Accumulation Functions (College Board AP® Calculus AB)

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 meters from that starting point,

    • is simply

  • Substituting in a value for 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:
  

  • here is the accumulation function

    • is a function of x

      • Its value changes as the value of changes

  • is the associated rate of change function

  • is the (fixed) starting point of the integral

  • is the (variable) ending point of the integral

  • is merely a 'dummy variable' used for evaluating the integral

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

      • This is true even if is defined as a function of

      • A dummy variable must still be used inside the integral

  • is calculating the accumulation of change as goes from to

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

  • If is a function of defined by

  • Then defined by is another function of