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.

16:9 · every frame verified for overlaps, spacing and edges before rendering

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.

Related animations

Newton's method: follow the tangent to the root

Starting from x0 = 2.5 on f(x) = x^2 - 2, this animation draws each tangent line and follows it down to the x-…

1/2 + 1/4 + 1/8 + ... = 1: filling a square

A unit square is progressively filled by repeatedly halving the remaining area, generating the series 1/2 + 1/…

Taylor polynomials of sin x

This animation overlays the sine curve with its Taylor polynomials of degree 1, 3, 5, 7, 9, and 11, drawn one …

The derivative as the limit of secant slopes

An animation tracks secant lines through the fixed point (2, f(2)) on f(x) = x^2/4 + 1/2 as the second point s…

Cauchy-Riemann Equations Worked Example

This animation walks through a complex analysis exam problem, finding where f(z) = u(x,y) + iv(x,y) is differe…

Riemann sums become the integral

This animation approximates the area under y = x^2 from 0 to 2 using rectangles, starting with four and doubli…