Accumulation Functions (College Board AP® Calculus BC): Revision Note

Accumulation functions

What is an accumulation function?

• An accumulation function is a function that outputs values which

  • represent an accumulation of change

  • over an interval from a given starting point

    • to a variable endpoint

  • E.g. if a strawberry harvester gathers half a kilogram of strawberries for each meter of strawberry plants (rate of change = 0.5 kilograms per meter)

    • Then the accumulation function from the harvester's starting point,

    • to a point $x$ meters from that starting point,

    • is simply $0.5x$

  • Substituting in a value for $x$ gives you the amount of strawberries harvested up to that point

How do I write an accumulation function as a definite integral?

• Most often you will see accumulation functions written as definite integrals:
  $g\left(x\right)={\int }_a^{x}f\left(t\right)dt$

  • $g$ here is the accumulation function

    • $g$ is a function of x

      • Its value changes as the value of $x$ changes

  • $f$ is the associated rate of change function

  • $a$ is the (fixed) starting point of the integral

  • $x$ is the (variable) ending point of the integral

  • $t$ is merely a 'dummy variable' used for evaluating the integral

    • Any letter except x can be used for the dummy variable inside the integral

• $g\left(x\right)$ is calculating the accumulation of change as $t$ goes from $a$ to $x$

• Note here that a definite integral is being used to define a new function of x

  • If $f$ is a function of $x$ defined by $f\left(x\right)$

  • Then $g$ defined by $g\left(x\right)={\int }_a^{x}f\left(t\right)dt$ is another function of $x$