Pauls Online Notes
Pauls Online Notes
Home / Calculus III / Multiple Integrals / Area and Volume Revisited
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 4-10 : 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 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,

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