Accumulation Functions — ∫ from a to x

g(x) = ∫ from a to x f(t)dt is an accumulation function. It accumulates area.

Key properties:

g'(x) = f(x) (by FTC Part 2)

g''(x) = f'(x)

g(a) = 0 (integral from a to a)

Given a graph of f:

Where is g increasing? Where f > 0 (positive area accumulating).

Where is g decreasing? Where f < 0 (negative area accumulating).

Where does g have a max? Where f crosses from positive to negative.

Where does g have inflection points? Where f has extrema.

AP exam loves this. "Given the graph of f, find g(4) where g(x) = ∫ from 0 to x f(t)dt." Calculate the area under f from 0 to 4 using geometry (triangles, rectangles).

Reference:

Wikipedia: FTC

https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus