Accumulation functions

Math

Definition

Functions defined as a definite integral with a variable upper limit, such as F(x) = ∫ from a to x of f(t) dt. They represent the accumulated area under f(t) from a to x and are central to the Fundamental Theorem of Calculus.

Examples

  • F(x) = ∫ from 0 to x of (2t + 1) dt represents the total area accumulated under the line 2t + 1 from 0 to x
  • If velocity is v(t), then ∫ from 0 to x of v(t) dt gives the total displacement from time 0 to time x
Key Fact

If F(x) = ∫ from a to x of f(t) dt, then F′(x) = f(x)

Related Terms

Fundamental theorem Antiderivatives Area between curves Riemann sums

Study This Concept

Practice accumulation functions with free review games in these units:

AP Calculus AB Unit 8: Applications of Integration