# The area under a graph.

Theorem 8.5.1   Let 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 .

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:

Proposition 8.5.3   Let 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 .
The two graphs intersect at and , as shown on Figure 3. Moreover, the graph is under for . Thus:

Noah Dana-Picard 2007-12-28