Theorem. The other half says a definite integral ∫ₐᵇ f equals F(b) − F(a) for any antiderivative F.
Why should accumulation and antidifferentiation be two faces of the same idea — and what continuity assumptions quietly sit underneath?