Fundamental theorem

Math

Definition

The Fundamental Theorem of Calculus connects differentiation and integration. Part 1 states that if F(x) = ∫ from a to x of f(t) dt, then F′(x) = f(x). Part 2 states that ∫ from a to b of f(x) dx = F(b) − F(a), where F is any antiderivative of f.

Examples

  • To evaluate ∫ from 1 to 3 of 2x dx, find the antiderivative x² and compute 3² − 1² = 8
  • If F(x) = ∫ from 0 to x of cos(t) dt, then F′(x) = cos(x) by Part 1
Key Fact

∫ from a to b of f(x) dx = F(b) − F(a), where F′(x) = f(x)

Related Terms

Antiderivatives Riemann sums Accumulation functions Area between curves

Study This Concept

Practice Fundamental Theorem with free review games in these units:

AP Calculus AB Unit 6: Integration and Accumulation