See the diagram for the areas we shall label as A, B, C, and D. For this simple case we can calculate the area of each labeled trapezoid from 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. 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 1 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 = 2.5,
C = 3.5, and D = 4.5. We shall consider areas below the x axis 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:

This agrees with the magnitude of our geometric result for D. We shall interpret area under the x axis as negative.

Now consider an integral with one limit of zero. Determine the sum of areas C and D:

This agrees in magnitude with our geometric results for C and D.

Now try an integral with the lower limit negative:

Again, this agrees with our geometric results.

Now try a negative limit and a limit of zero:

, as we expected.

Finally try both limits negative:

, as we expected.