Areas and volume ambiguity

Question

For my question, I came up with this very simple analogy for the original question I have in mind.

Case 1 let's say there is a rectangle with the length $X$ and width $Y$, so the area will be $XY$. If I have $2$ of these rectangles The combined area would be $2XY$.

Case 2 Let's say I have a cuboid with length $X$, width $Y$ and height $2$, so the volume will be $2XY$.

Now my question is $2XY$ here as $2$ interpretations, so how am I supposed to distinguish that multiplying with $2$ gives me a $2-$dimensional answer or takes me into the 3rd dimension.

I came up with this problem when I was studying the integration by cylindrical shells method. That why do we get the thickness of the shell when we multiply it with delta $x$.

Answer 1

In the first case the factor $2$ is a dimensionless number obtained by addition of two surfaces

$$XY+XY=2XY \quad [L^2]+[L^2]=[L^2]$$

while in the second case $2$ is a length times a surface, that is

$$2 \cdot X \cdot Y = 2XY \quad [L]\cdot[L^2]=[L^3]$$

Without specifying the dimensions involved is not possible establish a distinction between these expressions.

Answer 2

@user's answer is correct.

In the first case, the total area is $XY$. In the second case, the volume is $2XY [L]$, where $[L]$ is the unit of length you are using, e.g. meter, foot etc. This is assuming the $X$ and $Y$ include units along with the numerical value.

If $X$ and $Y$ are just numbers, the units are even more clear: $2XY\ [L]^2$, $2XY\ [L]^3$