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Area and Volume Revisited
This section is here only so we can summarize the geometric
interpretations of the double and triple integrals that we saw in this
chapter. Since the purpose of this
section is to summarize these formulas we aren’t going to be doing any examples
in this section.
We’ll first look at the area of a region. The area of the region D is given by,
Now let’s give the two volume formulas. First the volume of the region E is given by,
Finally, if the region E
can be defined as the region under the function 
and above the region D in xy-plane then,
Note as well that there are similar formulas for the other
planes. For instance, the volume of the
region behind the function and in front of the region D in the xz-plane is given by,
Likewise, the the volume of the region behind the function and in front of the region D in the yz-plane is given by,