is not easy. However, if we assume that f(x) > 0 for all x then, using the fact that the definite integral of f on the interval [a, b] is the area bounded by f and the x-axis between x = a and x = b, then it should be intuitively clear that this area is the sum of the area of the red region and the area of the yellow region in the following. Since the latter two areas are the definite integrals of f on the intervals [a, c] and [c, b], respectively, the property follows.

The following flash movie gives a justification in the special case when c is the midpoint of a and b. Use the following graph as the motivation of this justification.