See the diagram for the areas we shall label as A, B, C, and D. These areas are approximately triangles or trapezoids, but we'll find that they have slightly different area than those figures due to the curved boundary. We can calculate the area of each labeled portion from the geometry. Then we shall use the Fundamental Theorem of Calculus to also determine the areas and combinations of them.

There may be negative or positive numbers for limits, as well as zero. We shall consider all combinations. Areas below the x axis will be negative.

In each case, we shall be able to check our result by using the areas from the geometrical analysis.

From geometry, the area of a trapezoid is , where the h's are the heights of the two sides and w is the width. For example, for area A, the heights are 3 and 2, and the width is 1 unit. Hence the area A is

. We can find the rest in a similar way with the results being B = 3.5,
C = 1.5, and D = 2.5. Since D is below the x axis, we shall consider its area as negative.

Now let's use the definite integral to calculate combinations of these areas:

First, we determine area D, which is the simplest because the limits of the integral are both positive:

As expected, the magnitude is close to, and slightly less than the geometric result of 2.5 above. It is negative, indicating that the area is below the x axis. Note that here we have a negative area even though the limits are both positive. In general, the sign of a definite integral is not determined by the signs of its limits directly. A geometric interpretation is best.

Now try an integral with a limit of 0: Find area C + D:

Again, the magnitude is slightly less than the graphical result of 1 above. It is negative, indicating that more area is below the x axis.

Next try an integral with negative and 0 limits: Find the area A + B:

which compares with a magnitude of 7 from our graphical determination. This time the result from integration is slightly larger than the geometric approximation because the curvature of the boundary encloses more area than the trapezoid.

Finally, we consider an integral with two negative limits:

which compares in the right way with the graphical result of 3.5.