SOLUTION: Substituting into equation (6.7), we have
^{}
The area above a curve and below the X axis, as shown in
figure 611, will, through integration, furnish a negatvie answer.
Figure 611.Area above a curve.
If the graph of y = f(x), between x = a and x = b, has
portions above and portions below the X axis, as shown in figure 612, then
^{
}
is the sum of the absolute values of the positive areas
above the X axis and the negative areas below the X axis, such that
where
Figure 612.Areas above and below a curve.
EXAMPLE: Find the areas between the curve
y=x
and the X axis bounded by the lines
x= 2
and
x=2
as shown in figure 613.
Figure 613.Negative and positive value areas.
SOLUTION. These areas must therefore, we write be computed
separately;
and the absolute value of  2 is
Then,
Adding the two areas,
, we find
NOTE: If the function is integrated from  2 to 2, the
following INCORRECT result will occur:
This is obviously not the area shown in figure 613. Such
an example emphasizes the value of making a commonsense check on every solution. A sketch of
the function will aid this commonsense judgement.
EXAMPLE: Find the total area bounded by the curve
the
X axis, and the lines
x=
2
and
x=4
as
shown in figure 614.
SOLUTION. The area desired is both above and below the X axis;
therefore, we need to find the areas separately and then add them together
using their absolute values.
Figure 614.Positive and negative value areas.
Therefore,
and
Then, the total area is
