6.5 Interpreting the Behavior of Accumulation Functions Involving Area

Accumulation functions from an integral

An accumulation function g(x) = ∫ₐˣ f(t) dt represents the net signed area from a to x. By the Fundamental Theorem of Calculus, g'(x) = f(x). So the behavior of f (its sign, sign changes, and value) directly controls whether g is increasing, decreasing, or has extrema — even though g itself is defined by an integral, not a formula.

Reading graphs of f to analyze g

If f(x) > 0, g is increasing; if f(x) < 0, g is decreasing. When f changes from positive to negative, g has a local max; negative to positive gives a local min. Since g'(x) = f(x), g''(x) = f'(x), so g is concave up where f is increasing and concave down where f is decreasing — inflection points of g occur where f has local extrema.

Evaluating accumulation at specific points

To find g(x) at a particular x-value, compute the signed area between a and x using geometry (triangles, rectangles, semicircles) or given values. Areas above the x-axis count positively; areas below count negatively. Comparing g at different x-values requires adding/subtracting these signed areas carefully, especially across sign changes of f.

Key terms

Accumulation function

g(x) = ∫ₐˣ f(t) dt, representing net signed area from a to x; g'(x) = f(x).

When is g increasing/decreasing?

g increases where f(x) > 0 and decreases where f(x) < 0, since g' = f.

Local extrema of g

g has a local max where f changes + to −, and a local min where f changes − to +.

Concavity of g

g is concave up where f is increasing, concave down where f is decreasing, since g'' = f'.

Signed area and g(x)

g(x) equals the net signed area between f and the x-axis from a to x — areas below the axis subtract from the total.

Try an AP-style question

Where does g have its relative maximum on (0, 6)?

1. x = 1.5
2. x = 0
3. x = 6
4. x = 3