See the diagram for the areas we shall label as A, B, C, and D. These areas are approximately trapezoids, but we'll find that they have slightly less area than a trapezoid due to the curved boundary. 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 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 = 2.5,
C = 4.5, and D = 8.5.

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, this result is close to, and slightly less than the geometric result of 8.5 above.

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

Again, this is slightly less than the graphical result of 13 above.

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

,

which compares with 5 from our graphical determination.

Finally, we consider an integral with two negative limits:

,

which compares in the right way with the graphical result of 2.5.