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 2 and 1, 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 = 0.5,
C = 0.5, and D = 1.5. We shall interpret area below the x axis as negative.

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

First, we determine the area A + B + C + D:

This agrees with our geometric result for the 4 trapezoids, allowing for the negative character of the areas A and B.

Now consider another integral with an area of zero. Determine the sum of areas B and C:

, as we expected.

Now try an integral with unequal areas above and below the x axis:

Again, this agrees with our geometric results.

Now try an example with more area below the axis than above:

, as we expected.