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.
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
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