Accumulation Functions

Real Analysis · Axiom Academy

Sweep area under a curve and you don't just get a number — you get a brand-new function. Watch where it comes from, then drive it yourself. Pick a curve f and a fixed starting point a . Now let a second point x slide to the right and keep track of the area trapped under the curve from a to x . That running area isn't a single number — it changes as x moves, so it is a function of x . We call it the accumulation function. Press play and watch the sweep line travel from a . Everything it passes fills in as area under the curve, and the running total F(x) climbs to the exact area accumulated so far. F(x) is just the accumulated area — the running total the sweep line leaves behind as x moves. Watch the area become its own graph Drag x along the left graph. The shaded area under f from a to x is computed for real, and that same value is plotted on the right as the point — tracing out the accumulation function. Notice F keeps rising: the area only grows, because f stays positive here. The height of the right-hand curve at each x is exactly the shaded area on the left. Area, turned into a function. Move x and look at the thin strip the curve adds next: its area is about height width, . So the area grows at the rate f(x) — which means the slope of the F -curve at x should equal the height of f at x . The tangent's slope and f(x) are read out below; slide x around and watch them stay equal.

This is the written version of the interactive lesson above. See the full Real Analysis course.