The Fundamental Theorem of Calculus states that the integral of a function f(x) from a to x is the antiderivative F(x) of f(x), and that the definite integral of f(x) from a to b can be evaluated as the antiderivative F(x) evaluated from b to a. Specifically, if F(x) is defined as the integral of f(t) from a to x, then F(x) is the antiderivative of f(x), and the definite integral of f(x) from a to b equals F(b) - F(a).

1. The Fundamental Theorem of Calculus
Let f (x) be a continuous function on the interval [a, b].
FTC I: If x
F (x) = f (t) dt
a
then F (x) = f (x).
This says that F (x) is an antiderivative of f (x).
FTC II: If F (x) is an antiderivative of f (x) then:
b
b
f (x) dx = F (x) a = F (b) − F (a)
a
y
x