AP Calculus AB and BC glossary

Accumulation function

Also called: Integral function

An accumulation function is a definite integral whose upper bound is the variable, so it defines a new function measuring how much has accumulated from a fixed starting point. Its derivative is the integrand evaluated at the upper bound.

g(x) = \int_a^x f(t)\,dt \implies g'(x) = f(x)

Analysing one is the same job as analysing any function, done with $f$ as the derivative. Where $f$ is positive $g$ increases, where $f$ crosses from positive to negative $g$ has a local maximum, and where $f$ increases $g$ is concave up.

The dummy variable inside must differ from the bound. Writing $\int_a^x f(x)\,dx$ uses $x$ for two different roles and is why $t$ is the conventional choice inside.

With a chain rule

If the upper bound is a function rather than $x$ alone, the derivative picks up a factor: $\frac{d}{dx}\int_a^{u(x)} f(t)\,dt = f(u(x)) \cdot u'(x)$ .

Appears in: Unit 6: Integration and Accumulation