Theorem 8.5.1Let
be a function defined on an interval
. Suppose that:

has a primitive on
;

.

Then the area of the region defined by
and
is equal to
.

Figure 1:
The area under the graph of a function.

Example 8.5.2
The area
of the plane region bounded by the
axis, the
axis, the parabola whose equation is
and the line whose equation is
is given by:

Figure 2:
The area under a parabola.

Proposition 8.5.3Let
and
be the respective graphs of the functions
and
, both defined and
continuous on the interval
, where
. Suppose that
.
Then the area of the plane region bounded by the graphs
and
, and by the lines whose equations are
and
is given by:

Example 8.5.4
Take
and
, for
.

Figure 3:
The area between two curves.

The two graphs intersect at
and
, as shown on Figure 3. Moreover, the graph
is under
for
.
Thus: