Integral as Accumulated Area

Sweep x and watch the shaded Riemann rectangles under a curve become the height of the area function F(x) below — with the Fundamental Theorem F′(x)=f(x) shown live.

This interactive integral visualization shows the definite integral as accumulated area under a curve. As you sweep the upper limit $x$, Riemann rectangles fill in the region beneath the function $f(x)$, and their running total becomes the height of the area function $F(x)$ traced on the panel below — so you watch area turn directly into a new curve.

The animation makes the Fundamental Theorem of Calculus visible. The accumulated area $F(x) = \int_a^x f(t),dt$ grows fast where the curve is tall and slowly where it is short, and its rate of growth at any point is exactly the height of the curve there — that is, $F'(x) = f(x)$. Seeing the area curve and the original function move together is what makes this deep link between integration and differentiation intuitive rather than abstract.

Adjust the number of Riemann rectangles to watch the approximation tighten toward the true area as the partition gets finer, and step through the sweep to connect each shaded strip to one point on the accumulation curve. Regions where the function dips below the axis subtract area, so the accumulated total can fall as well as rise. It's a clear, visual introduction to integration for anyone learning calculus.