Now when the function f has some regions where it is positive and others where it is negative, we may compute the definite integral by integrating over the regions where the function is positive and add that to the integral over the region where the function is negative. This leads to the observation that where is the area bounded by the region where the function is positive and is the area bounded by the region where the function is negative. So in this most general case, the definite integral can still be thought of as measuring area, but it does so by measuring some areas as negative and others as positive. With this interpretation, we can try to convey graphically the main idea behind the Fundamental Theorem of Calculus. Below, you will see the graph of a function f on the left.
Yellow area contributes positively, while orange area contributes negatively.
Notice that as you first drag the ball to the right away from the origin, you are in a region where is positive. Consequently, as you increase x , the new area which is exposed is weighted positively in the definite integral. This means that the function is increasing in that region. When you reach the point at which , you have uncovered all of the positive area. This produces a maximum value for .