Examples

Example 3.1.2   Find the area enclosed by , , , .

Area

We have reduced the problem to a computation:

The above example illustrates the simplest case. In practice more interesting situations often arise. The next example illustrates finding the boundary points when they are not explicitly given.

Example 3.1.3   Find area enclosed by the two parabolas and .

Problem: We didn't tell you what the boundary points are. We have to figure that out. How? We must find exactly where the two curves intersect, by setting the two curves equal and finding the solution. We have

so , hence the intersect points are at and . We thus find the area by computing

Example 3.1.4   A common way in which you might be tested to see if you really understand what is going on, is to be asked to find the area between two graphs and . If the two graphs are vertical, subtract off the right-most curve. Or, just switch and '' everywhere (i.e., reflect about ). The area is unchanged.

Example 3.1.5   Find the area (not signed area!) enclosed by , , and .

Write , so that we can obtain the graph of the parabola by shifting the standard graph. The area comes in two pieces, and the upper and lower curve switch in the middle. Technically, what we're doing is integrating the absolute value of the difference. The area is

Something to take away from this is that in order to solve this sort of problem, you need some facility with graphing functions. If you aren't comfortable with this, review.

William Stein 2006-03-15