This is the standard proof of the Fundamental Theorem of Calculus. Type any differentiable F; the page computes f = F′ numerically. The top graph is F, the bottom graph is f, and [a, b] is cut into n equal pieces.
1. Telescope. F(b) − F(a) is the sum of the changes of F over the pieces: the colored rises in the staircase, restacked at the right. This is exact for every n.
2. Mean Value Theorem. On each piece the secant line has some parallel tangent line, touching at a point ci. So the change over that piece is exactly f(ci) Δx. (If you averaged 70 mph between two speed cameras, your speedometer read exactly 70 at some instant.)
3. Riemann sum. In the bottom graph, the rectangle over each piece has height f(ci), so its area equals that piece’s rise. The rectangles form a Riemann sum for ∫f with sample points ci, and it equals F(b) − F(a) exactly.
4. Limit. As n grows, every Riemann sum of a continuous f approaches ∫f, but this one never moves: it is F(b) − F(a) for every n. So the integral is F(b) − F(a). For comparison the panel shows the left and midpoint sums, which do move.
When it fails. The ⚠ presets break a hypothesis. For −1/x on [−1, 1], F is undefined at 0 and the piece containing 0 has a negative rise while f = 1/x² is positive, so no MVT point exists; ∫−11 1/x² dx is not −2. For |x| on [−1, 2], the corner at 0 leaves the piece straddling it with no MVT point (try n = 2 or 4); when 0 is an endpoint of a piece, as for n = 3, every piece is fine. Such pieces are marked in red.
Hover over a piece for its numbers. Keys: ← → change n, 1–4 choose the step. The address bar keeps the settings.