If f is continuous on a closed interval [a, b] and F' = f then

Proof

Notation:

This theorem implies that all techniques available for computing antiderivatives can be used to compute definite integrals, ie areas under curves.

Use of the theorem to compute areas:

_{Example 1}

_{Example 2}

_{Example 3}

_{Example 4}

In particular, we can apply the substitution method to computing areas. 

Substitution:

_{Example 5}

_{Example 6}

_{Example 7}