The fundamental theorem of calculus: area has a derivative

A curve f is shown alongside a second graph tracking A(x), the running area under f from 0 to x. As x sweeps rightward, the shaded region grows while A(x) traces its accumulated value in real time. At x = 2, a tangent line to A has slope exactly f(2), demonstrating that A'(x) = f(x). Useful for introducing the Fundamental Theorem of Calculus and connecting accumulation with rate of change.

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The prompt that made it

The shaded area under f accumulates as x sweeps right while A(x) = integral from 0 to x is traced on a second axes; the tangent to A at x = 2 has slope f(2), so A'(x) = f(x).

Make your own version

Make the next one in this series

• Show the same construction with a negative region of f and how A(x) decreases there.
• Animate the Riemann sum approximating A(x) as rectangle width shrinks to zero.
• Compare A(x) for two different starting points to show how the constant of integration shifts the graph.

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