Accumulation function

A function $g(x)=\int_a^x f(t)\,dt$ that reports the running total of $f$ 's signed area up to $x$ .

An accumulation function $g(x)=\int_a^x f(t)\,dt$ reports the net signed area under $f$ from a fixed starting point $a$ up to a VARIABLE endpoint $x$ — as $x$ moves, $g(x)$ tracks the running total. This is the object the Fundamental Theorem of Calculus, Part 1 differentiates: $g'(x)=f(x)$ .

Examples

For $f(t)=2t$ and $a=0$ , $g(x)=\int_0^x 2t\,dt=x^2$ — the running total of area swept out as $x$ grows.