Lecture: Section 6.5

The Fundamental Theorem of Calculus

Part 1:  Let f(x) be continuous on [a, b] and define

Then g’(x) = f(x).  That is,

Part 2:  Let f(x) be continuous on [a, b] and let F(x) be any antiderivative of f(x).  Then

Examples:  Evaluate the definite integrals:

Remember: When we find this with rectangles, the height of each rectangle is f(x).  When x < 0, f(x) is negative and the rectangles would have negative area.  So negative definite integrals are possible.  Also, when we start looking  more at area, we have more to think about.

Definition:  The average value of a continuous function f(x) on the interval [a, b] is given by

Example:  Find the average value of the function f(x) = x² + x on the interval [0, 3].

This says that the average value of the function f(x) = x² + x on the interval [0, 3] is 9/2.

Let’s look at a graph

The red area (which represents the integral) is the same as the area over the same interval with height of the average value of thefunction.