When finding the area under the graph, we know that the intervals [0,3] and [5,8] must result in positive areas and the interval [3,5] must result in a negative area. Now, when finding the max/min areas, let's think about possible non-continuous graphs; these could look like 3 separate rectangles with the length of the given intervals and the height being either the max/min for the intervals or something very close to zero (I will use ±0.0001 to illustrate this in the examples). Note that the curves needed to create a continuous function will have less area than these rectangles.
Max area: Using the max value of 10 for the positive intervals and a minute negative value for the [3,5] interval, we get A = 6•10 + 2•(-0.0001) = 59.9998.
Min area: 6•0.0001 + 2•(-6) = -11.9998
As the minute areas approach zero, we find that the rectangular areas give a maximum at 60 and minimum at -12. But, as stated above, the actual function must have less area (positive or negative) than the rectangular area. This means A(x) is less than 60 (not "less than or equal to") and more than -12.
