Calculus III - Area and Volume Revisited

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Section 15.10 : 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,

\[{\mbox{Area of }}D = \iint\limits_{D}{{dA}}\]

Now let’s give the two volume formulas. First the volume of the region \(E\) is given by,

\[{\mbox{Volume of }}E = \iiint\limits_{E}{{dV}}\]

Finally, if the region \(E\) can be defined as the region under the function \(z = f\left( {x,y} \right)\) and above the region \(D\) in \(xy\)-plane then,

\[{\mbox{Volume of }}E = \iint\limits_{D}{{f\left( {x,y} \right)\,\,dA}}\]

Note as well that there are similar formulas for the other planes. For instance, the volume of the region behind the function \(y = f\left( {x,z} \right)\) and in front of the region \(D\) in the \(xz\)-plane is given by,

\[{\mbox{Volume of }}E = \iint\limits_{D}{{f\left( {x,z} \right)\,\,dA}}\]

Likewise, the volume of the region behind the function \(x = f\left( {y,z} \right)\) and in front of the region \(D\) in the \(yz\)-plane is given by,

\[{\mbox{Volume of }}E = \iint\limits_{D}{{f\left( {y,z} \right)\,\,dA}}\]