This takes a little explanation. I realize that double integrals can be used to calculate both an area or a volume but should I assume that in the case of calculating the area I am really calculating the volume and multiplying it by a height of 1 which just gives you the area?
Or are they really two different techniques that depend solely on the context of the assigned problem ?
Intuitively, integration is defined as a suitable limit of a sum. Depending on what we are adding in the sum this limit can be different things.
Also for a single integral of a function $y=f(x)$ we can have an area as in $$ A=\int_a^b f(x)dx $$ or arc length as in $$ L=\int_a^b \sqrt{1+{f'(x)}^2}dx $$
Under normal circumstances, the triple integral evaluates Volume as it spans three dimensions and double integral evaluates area as it spans two dimensions. A case in point is something like this what @chappers had already indicated.
For example, the volume of a solid between two planes could be written as
$\int_{0}^{2\pi}\int_{0}^{r}\int_{z_{bottom}}^{z_{top}} dV = \int_{0}^{2\pi}\int_{0}^{r}\int_{z_{bottom}}^{z_{top}} rdzdrd\theta = \int_{0}^{2\pi}\int_{0}^{r}\left[z_{top} - z_{bottom}\right]rdrd\theta$
The last one is a double integral but the first two are triple integral all measuring the volume.
