Fundamental Theorem of Calculus for AP Calculus

Practice problems for these concepts can be found at: Definite Integrals Practice Problems for AP Calculus

First Fundamental Theorem of Calculus

If f is continuous on [a, b] and F is an antiderivative of f on [a, b], then

Example 1

Example 2

Example 3

Example 4

If f' (x )=g (x ) and g is a continuous function for all real values of x, express in terms of f.

Let u =3x; du =3dx or =dx.

Example 5

Cannot evaluate using the First Fundamental Theorem of Calculus since is discontinuous at x =1.

Example 6

Using a graphing calculator, evaluate

Using a TI-89 graphing calculator, enter and obtain 2π.

Second Fundamental Theorem of Calculus

If f is continuous on [a, b] and F (x )= f (t)dt, then F' (x )= f (x ) at every point x in [a, b].

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Practice problems for these concepts can be found at: Definite Integrals Practice Problems for AP Calculus