This is a typical specification of a region. If we want to compute the volume of this region, or integrate some function f(x,y,z) over it, we have to trnaslate this collection of equalities -- equations, that is -- into a collection of inequalities, because these are what we use in limits.
As we shall see in this note, one way of specifying this region by inequalities is:
Unfortunately, problems usually arise in the equality form. Usually we are told that a region is bounded by some collection of surfaces -- three in this case -- and, these surfaces in turn are specified by giving their equations. Hence, we have a problem of translation. This will be done with an intermediate step. We shall go from:
The key questions involved in this are:
These are the questions we now seek to answer for this example.
First, we shall always work with two dimensional drawings in which each curve or line is labeld with its equation. With such drawings, you cannot go wrong.
Now which two dimensions? Well, notice that two of our three equations involve only y and z. So lets first draw a side view in the y,z plane:
Where did the line z = 2 come from? That is the side view of the top of the region. It came form the equation z = 2 - x^2. To see how, draw the next picture, an "end view" in the x.z plane:
