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.

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