Accumulation functions
Also known as: area accumulation function, integral function
An accumulation function is a function defined by a definite integral with a variable upper limit, such as F(x) = ∫ from a to x of f(t) dt. It measures the net area accumulated under a curve from a fixed starting point up to x.
An accumulation function turns a definite integral into a function of its upper limit. Given a function f and a fixed starting value a, the accumulation function is F(x) = ∫ from a to x of f(t) dt — for each input x, it outputs the net signed area under f between a and x. As x moves right, F accumulates area: positive where f is above the t-axis, negative where f is below it.
The key to analyzing accumulation functions is the Fundamental Theorem of Calculus, which says F′(x) = f(x). In other words, the graph of f is the derivative of the accumulation function. That means you can read F's behavior straight off f's graph: F is increasing where f is positive and decreasing where f is negative; F has a local maximum or minimum where f changes sign; F is concave up where f is increasing; and F has an inflection point where f changes from increasing to decreasing or vice versa.
For example, if F(x) = ∫ from 0 to x of f(t) dt and f is positive on (0, 3) and negative on (3, 5), then F increases until x = 3, peaks there, and decreases afterward — even if you never find an antiderivative. Values of F are computed by adding up geometric areas (triangles, rectangles, semicircles) on the graph of f.
Accumulation functions are a signature topic on the AP Calculus AB exam (units 6.4 and 6.5). Both multiple-choice and free-response questions present a graph of f and ask about the values, extrema, and concavity of its accumulation function, so mastering the F′ = f relationship is essential.
Key takeaways
- An accumulation function has the form F(x) = ∫ from a to x of f(t) dt, giving the net area under f from a to x.
- By the Fundamental Theorem of Calculus, F′(x) = f(x).
- F increases where f is positive, decreases where f is negative, and has extrema where f changes sign.
- F is concave up where f increases and has inflection points where f changes direction.
- AP Calculus AB tests accumulation functions using graphs of f and geometric area calculations.